Article pubs.acs.org/jced
Cluster Structure of Dissolved Gas Nanobubbles in Ionic Aqueous Solutions Paper presented at the 18th Symposium on Thermophysical Properties, Boulder, CO, June 24 to 29, 2012. Nikolay F. Bunkin,* Alexey V. Shkirin, and Valery A. Kozlov A.M. Prokhorov General Physics Institute, Moscow, Vavilova 38, 119991 Russia ABSTRACT: Aggregation of nanodispersed particles plays an essential role in the technology of colloidal systems. In this work the formation of clusters composed of long-living gas nanobubbles in electrolyte aqueous solutions has been studied experimentally by combining three distinct but complementary laser techniques: dynamic light scattering (DLS), laser phase microscopy, and polarimetric scatterometry. We propose a mathematical approach to modeling the structure of spherical particle clusters that is based on the solution of inverse problem of optical wave scattering with allowance for cluster−cluster aggregation. In this way we found the characteristic size of nanobubbles and fractal properties of their clusters in aqueous solutions of NaCl. The described method can be applied to the exploration of clustering in a wide class of disperse systems of spherical particles.
1. INTRODUCTION The processes of aggregation, taking place to a greater or lesser degree, are a common feature of all colloidal systems, and they may result in partial clustering of dispersed particles, especially if the particles are strongly charged due to the presence of free ions in the dispersion medium. The clusters, being produced in dispersions, can have various morphologies, depending on what is the prevailing type of aggregation: “particle to cluster” or “cluster to cluster” aggregation. Hence, it is important to determine the existence of cluster structures and their parameters. In this work, it is shown that complementary use of several laser diagnostic methods allows us to retrieve effectively the information about nanoparticle clusters. The particular subject of our study is the clusters of quasistable gas nanobubbles in an equilibrium ionic solution, which is kept under normal conditions, that is, at room temperature and atmospheric pressure. The existence of long-living bubbles plays, for instance, the principle role in the interpretation of the ultrasonic cavitation phenomenon. Actually, if we focus an ultrasonic wave of high enough intensity in a liquid, we will see a track of vapor-gas bubbles in the focal volume of an ultrasonic lens. The rupture strength of a liquid can be expressed as σn1/3 l ∼ 109 Pa, where σ is the surface tension coefficient and nl is the number density of molecules of the liquid (all numerical estimates will be hereinafter made for water). At the same time, experimental data indicate that the cavitation can be induced even at the amplitudes of a sound wave of about 1 atm. It follows from here that long-lived (quasi-stable) centers of the cavitation must be present in the liquid for the cavitation effect to occur. The steady gas bubbles are evidently related to such centers. We mean here micrometer-sized bubbles, since the bubbles of smaller size are squeezed by the surface tension forces and disappear, whereas larger bubbles (whose size achieves 1 mm) promptly float up. © 2012 American Chemical Society
The mechanical equilibrium condition for a micrometer-sized bubble is given by the known Young−Laplace equation: Pin = Patm +
2σ > Patm R
(1)
where R is the radius of the bubble, Pin is the pressure of gas inside the bubble, and Patm is the pressure of atmospheric gas above the surface of the liquid, that is, the atmospheric pressure. Let us note that in the formula eq 1 we did not allow for the hydrostatic pressure associated with the weight of the liquid volume above the bubble: we can obviously ignore it, since all experiments on the cavitation are carried out, as a rule, under laboratory conditions, that is, when the hydrostatic pressure is much less than the atmospheric pressure. Equation 1 implies that the pressure of gas inside micrometer-sized bubble is always higher, than the pressure of the same gas above the liquid surface. Thus, the solution of gas in liquid, whose content, according to the Henry law, is controlled by the pressure Patm, appears to be unsaturated with respect to the pressure Pin of the same gas inside the bubble. It follows from here that such bubble is diffusively nonstable: the gas escapes from such a bubble by diffusion kinetics, and the bubble eventually disappears. This fact was analyzed (probably for the first time) in ref 1. As was shown in this study, if we deal with a bubble with the initial radius 10−3 cm, the lifetime of such bubble does not exceed 10 s. This time drastically falls with decreasing the radius of bubble. For example, if the bubble radius is about 100 nm, its lifetime does not exceed 10 μs in a wide range of temperatures, see, for example, ref 2. Hence, to Received: June 29, 2012 Accepted: August 26, 2012 Published: September 13, 2012 2823
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laser beams employed.21−26 It is worth saying that the ionstabilized bubbles and the clusters composed of such bubbles should play an important role in living processes. In our opinion, the breathing of marine organisms (for instance, fish) is performed with the help of clusters composed of the ionstabilized bubbles. Indeed, as is widely known, for the breeding of fish under artificial conditions (as related, for example, to aquarium fish) a certain mineralization level of the aquarium water is required, since the fish perish in distilled (ion-free) water. On the one hand, if we suppose that in the respiration process of aquatic organisms the extraction of molecules of the dissolved oxygen out of the water matrix occurs, then we should reckon that the additional mineralization is not necessary. On the other hand, the process of extraction is energy and time-consuming. Implying that at breathing a fish captures the clusters of the ion-stabilized bubble by gills, the problem associated with the extraction of molecules of dissolved oxygen seems to be solved. But, as was shown in our recent experimental work,18 for the ion-stabilized bubbles and the clusters of such bubbles to exist the increased content of ions is necessary. Besides, in the same work it was shown that such clusters play a negative role in living tissues at a sharp decrease of arterial pressure. As follows from our results, these clusters are nucleated at the erythrocyte membranes at a steep reduction of pressure that essentially worsens the mechanic and elastic properties of the erythrocytes. The erythrocytes, whose membranes carry these bubble clusters, cannot take part in the gas exchange processes that can be one of reasons of the caisson disease.
observe regularly the cavitation effect, we should require the surface tension to be somehow compensated. One of the mechanisms for the neutralization of excess pressure inside a bubble can be associated with solid impurities arisen from the outside; that is, bubbles are generated at solid (possibly hydrophobic) microparticles suspended in the liquid. It may occur that the gas bubble, attached to a solid interface, becomes stable both mechanically and diffusively. The model where the stable heterogeneous centers of the cavitation arise due to the presence of solid impurities, is widely accepted; see, for example, refs 3 to 6. It is necessary to note, however, that even fine filtration of liquid and removing the solid impurities cannot completely suppress the cavitation effect. Additionally, it is well-known (see, e.g., ref 7) that the cavitation ability in aqueous media increases at adding various salts; albeit new solid particles are definitely not inserted into water sample together with the salt. Thus, it should be recognized that the aboveconsidered mechanism of microbubbles stabilization is inconsistent. If liquids are considered to be under normal conditions and saturated by dissolved gas (for example, atmospheric air), the mechanism of stabilization is stipulated by the selective adsorption of ions of the same sign on the bubble interface. It is necessary at once to note that the adsorbed ions (here we speak about the ions of inorganic salts) can give rise to a slight growth of the surface tension coefficient (see, e.g., tables in ref 8 and also refs 9 and 10), which should be taken into account in numeric estimations. This ion-stabilized bubble model was first put forward in the study11 and then was developed in subsequent works.12,13 The mechanical stability condition for a ion-stabilized bubble is expressed as: Pin + Pe = Patm + 2σ /R
2. EXPERIMENTAL INVESTIGATION OF ION-STABILIZED BUBBLE CLUSTERS In this section we report the experimental study of clusters composed of long-living (quasi-stable) gas nanobubbles in an aqueous solution of NaCl. The study was done with the help of three distinct but complementary laser techniques: modulation interference microscopy (phase microscopy), dynamic light scattering, and polarimetric scatterometry. 2.1. Experimental Setup. The modulation interference microscope was described comprehensively in our previous studies.15,16,18 This microscope allows us not only to find the sizes of the objects under study, but also to determine their refractive index n (the optical density) with respect to ambient medium. In this experiment the phase shift δ between the object and the reference waves of the interferometer is measured. In this process the object wave passes through the cell with the liquid sample having the refractive index n0 (see Figure 1 and ref 15), then the wave is reflected from the mirror substrate of the cell, again passes through the same liquid sample, and finally is input to a pixel of the CMOS-matrix. The cell with a thin liquid sample has a mirror substrate and is covered by a transparent (made of glass) cover-plate. The phase shift δ between the object wave, which passes through the transparent particle located in the liquid with the refractive index n0, and the reference wave is measured. The refractive index n of that particle can be higher or lower than n0. In the case, where n < n0 (e.g., gas bubble) the value of δ < 0, while at n > n0 (solid particle) δ > 0. In the case of a compound particle (e.g., a gas bubble is adhered to a solid particle) the situation is possible, when δ = 0; that is, such compound particles are not resolved in our experiment due to their very low phase contrast. The reference beam strikes the same pixel, so we actually measure the interference pattern intensity I. In the absence of
(2)
Here Pe = −(∂Φe/∂V)T is the ponderomotive pressure, caused by the presence of the charge Q0 at the spherical interface of the ion-stabilized bubble, and Φe is the energy of electrostatic field of the system in question. This expression is evident from the general thermodynamic definition of pressure P = −(∂Φ/∂V)T, where Φ is the free Helmholtz energy of the system. In our case, the system is just a spherical area of radius r ≥ R, surrounding the charged bubble of radius R. The electric field inside this area is nonzero (the field inside the ionstabilized bubble is equal to zero because of the spherical symmetry). For that reason, the Helmholtz energy and the volume are found as Φe = (1/2ε)∫ rR[Q2(x)/x2]dx, and V = (4π/3)(r3 − R3). The pressure applied by the bubble to the surrounding liquid (i.e., at r = R) is Pe(R ) = −
2 1 ∂Φe 1 Q 2(R ) 1 Q0 = = 8πε R4 8πε R4 4πR2 ∂R
(3)
It is clear that the pressure Pe expands the bubble and thus is capable of compensating the squeezing surface tension forces at a certain radius R. Equation 3 implies that the pressure applied to the outer surface of the spherical region of radius r is P(r) = −(1/4πr2)(∂Φe/∂r) = (1/8πε)Q2(r)/r4; that is, this pressure falls with growing r. References 14 to 20 describe the experiments on ion-stabilized bubbles in equilibrium aqueous solution of salts, saturated by dissolved air. Furthermore, some argument, supporting this hypothesis, was inferred from observations of the low-threshold laser-induced breakdown in water and aqueous electrolytic solutions transparent for the 2824
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Figure 1. Schematic diagram, illustrating the principle of phase microscopy.
any particles in the liquid the phase shift δ0 can be set arbitrarily. The parameters of the measuring system are chosen in such a way that δ0 = πm + π/4, m = 0, 1, 2, ..., and the intensity I is a uniform function across the matrix. The interference pattern intensity is apparently I ∼ cos(δ + δ0). Therefore at the initial value of δ0, adjusted as was indicated, the measured intensity I is highly sensitive to changing the argument (δ + δ0) associated with the presence of particles. When the object wave passes through a spherical particle of radius R, the phase shift is given by 4π R (n − n 0 ) δ= (4) λ where n is the refractive index of the test particle. Thus, by measuring the value of δ, we can subdivide the colloid particles into those having high and low (with respect to water) optical (and hence the material) density. 2.2. Preparation of Samples. Water samples were carefully purified in an ion-exchange resin rectification column; the specific resistance was 5 MΩ·cm, which was measured just before each series of experiments. Additionally, we tested the magnitude of the pH, which was equal to 6.7, that is, water samples possessed by slightly acidic properties. The aqueous samples were prefiltered as well; we used a porous membrane filter with the size of a pore of ∼100 nm. The samples of water were stored in nontransparent blackened (to avoid the influence of daylight) flasks made of fused silica or Teflon and supplied with press caps to keep the sample away from penetration of carbon dioxide and dust particles; note that the magnitudes of specific resistance and pH were not changed during storage in the flasks. We also prepared aqueous NaCl solutions with different concentration. The salt was of analytical grade (the mass fraction of NaCl is greater than 99 %) and used as purchased. 2.3. Phase Microscopy Experiment. In the phase microscopy experiment we compared the spatial phase profiles measured for a colloid silica particle at the micrometer scale (R = 0.63 μm), and for long-living particle of low optical density, spontaneously arisen in the liquid and having a similar size. In Figure 2 we present a photo of the colloid silica particles in the white-light microscope (a) and 3D distribution of the optical density for one of them (b). In the last diagram the optical density is plotted along the Zaxis, whereas the particle itself is localized in the XY-plane. It is worth noticing that the optical density is scaled (conventionally) in nanometers, that is, in the units of wavelength of the radiation used (λ = 532 nm). In Figure 3a the white light microscope photo of the particles spontaneously arisen in aqueous NaCl solution (1.06 mol·kg−1) is shown; in the absence of ionic admixtures such particles are generated very
Figure 2. Monodisperse colloid silica particles (R = 0.63 μm) in the microscope white light (a) and 3D optical density profile for a single particle (b).
weakly, see refs 17 and 18. Let us notice that the emergence of nanobubble clusters in initially distilled water, which we reported in ref 15 in our current opinion, was caused by long-term storage in glassware, which gave rise to pH change, that is, to increase in ionic content, see, for example, ref 27. We note that in accordance with the model of ionic adsorption described in ref 28 the Cl− anions are capable of adsorption at the ion-stabilized bubble interface and thus can result in the nanobubble stabilization. In Figure 3b the phase profile of such particles is illustrated. As follows from the comparison of the graphs in Figures 2b and 3b, the optical density for colloid silica is higher than that for water (as should be), whereas the optical density for the long-living spontaneously arisen particle at the micrometer scale appears to be lower. Indeed, the optical density of spontaneously arisen 2825
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Figure 4. Optical density profiles for a colloid silica particle of diameter D = 1.25 μm (gray color) and for a particle spontaneously arisen in aqueous NaCl solution (1.06 mol·kg−1, black color). The maximum and minimum values of optical density Δh are measured from a conventional zero level, which is not coincident with the X-axis.
Figure 3. Particles, spontaneously arisen in aqueous NaCl solution (1.06 mol·kg−1) in the microscope white light (a) and 3D optical density profile for a single particle (b). The particles in the photo (a) are seen very faintly since the steep interphase boundary is actually absent.
particle has a concave shape, while the optical density of the colloid silica particle is of a convex shape. In Figure 4 the optical density profiles for the monodisperse colloid silica particle and the particle of low optical density spontaneously arisen in aqueous NaCl solution (1.06 mol·kg−1) are shown; the profiles were measured along the diameters of the particles, and we assumed that these diameters are approximately equal to one another. The quantitative estimate of the refractive indices of the particles (assuming their spherical shape) is expressed as ⎛ Δh ⎞ ⎟ + n n = γ⎜ 0 ⎝ 2R ⎠
Figure 5. Time-correlation function GI(τ) for the light scattered at the angle θ = 60°, as was studied in 1.06 mol·kg−1 aqueous NaCl solution.
case of colloid silica nSiO2 = 1.46, which is in agreement with literature. At the same time, for the spherical particle of low optical density depicted in Figure 4 (Δhb = −20 nm) it occurred that nb = 1.26. We see that the spontaneously arisen particles cannot be purely gaseous spheres. Such particles can therewith be the clusters composed of the long-living (or stable) gas nanobubbles with the refractive index n = 1; the presence of liquid films between the gas cores in a cluster causes a slight rise of refractive index, that is, nb > n = 1. It should be noticed that in the case of the colloidal silica particles the assumption of sphericity is quite reasonable, whereas for the particles, spontaneously arisen in salt solution, this supposition should be tested in some additional experiments. Indeed, at the violation of sphericity we should introduce a certain correction factor into the denominator of eq 5; in the case of spheroid, flattened along the vertical axis, this would result in reducing the value of nb. However, if it were indeed monolithic gas particles, having the refractive index n = 1, its vertical size would
(5)
where n0 = 1.332 (for water), Δh is the maximum/minimum value of the optical density h, scaled in nanometers; for more optically dense (with respect to water) silica particles ΔhSiO2 > 0, while for spontaneously arisen “bubbles” Δhb < 0, and the values of |Δhb| vary mainly in a range of (15 to 50) nm. Here γ is a correction dimensionless factor, which was found in the process of the device calibration; in our case γ = 2.73. It can be easily deduced from the graphs in Figure 4 and eq 5 that in the 2826
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be about 80 nm, while horizontal size is still about 1 μm. It is clear that the Archimedes force cannot squeeze a micrometersized bubble at a solid cover plate (see Figure 1) to that extent. Thus the particles of low optical density cannot be monolithic gas particles; that is, they should contain liquid films inside. To summarize, the genuine value of the refractive index of such particles is still unknown to us, albeit in our analysis it is quite sufficient to know that their refractive index is smaller than that for water, but higher than that for pure gas medium. Such particles are apparently can be associated with the clusters of gas nanobubbles. 2.4. Experiment with Dynamic Light Scattering. To clarify whether gas nanobubbles occur spontaneously in salt solutions we first of all carried out an experiment with dynamic scattering of light (DLS), as in refs 29 to 32. The DLS technique is based upon photonic correlation spectroscopy and is widely used for determining the sizes of nanoparticles in liquids. The idea of this technique consists in measuring the intensity I of light scattered in a liquid sample at a certain angle θ and consequent calculation of the time-correlation function GI(τ) ≡ ⟨I(t)I(t + τ)⟩.33 If we deal with particles of various sizes, the time correlation function GI(τ) is just a sum of several exponents. We performed the decomposition of the obtained function into the sum of exponents by means of a specialized computer program DynaLS,30 which uses the histogram method, including regularization. It should be kept in mind that this decomposition gives us nothing but the distribution of intensity of light scattered by particles of a given size. It is possible to easily get the particle size distribution if the intensity indicatrix of the light scattered by the particles is known. When applying the program, the boundaries of the time interval, where the decomposition is done, should correspond to the boundaries of the particle radius interval, where the intensity distribution is built. Furthermore, eqs 6 to 8 are adequate, and the DynaLS works correctly if the scattered light has Gaussian statistics. This implies that the mean number of particles of each size must exceed 20 to 30. In our measurements this condition was met. Below we present the results of the DLS experiment with 1.06 mol·kg−1 NaCl aqueous solution at the scattering angle θ = 60°. The function GI(τ) for that sample is illustrated in Figure 5. The analysis of the function GI(τ) gives us the following distribution of the scattered light intensity over the particles sizes. In this figure we clearly see two characteristic modes, related to several tens of nanometers and approximately one micrometer. Besides, we can see the one nanometer-sized particles and the particles at the scale of several hundred of micrometers. In our opinion, they are related respectively to the photomultiplier autocorrelations and to the convective motions of micrometer-sized particles. Let us remark that when explaining the graphs in Figure 6 (e.g., if we will estimate the concentration of the scattering particles), the right-hand peak related to particles at the scale of (105 to 106) nm should not be taken into account. Indeed, the particles of such size (0.1 to 1) mm should be clearly seen by naked eye, but we did not see these particles. Note that the water samples have been preliminarily thoroughly filtered by a porous membrane filter with a pore size of 200 nm, and chemically pure salt was added subsequently to this water. The occurrence of this peak is rather associated with a certain inhomogeneity of the particles distribution over the liquid volume, see ref 29. Moreover, we
Figure 6. Distribution of the intensity of the light scattered at the angle θ = 60° over the sizes of particles inside the sample of aqueous 1.06 mol·kg−1 NaCl solution.
always should account for weak convective counter-flows inside liquid sample, which result in slow stochastic oscillations of the mean scattered intensity. In computer processing of the experimental data these slow oscillations are identified as intensity fluctuations having very long time of coherence, which implies the presence of huge particles. As these particles are not seen, the right-hand peak in Figure 6 must be ignored. Thus we can assert that in the volume of aqueous salt solution the particles at the micrometer scale are indeed present; apparently the low optical density particles observed in the phased microscopy experiment, and the micrometer-sized particles found in the DLS experiment are the same particles (in fact, the liquid samples were simultaneously studied in both experiments). The main result obtained in the DLS experiment is the presence of rather broad size distribution for the nanometer-sized particles. However we still cannot say anything about the internal structure of the detected micrometer-sized particles. To answer this question, we carried out an additional experiment with laser polarimetric scatterometry. In this experiment we measured the angular dependencies of the scattering matrix elements. 2.5. Scattering Matrix Measurements. Let us remind that the scattering Fij(θ) matrix is a (4 × 4) matrix, transforming the Stokes vector of incident wave into the Stokes vector of wave scattered at the angle θ. It is known (see, e.g., refs 34 to 36) that the analysis of angular dependences of the scattering matrix elements not only gives the information about the effective size and geometric shape of scatterers, but also allows us to distinguish monolithic particles from the particles of cluster type (see below). The experimental setup used by us was described in detail in refs 15 to 18. Note that in this experiment (as well as in the microscopy experiment) we employed the second harmonic of CW YAG:Nd3+-laser (λ = 532 nm). The setup was calibrated by aqueous monodisperse suspensions of colloidal silica and polystyrene latex; the angular dependences of the scattering matrix elements obtained in these calibration experiments were in good agreement with the corresponding theoretic graphs, see refs 15 and 18 which allows us to claim that our experimental results are quite reliable. In Figure 7 the angular dependences of the scattering matrix elements for distilled water (black circles) and aqueous NaCl solution (1.06 mol·kg−1, gray circles) are plotted. The element F11(θ) is associated with the scattering indicatrix; other dependences exhibited in this figure are normalized to this element. As is known,35 the scattering matrix for the particles possessing a symmetry plane (e.g., spherical particles) has a block-diagonal form. Additionally, as was shown in our numeric 2827
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Figure 7. Experimental data: black circles (doubly distilled water); blue circles (aqueous NaCl solution, 1.06 mol·kg−1). Theoretic curves: gas spheres with the log-normal parameters: 1 (Reff = 100 nm, veff = 0.01), 2 (Reff = 500 nm, veff = 0.1), 3 (Reff = 1 μm, veff = 0.1); 4, Rayleigh particles.
Figure 8. Angular dependences of the scattering matrix elements for the ensembles of the clusters of ion-stabilized bubbles having the parameters ⟨N⟩ = 420, Reff = 50 nm, veff = 0.02; here α = −1.1, D = 2.35 (short dashes), α = −1.4, D = 2.45 (gray color), α = −1.7, D = 2.56 (long dashes). The scattering by the Rayleigh particles is shown by solid black curves. The experimental dependences for aqueous NaCl solution (1.06 mol·kg−1) are given by the open circles.
μm, veff = 0.1), 4, Rayleigh particles; Reff and νeff are the effective radius and the dimensionless effective width of the log-normal distribution, respectively. Note that the log-normal distribution parameters have been chosen in accordance with the experimental data obtained with the help of the microscope. Specifically, we plotted the histograms of the particles, spontaneously arisen in the liquid, over their sizes. As follows from the dependences exhibited in Figure 7, the element F11(θ) can basically correspond to the optical wave scattered by the monolithic micrometer-sized gas bubbles. At the same time, this element definitely cannot be connected with the optical wave scattered by Rayleigh particles or the gas nanobubbles having the log-normal parameters Reff = 100 nm and veff = 0.01.
modeling study (see our recent work28), the scattering matrix for clustered particles composed of spherical Rayleigh-type monomers has a block-diagonal structure as well. This is why it is not necessary to display here the graphs for all 16 elements, and we restrict ourselves by representing the graphs for the elements F11(θ), f12(θ), f 22(θ), f 33(θ), f 34(θ), and f44(θ), where f ij(θ) = Fij(θ)/F11(θ). In the graphs experimental uncertainties are depicted as error bars. The bar length grows with increasing the scattering angle due to reducing the signal-to-noise ratio. In the same diagrams we exhibit the result of numeric modeling for monolithic gas spheres at the micrometer scale with the following parameters of the log-normal distribution: 1 (Reff = 100 nm, veff = 0.01), 2 (Reff = 0.5 μm, veff = 0.1), 3 (Reff = 1.0 2828
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However, as is seen in other diagrams of the figure, the experimental dependences for the rest elements are best approximated by the Rayleigh particles or gas nanobubbles. It is known that such behavior is specific for clustered scatterers (see, e.g., refs 38 to 40): the particle itself has a size exceeding the wavelength of the incident light, that is, it is a Mie scatterer (as is revealed in the behavior of the element F11(θ) − the scattering indicatrix), but this particle should be composed of monomers, whose size is essentially smaller than the wavelength, that is, these monomers are Rayleigh-like particles and, in our particular case, nanosized bubbles. The question arises of what is the scenario of formation of such clustered particles? The data obtained with the phase microscope allow us to suppose that the clusters composed of ion-stabilized bubbles should not have vary dense packing, but their structure should be quasi-isotropic and somewhat localized in space. This property of clusters in our case is caused by the isotropic Coulomb attraction between the positive and the negative compound-particles. We have modified the hierarchic model for the clustering of spherical particles.37 As was indicated in ref 28, the radius of ionstabilized bubble is a random value lying in the range (10 to 100) nm. Thus the problem consists in the numeric modeling of clusters composed of spherical monomers having a certain distribution over their sizes. As earlier, we approximated this distribution by a log-normal function. In our model we iteratively generated sequences of clusters, starting with N separate spherical particles. The values of radii of these spheres are just the realizations of the stochastic process with a specified distribution. At each step of the iteration routine two clusters were randomly chosen; these clusters aggregate to one another, forming thus a new cluster. The probability P of choosing the given cluster obeys the power law in the form P = C·V−α, where V is the cluster volume, α is the parameter of the model, and C is the normalizing factor. In this hierarchic cluster−cluster aggregation model the average fractal dimension D of generated ensembles of the clusters depends monotonically upon the parameter α. This parameter is just a new “degree of freedom” that allows us to bring the angular dependence of the scattering indicatrix (the element F11(θ)) into the maximum proximity with the experimental points. To simulate the structure of the clusters composed of the ion-stabilized bubbles, we tried to find a solution of the inverse scattering problem for stochastically generated 103 hierarchictype clusters. For that we calculated numerically a set of the scattering matrices as the averages over computer-generated random cluster samples, whose statistical parameters were preset over a uniform grid. A solution was found by minimizing the divergence between the measured angular profiles of the scattering matrix and the calculated ones for an entire sample of clusters. In Figure 8 the results of the simulation of scattering matrix in accordance with this routine are shown. Here Reff and veff are the effective radius and effective width of the log-normal distribution over sizes of the ion-stabilized bubbles, D is the average fractal dimension of the clusters composed of ionstabilized bubbles, ⟨N⟩ is the average number of ion-stabilized bubbles in cluster, and σN is the dispersion of number of the ion-stabilized bubbles in cluster. As follows from the graphs, the calculated theoretic dependences are very close to the experimental ones for all explored elements of the scattering matrix. Thus the data obtained in the microscopy and scatterometry experiments allow us to conclude that the
particles, spontaneously arisen in salt solutions, are indeed hierarchic clusters composed of ion-stabilized bubbles. The angular profiles of the scattering matrix elements of cluster ensemble as dependent upon the statistical parameters Reff, veff, ⟨N⟩, σN, and α were investigated by the numeric experiments. For veff = (0.01 to 0.04) and σN = (60 to 90) the results of calculations were steady; that is, the changes of the element F11(θ) were not essential. This is why in the computer simulation we put veff = 0.02 and σN = 70. For arbitrary values of Reff in the range (33 to 55) nm it appears possible to find such values of ⟨N⟩ and α that the mean-square deviation of the theoretic diagrams for the element F 11 (θ) from the corresponding experimental data does not exceed the experimental error: σF11 ≤ σexp = 0.01. In ref 28 it has been shown that the theoretic radii R of ionstabilized bubbles radii may possess the values (22, 26, 45, 80, and 84) nm. If R = 45 nm is taken as the mean radius of lognormal distribution, then the value of effective radius Reff will be about 50 nm, which falls within the range for the effective radii. In Figure 9 a stochastic computer model of the hierarchic cluster with the parameters ⟨N⟩ = 420, Reff = 50 nm, veff = 0.02,
Figure 9. Mutually perpendicular projections of a stochastic realization of the hierarchic ion-stabilized bubble cluster with the parameters ⟨N⟩ = 420, Reff = 50 nm, veff = 0.02, α = −1.4 (D = 2.45).
and α = −1.4 (D = 2.45) is displayed. We suppose that the detected in 1.06 mol·kg−1 aqueous solution of NaCl clusters of ion-stabilized bubbles look similar to this.
3. CONCLUSIONS Experiments with three independent techniques (phase microscopy, DLS, and polarimetric scatterometry) have revealed the existence of spontaneously forming particles at the micrometer scale in aqueous solutions of NaCl. The refractive index of such particles is less than that for the ambient liquid, but slightly higher than that for gas medium, and the measured angular profiles of the scattering matrix elements are characteristic for micrometer-sized clusters composed of small monomers: the cluster itself is a Mie particle, whereas the monomers are Rayleigh-like particles. Additionally, spontaneously generated free nanoparticles, ranging in size from approximately 30 nm to approximately 100 nm, are also detected in these solutions. Thus, it is straightforward to assume that the observed micrometer-sized particles have resulted from the coagulation of dissolved-gas nanobubbles. 2829
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(12) Bunkin, N. F.; Bunkin, F. V. Screening of Strongly Charged Macroparticles in Liquid Electrolyte Solutions. J. Exp. Theor. Phys. 2003, 96, 730. (13) Bunkin, N. F.; Bunkin, F. V. Adsorption and desorption of ions at the surface of liquid. Z. Phys. Chem. 2001, 215, 111. (14) Bunkin, N. F.; Lobeyev, A. V.; Lyakhov, G. A.; Ninham, B. W. Mechanism of low-threshold hypersonic cavitation stimulated by broadband laser pump. Phys. Rev. E 1999, 60, 1681. (15) Bunkin, N. F.; Suyazov, N. V.; Shkirin, A. V.; Ignatiev, P. S.; Indukaev, K. V. Nanoscale structure of dissolved air bubbles in water as studied by measuring the elements of the scattering matrix. J. Chem. Phys. 2009, 130, 134308. (16) Bunkin, N. F.; Suyazov, N. V.; Shkirin, A. V.; Ignatiev, P. S.; Indukaev, K. V. Cluster Structure of Stable Dissolved Gas Nanobubbles in Highly Purified Water. J. Exp. Theor. Phys. 2009, 108, 800. (17) Bunkin, N. F.; Shkirin, A. V.; Kozlov, V. A.; Starosvetskiy, A. V. Laser scattering in water and aqueous solutions of salts. Proc. SPIE 2010, 7376, 73761D. (18) Bunkin, N. F.; Ninham, B. W.; Shkirin, A. V.; Ignatiev, P. S.; Kozlov, V. A.; Starosvetskiy, A. V. Long-living nanobubbles of dissolved gas in aqueous solutions of salts and erythrocyte suspensions. J. Biophotonics 2011, 4 (3), 150. (19) Bunkin, N. F.; Bunkin, F. V. The New Concepts in the Optical Breakdown of Transparent Liquids. Laser Phys. 1993, 3, 63. (20) Bunkin, N. F.; Lobeyev, A. V. Bubbston-cluster structure under conditions of optical breakdown in a liquid. Quantum Electron. 1994, 24, 297. (21) Vinogradova, O. I.; Bunkin, N. F.; Churaev, N. V.; Kiseleva, O. A.; Lobeyev, A. V.; Ninham, B. W. Submicrocavity structure of water between hydrophobic and hydrophilic walls as revealed by optical cavitation. J. Colloid Interface Sci. 1995, 173, 443. (22) Bunkin, N. F.; Lyakhov, G. A. Microbubbles of dissolved gas in water: physical studies and possible applications in biological technologies and medicine. Phys. Wave Phenom. 2005, 13, 61. (23) Bunkin, N. F.; Bakum, S. I. Role of a dissolved gas in the optical breakdown of water. Quantum Electron. 2006, 36, 117. (24) Bunkin, N. F.; Kochergin, A. V.; Lobeyev, A. V.; Ninham, B. W.; Vinogradova, O. I. Existence of charged submicrobubble clusters in polar liquids as revealed by correlation between optical cavitation and electrical conductivity. Colloids Surf., A 1996, 110, 207. (25) Bunkin, N. F.; Kiseleva, O. I.; Lobeyev, A. V.; Movchan, T. G.; Ninham, B. W.; Vinogradova, O. I. Effect of salts and dissolved gas on optical cavitation near hydrophobic and hydrophilic surfaces. Langmuir 1997, 13, 3024. (26) Bunkin, N. F.; Ninham, B. W.; Babenko, V. A.; Suyazov, N. V.; Sychev, A. A. Role of Dissolved Gas in Optical Breakdown of Water: Differences between Effects Due to Helium and Other Gases. J. Phys. Chem. B 2010, 114, 7743. (27) Duval, E.; Adichtchev, S.; Sirotkin, S.; Mermet, A. Long-lived submicrometric bubbles in very diluted alkali halide water solutions. Phys. Chem. Chem. Phys. 2012, 14, 4125. (28) Bunkin, N. F.; Yurchenko, S. O.; Suyazov, N. V.; Starosvetskiy, A. V.; Shkirin, A. V.; Kozlov, V. A. Modeling the cluster structure of dissolved air nanobubbles in liquid media. Classification and Application of Fractals; Hagen, W. L., Ed.; Nova Science Publishers: New York, 2011. (29) Kovalenko, K. V.; Krivokhizha, S. V.; Masalov, A. V.; Chaikov, L. L. Correlation spectroscopy measurements of particle size using an optical fiber probe. Bull. Lebedev Phys. Inst. 2009, 36, 95. (30) Chu, B. Laser Light Scattering; Academic Press: New York, 1974. (31) Berne, B. J.; Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology and Physics; Wiley-Interscience: New York, 1976. (32) http://www.photocor.com/download/Alango/dynals-whitepaper.htm. (33) Bendat, J. S.; Piersol, A. G. Random Data: Analysis & Measurement Procedures; Wiley-Interscience: New York, 2000. (34) Gerrard, A.; Burch, J. M. Introduction to Matrix Methods in Optics; Wiley: London, 1975.
We have proposed a mathematical approach to modeling the clusters of spherical particles that is based on the approximation of experimental angular dependences of the scattering matrix elements by the theoretical ones, which are calculated as the average over a stochastic cluster ensemble. The hierarchical computational model of fractal growth allowing for cluster− cluster aggregation is elaborated. In this model the fractal dimension of generated random realizations of clusters has monotonic dependence on the model parameter, defining the probability of participation of a cluster in coagulation act according to its volume. The variation of this parameter enables us to approximate uniformly the scattering indicatrix within a given angular range. With the help of this aggregation model the inverse light scattering problem of determining the fractal properties of nanobubble clusters in water media was solved. It was found that in aqueous 1.06 mol·kg−1 solution of NaCl the mean number of monomeric nanobubbles in one cluster is 420, the effective radius and effective relative width of log-normal size distribution of monomers are 50 nm and 0.02, respectively, the mean fractal dimension of nanobubble clusters D = 2.45.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Funding
This study has been supported by the Russian Foundation for Basic Researches, Grant No. 10-2-00377-A. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors wish to acknowledge the contribution made by P. S. Ignatiev, L. L. Chaikov, I. S. Burkhanov, and A. V. Starosvetskij to this work.
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REFERENCES
(1) Epstein, P. S.; Plesset, M. S. On the Stability of Gas Bubbles in Liquid-Gas Solutions. J. Chem. Phys. 1950, 18, 1505. (2) Ljunggren, S.; Eriksson, J. C. The lifetime of a colloid-sized gas bubble in water and the cause of the hydrophobic attraction. Colloids Surf., A 1997, 129−130, 151. (3) Crum, L. A. Tensile strength of water. Nature (London) 1979, 278, 148. (4) Crum, L. A. Acoustic cavitation thresholds in water. Cavitation and Inhomogeneities in Underwater Acoustics; Lauterborn, W., Ed.; Springer-Verlag: New York, 1980. (5) Crum, L. A. Nucleation and stabilization of micro-bubbles in liquids. Appl. Sci. Res. 1982, 38, 101. (6) Crum, L. A. Nucleation and stabilization of microbubbles in liquids. Mechanics and Physics of Bubbles in Liquids; van Wijngaarden, L., Ed.; Martinus Nijhoff Publishers: The Hague, 1982. (7) Sirotyuk, M. G. Experimental investigations of ultrasonic cavitation. High Intensity Ultrasonic Fields; Rozenberg, L. D., Ed.; Plenum Press: New York, 1971. (8) International Critical Tables, Vol. 4; Washbur, E. W., Ed.; McGraw-Hill: New York, 1928. (9) Wagner, C. Oberflächenspannung verdünnter elektrolytlösungen. Phys. Z. 1924, 25, 474. (10) Onsager, L.; Samaras, N. T. The surface tension of DebyeHiickel electrolytes. J. Chem. Phys. 1934, 2, 528. (11) Bunkin, N. F.; Bunkin, F. V. Bubbstons: stable microscopic gas bubbles in very dilute electrolytic solutions. J. Exp. Theor. Phys. 1992, 74, 271. 2830
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(35) Mishchenko, M. I.; Travis, L. D.; Lacis, A. A. Scattering, Absorption, and Emission of Light by Small Particles; Cambridge University Press: Cambridge, 2002. (36) Doicu, A.; Wriedt, T.; Eremin, Y. A. Light Scattering by Systems of Particles; Springer-Verlag: Berlin, 2006. (37) Menguc, M. P.; Manickavasagam, S. Characterization of size and structure of agglomerates and inhomogeneous particles via polarized light. Int. J. Eng. Sci. 1998, 36, 1569. (38) Kimura, H. Light-scattering properties of fractal aggregates: numerical calculations by a superposition technique and the discretedipole approximation. J. Quant. Spectr. Rad. Trans. 2001, 70, 581. (39) Kolokolova, L.; Kimura, H.; Ziegler, K.; Mann, I. Lightscattering properties of random-oriented aggregates: Do they represent the properties of an ensemble of aggregates? J. Quant. Spectr. Rad. Trans. 2006, 100, 199. (40) Jullien, R. Aggregation phenomena and fractal aggregates. Cont. Phys. 1987, 28, 477.
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