Principles and Applications of Broadband Acoustic Resonance

Jan 18, 2012 - ... of compounds as a function of time, concentration, and solvent system. ... It is anticipated that BARDS will open up a new window i...
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Principles and Applications of Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS): A Sound Approach for the Analysis of Compounds Dara Fitzpatrick,*,† Jacob Krüse,‡ Bastiaan Vos,† Owen Foley,† Donncha Gleeson,† Eadaoin O’Gorman,† and Raymond O’Keefe† †

Department of Chemistry, University College Cork, Cork, Ireland Kinetox, Beilen, The Netherlands

‡

S Supporting Information *

ABSTRACT: The dissolution of a compound results in the introduction and generation of gas bubbles in the solvent. This formation is due to entrained gases adhered to or trapped within the particles. Furthermore, a reduction in gas solubility due to the solute results in additional bubble generation. Their presence increases the compressibility of the solvent with the added effect of reducing the velocity of sound in the solvent. This effect is monitored via the frequency change of acoustic resonances that are mechanically provoked in the solvent and are now used as an insightful analytical technique. An experimental set up was designed to study a large number of compounds as a function of time, concentration, and solvent system. This revealed the role of the various physical and chemical mechanisms in determining the observed response. It is also shown that this response is strongly dependent on the physical and chemical characteristics of the solute compound used, therefore resulting in a method for the characterization of compounds and mixtures. Additional factors such as morphology (polymorphism), particle size, and dissolution rate are shown to be key in the variation of the resulting response. A mathematical model has also been developed in parallel, which inter-relates the various processes involved in the observed response. It is anticipated that BARDS will open up a new window into transient dissolution processes and compound characterization.

T

solution. This is now shown to be a powerful method for the analysis and characterization of compounds and mixtures. The technique is designated as Broadband Acoustic Resonance Dissolution Spectroscopy (BARDS) due to the specific experimental approach used and the focus on dissolving compounds. First, the physical principles of the underlying acoustic phenomenon are described, followed by the practical instrumental implementation of the proposed method in an experimental set up. The resulting data obtained demonstrate which physical and chemical properties are involved in the observed acoustic response when adding a solute to a solvent. There are also applications with nonsoluble materials, e.g. silica, indicating that dissolution is not imperative for this investigation technique. Also, liquid−liquid interactions are observable via BARDS, e.g., the addition of acetonitrile to water. However, only solute−solvent interactions will be presented here. Mathematical modeling will be used to inter-relate the various chemical and physical processes involved, to gain more

he instant addition of a solute into a solvent invariably results in the visible appearance of minute air bubbles in the liquid. The instantaneous presence, generation, and subsequent disappearance of these bubbles can be detected indirectly, in a sensitive way and in real time by using a specific acoustic phenomenon, which relates the total bubble volume present to the sound velocity in the liquid. The acoustic effect of bubbles, in this case specifically generated by dissolving table salt, has been independently observed by one of the authors (DF). However, the acoustic effect of air bubbles has been described and discussed previously by Frank S. Crawford in a series of papers in the early 1980s.1,2 Other observations on the acoustic phenomenon and citations to its relevance also exist in the current literature.3−10 It is not until now that its significance as an investigative tool for the analysis of chemical compounds has been realized. This is achieved by monitoring the acoustic phenomenon during the dissolution of a compound e.g., NaCl, as the gas solubility in solution reduces and the gas volume increases. The subsequent gas supersaturation results in additional gas bubbles which counter the gas supersaturation. Careful monitoring of the resonance frequency time courses in the liquid, after broadband acoustic excitation of the containing vessel, captures indirectly the change in gas/bubble volume in © 2012 American Chemical Society

Received: September 21, 2011 Accepted: January 18, 2012 Published: January 18, 2012 2202

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Figure 1. (A) BARDS spectrum of 1.4 M sodium carbonate dissolved in 100 mL of water. Note the steady state resonances for 30 s before the addition of the compound. (B) Schematic diagram (top view) of the prototype BARDS spectrometer.

results in an increase in volume and density of the solution and thereby, in theory, in an additional reduction in sound velocity. However, with the experiments presented in this paper and the concentrations used, this further reduction will usually be minor. In brief, two key parameters, density and compressibility, which normally oppose each other, with respect to the speed of sound, instead combine and reinforce one another to create an acoustic phenomenon in solution. The frequency minimum ( f min) of the BARDS response represents an equilibrium between the rate of formation of gas in solution and the rate of liberation of gas from the surface of the solvent. The solution is at its most compressible at this point.

insight into the sensitivity of the observed response for specific properties of the compounds and solvents. Finally, examples of promising applications of the technique in the analysis and characterization of chemical compounds will be presented.

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PRINCIPLES OF THE BARDS RESPONSE The sound velocity in a medium whether air or liquid phase is determined by eq 1 V(sound) =

1 K ·ρ

(1)

where ρ is mass density, and K is compressibility, which is the inverse of the bulk modulus, of the medium. Generation of micro gas bubbles in a liquid, representing only a small fraction of the total liquid volume, decreases the density in a negligible way, when compared with the large increase in compressibility.1,2 Therefore the net effect is a significant reduction of the sound velocity in the liquid. Frank S. Crawford derived the following relationship, eq 2, between the fractional bubble volume and the sound velocity in water vw = (1 + 1.49 × 104 ·fa ) (2) v

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EXPERIMENTAL SECTION Materials. The following materials used were of analar grade and were purchased from Sigma Aldrich and Riedel-de Haen: tripotassium phosphate, dipotassium hydrogen phosphate, potassium dihydrogen phosphate, glucose, mannose, sodium carbonate, copper sulfate, L-tartaric acid, D-tartaric acid, and glycine. Doubly distilled water was used for all experiments. Instrumentation. The prototype spectrometer used to investigate the BARDS response consists of a glass tumbler with a microphone, a magnetic stirrer, and follower (Figure 1B). The microphone is attached to the outside of the glass at 1.8 cm from the top rim using adhesive tape. Since physical contact to the glass is not necessary, alternatively a microphone positioned just above the top of the glass may be used. The glass, containing 100 mL of water is placed off center on the stirrer plate so as to allow the magnetic follower to gently tap the inner glass wall. In this way the follower acts as a source of broadband acoustic excitation, thereby inducing various acoustic resonances in the glass, the liquid, and the air column above the liquid. The induced acoustic resonances are registered by the microphone and converted to a spectrum using a PC with a sound card and generic software. The resonances of the liquid vessel are recorded in a frequency band of 0−20 kHz in a BARDS experiment. Experimental Procedure. In a typical experiment, the spectrometer records the steady state resonances of the system as a reference for thirty seconds when the stirrer is set in motion. The pitch of the resonance modes in the solution change significantly, when the solute is added, before gradually returning to steady state over several minutes (Figure 1A). The amounts added are expressed as molar concentrations in all figures unless, due to the type of material, gram/L is preferred. Thereby no corrections are made for effects of volume change due to the addition of a solute.

where vw and v are the sound velocities in pure and bubblefilled water, respectively. fa is the fractional volume occupied by air bubbles. The factor 1.49 × 104 in the formula was calculated as shown in eq 3 (vw)2 ρw ·

1 = 1.49 × 104 γp

(3)

where ρw is the density of water, γ is the ratio of specific heats for dry air, and p is the atmospheric air pressure. Equation 2 is based on the approximation presented originally by Wood.10 BARDS analysis of an induced acoustic response (Figure 1A) is focused on the lowest variable frequency time course, i.e., the fundamental resonance mode of the liquid. Its frequency is determined by the sound velocity in the liquid and the approximate but fixed height of the liquid level, which corresponds to 1/4 of its wavelength.1 Thus the frequency response takes the form of eq 4 freq =

freqw 1 + 1.49 × 104 ·fa

(4)

where f reqw and f req are the resonance frequencies of the fundamental resonance modes in pure and bubble-filled water, respectively. Addition and dissolution of a test compound 2203

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Figure 2. (A) The effect of temperature on the fundamental resonance profile of sodium carbonate dissolved in 100 mL of H2O. (B) The effect of sequential additions of 0.5 M sodium carbonate (SC) to the same aqueous solution. (C) The effect of a gradual increase in concentration of sodium carbonate.

generally do not have a direct relationship to the total bubble volume. Controlling Variables of BARDS. The effects of variables such as temperature, volume, mass, viscosity, density, and solvent have been systematically investigated. First, a noticeable effect of temperature, on the BARDS spectrum, is related to the solubility of the compound at a given temperature, its dissolution rate, and the viscosity of the solution. Figure 2A shows the BARDS response of sodium carbonate at 6 and 21 °C. The dissolution process is delayed at lower temperature, indicated by a delayed f min, and ΔT takes an additional 200 s. A change in viscosity with temperature plays a role in both processes. Sequential additions of sodium carbonate result in an increase in salinity and also in viscosity.12 The initial deflection to f min upon each addition of sodium carbonate in Figure 2B is in the same order of magnitude indicating a similar change in gas solubility/compressibility. The return rate to equilibrium decreases with each addition due to the increase in viscosity thus reducing the rate of gas liberation at the surface. It is also due to a decrease in dissolution rate of the solute at higher viscosity and also when approaching its solubility limit. Similar effects are observed when solvents of different viscosities are used for the analysis of a compound. Mechanical variables were investigated to ascertain their relevance, e.g., a magnetic follower is not only used for mixing but also to induce the dissolution vessel and liquid to resonate in the acoustic frequency range. Alternative methods of acoustic excitation have been investigated such as external induction sources and frequency generators. However, none have provided satisfactory spectra compared to those generated by internal resonance induction using the stirrer bar. The effects of various parameters on this induction of acoustic resonances have been investigated in order to validate the method. It has been found that neither the thickness of the stirrer bar nor the presence of a raised ridge at the center of the bar have any effect on the acoustic spectra. However, both increase of bar length and spinning rate affect the BARDS response in a similar way, e.g., a more rapid return to steady state frequency. This effect is especially observed for Na2CO3 and becomes more prominent at higher concentrations, closer to the solubility limit. The effects at comparable concentrations of NaCl, which is much more soluble than Na2CO3, are virtually absent. Therefore, when dissolution rate becomes a limiting factor in the BARDS response, it is sensitive to stirring and mixing efficiency.

In Figure 2A the frequency time course of the fundamental resonance is presented, as manually extracted data from the total acoustic response (e.g., Figure 1A). Spectra were recorded for a total of 400−800 s depending on the rate of return of the fundamental frequency to steady state. All experiments were performed in triplicate, and an average reading with error bars is presented. Under standardized conditions (constant volume, concentration, temperature, and stirring rate) the time courses of the observed acoustic profiles are shown to be highly reproducible. Mathematical Modeling. Mathematical modeling is used to inter-relate the various chemical and physical processes that are involved in BARDS and their kinetics. It describes the introduction and generation of air bubbles in the liquid and their subsequent dissipation. The underlying processes include dissolution of solutes, decrease in gas solubility plus initiation and growth of gas bubbles. The kinetic behavior of the various processes is described by a series of differential equations, which are numerically solved using the Berkeley Madonna software package.11

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RESULTS Spectral Information. The spectra are characterized by assigning specific features with designated nomenclature. The first thirty seconds of the spectrum in Figure 1A shows several steady state resonances of the vessel. The sample compound is added at the thirty second time point which results in a significant decrease in many of the resonant frequencies. The resonance line with the lowest frequency minimum is selected for comparison with other spectra as it is the most retrievable and also interpretable feature. This resonance line is designated the fundamental curve. The time taken to reach f min is designated as Δt. The origin of the fundamental curve can be identified by following the fundamental resonance line positively along the time axis from f min until it reaches steady state. At this time point it is possible to extrapolate negatively along the time axis to find the matching resonance (origin) in the first thirty seconds of steady state resonance. The origin of the fundamental resonance is designated as the ‘volume line’ due to its interdependence with the volume of solvent in the vessel. The approximate time taken for the fundamental curve to progress from f min to steady state is designated as ΔT. Other resonance curves and lines are attributed to overtones, additional modes, and interferences or else originate from the air column above the liquid. These additional resonances 2204

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Figure 3. Fundamental acoustic concentration profiles of (A) K3PO4, (B) K2HPO4, and (C) KH2PO4 are shown respectively (replicates n = 3 replicates for each concentration profile, solvent = 100 mL of H2O).

Figure 4. (A) The behavior of copper pentahydrate (Penta H) and its anhydrous form (AH) are shown to be significantly different. Mixture ratios of the compounds are also shown. (n = 3 replicates for each concentration, solvent = H2O.) (B) Fundamental acoustic profiles of epimers D-glucose and D-mannose and mixture ratios of both compounds dissolved in 100 mL of H2O. (C) Fundamental profiles of particle size distributions of γ glycine dissolved in 100 mL of H2O.

(hydrates). The pentahydrate of copper sulfate (0.4 M) shows resonance f min = 6.5 kHz, whereas an equimolar amount of the anhydrous compound has a greater deflection to f min = 1.8 kHz due to a significantly different change in the compressibility of the aqueous solvent. Analysis of Epimers. Figure 4B demonstrates the ability of BARDS to differentiate between epimers of D-glucose and Dmannose. The compounds, of the same molecular weight, differ only in the orientation of one chiral center, which results in two different crystalline forms. The acoustic profiles of both compounds are significantly different. The mixtures of the two compounds produce profiles intermediate to the pure forms. Particle Size Effects. Figure 4C illustrates the effect of particle size on the acoustic profile of γ glycine (1 M), a simple amino acid, for four different size distributions. In general, the larger the particle size the greater the acoustic response. It is possible to differentiate between the particle size distributions at Δt = 70s. In the graph, the differences in the BARDS response for various particle sizes will predominantly lie in the magnitude of f min. The differences in the profiles at Δt = 70 s may still result from those at Δt = 0 s. Effects of the Geometry of the Vessel. It should be noted from eq 4 that in theory the ratio of f reqw and f req is not dependent on the geometry of the glass or the height of the liquid column, enabling the comparison of results obtained in various volumes and vessel geometries through normalization.

The effect of an incremental increase in concentration is shown in Figure 2C for sodium carbonate. The acoustic response is observed to increase with increasing concentration. An explanation of the observed response is given in the following sections and is successfully simulated in Figure 6B and C. BARDS Qualitative Analysis. A homologous series of anhydrous phosphate salts were chosen in order to illustrate qualitative applications of the technique in showing differences between these related compounds. Figure 3 contains the acoustic profiles (n = 3 replicates for each concentration) of tri-, di-, and monopotassium phosphate for a range of concentrations. The behavior of the potassium phosphate salts can be seen to alter significantly as a potassium cation is replaced by a hydrogen cation in the compound. These changes can be characterized in several ways, e.g., the concentration required to achieve a specific decrease in the resonant frequency to f min. Another feature is the concentration which indicates a transition in behavior of the profiles from ‘V-shaped’ to ‘Ushaped’ curves. The magnitude of the acoustic response and the rate of return to steady state frequencies are also characteristic for each compound. Discrimination of Pseudo Polymorphs (Hydrates vs Anhydrous)). Another useful application is illustrated by a demonstration of the qualitative differences between anhydrous and hydrated compounds e.g., copper sulfate. Figure 4A shows the acoustic profiles of copper sulfate’s pseudo polymorphs 2205

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Figure 5. (A) Comparison of L-tartaric acid spectra in two different volumes and the normalization of 100 mL data. (B) Comparison of D-tartaric acid spectra in two different volumes and the normalization of 100 mL data. (C) Comparison of sodium carbonate spectra in two different volumes and the normalization of 100 mL data. Note the significant correlation between the 25 mL data and the normalized 100 mL data.

Figure 6. (A) Discrimination of 0.14 M piracetam polymorphs (II and III) and a 50:50 mixture ratio of both dissolved in 25 mL of H2O. (B) Simulation of the increase in concentrations of sodium carbonate shown in Figure 2B. (C) Simulation of the sequential addition of sodium carbonate to the same aqueous solvent. Concentrations are the same as those in Figure 2C.

However, the time course of the fractional volume occupied by air bubbles (fa) may be affected by changes in vessel geometry, possibly causing changes in the physical and chemical kinetics. Figure 5A shows the comparison of tartaric acid measured in volumes of 100 mL (vessel A) and 25 mL (vessel B). Vessel A produces a volume height of ∼4.8 cm with a resonant frequency of the volume line = 14.5 kHz. Vessel B produces a volume height of ∼2.7 cm with a resonant frequency of the volume line = 8.5 kHz. A normalization factor derived from the ratio of the volume line frequencies was used to convert the 100 mL data to correlate with the experimentally measured 25 mL data. There is significant correlation between the normalized data and the experimentally measured data. Figure 5C shows a similar correlation when sodium carbonate is measured using the same methodology in the same vessels. In theory, the surface area/volume ratio may be expected to have an effect on the spectra. However, it was demonstrated not to be a dominant factor in the methodology used in the experiments of Figure 5A, B, and C where the surface area/ volume ratio differs by a factor of 1.7 between the 100 and 25 mL experiments. Polymorph Analysis. The two polymorphic forms of piracetam (II and III) yield significantly different spectra from each other as shown in Figure 6A. This illustrates the discrimination of crystal morphology. A mixture of the two forms also has a unique acoustic signature which is intermediate to both forms. BARDS Simulations. Figure 6B shows examples of modeling the BARDS effect for a set of concentration profiles of sodium carbonate. Further simulations of the BARDS experiment with four sequential additions of 0.5 M sodium

carbonate (Figure 2C) are presented in Figure 6C. There exists significant correlation in the behavior of the theoretical and experimental profiles. This supports the mechanistic approach now outlined in the next section.

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MODELING AND INTERPRETATION OF BARDS A physically and chemically based mathematical model has been developed. This describes and inter-relates the various processes involved in the generation of a BARDS response. The model has been tested and optimized with BARDS data obtained for a variety of compounds using a range of concentrations and in some cases also various solvents (Figure 6B and C). The main objectives are to gain insights into the role and contribution of each of the various processes involved in the acoustic response, by comparing simulations with experimental results. However it should be noted that the model, with its large number of parameters, is not designed to fit to experimental data. Literature data or reasonable estimates have been used for several of the parameters in the model, e.g., solubility’s of compounds and gases and solution viscosities. The values of kinetic parameters have also been based on experimental results. Several processes can be distinguished in the generation of the BARDS response upon addition of a compound/solute to a solvent. Detailed information regarding each of these processes is now elaborated upon. Dissolution Rate of the Added Compound. The quantitative description of the dissolution process is based upon the approach of Hixson and Crowell,13 which in addition to the diffusion limitation around the dissolving particle, also 2206

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with increasing amounts of dissolved substances results in gas oversaturation in the solution

takes into account the reduction in particle surface area during the dissolution process. This is in contrast to the NoyesWhitney approach.14 The effect of the dissolved solute concentration on the dissolution rate, which plays a role near the solubility limit, so under nonsink conditions, is also implemented in the dissolution rate modeling. In the modeling of the dissolution process it is assumed that the added solute consists of spherical particles of identical diameter. The number of particles added, the amount of solute per particle, and the surface area per particle are calculated from the amount of solute added, the density of the particle, and the initial particle diameter. The time course of the amount of nondissolved solute per particle during the dissolution process is obtained by integration of eq 5 ⎛ D·Ps ⎞ dA ⎟ · (C =−⎜ − Cvol) ⎝ L ⎠ solb dt

Gos = Gsol(0) − Gsol(Cvol)

where Gos is the gas oversaturation in the solution, Gsol(Cvol) is the gas solubility in solution as a function of the concentration of dissolved solute, and Gsol(0) is the solubility of gas in water. If it is assumed that Gos does not decrease, e.g. via bubble growth or gas exchange at the liquid surface, the time dependent change in gas oversaturation can be calculated using eq 9 dGos dGsol dCvol =− . dt dCvol dt ⎛ D·Ps ⎞ dCvol P ⎟ · (C =⎜ − Cvol) · num ⎝ L ⎠ solb dt V

(5)

2Dw (ηbulk + ηsat )

(10)

Reduction of the Actual Gas Oversaturation via Gas Exchange at the Gas/Liquid Surface of Bubbles Present in the Solution. Net transport of dissolved gases will occur from the oversaturated solution into gas bubbles present in the solution, to restore the equilibrium in agreement with Henry’s Law constants for these gases. The reduction of gas oversaturation via this mechanism is assumed to be a first order kinetic process

(6)

where Avol is the amount of dissolved solute in the solution, Pnum is the number of particles in the solution, and V is the volume of the solution. The diffusion coefficient (D) in the barrier is calculated by eq 7 D=

(9)

where

where A is the amount of solute present in a single particle at time t, Ps is the surface area of the particle at time t, L is the thickness of the diffusion barrier around the particle, D is the diffusion coefficient in the barrier, Csolb is the maximum solubility concentration of the solute, and Cvol is the concentration of dissolved solute in the solution given by eq 6 P ·A Cvol = num vol V

(8)

dGos = −Gos ·kbg dt

(11)

where kbg is the bubble growth rate constant. It should be noted, especially in the case of bubbles with a small diameter (d), the pressure (p) inside these bubbles will be significantly increased, according to the Young−Laplace equation19

(7)

where Dw is diffusion coefficient in pure solvent (H2O), ηbulk is the viscosity in the bulk solution at a given concentration of dissolved solute, and ηsat is the viscosity at the maximum solubility of solute, Csolb; both are values relative to the viscosity of the pure solvent. During the dissolution process the changing particle surface, Ps, is calculated from the amount of solute still present in the particle, A, and the density of the solute. According to eq 5 the dissolution rate will decrease as Cvol approaches Csolb. Furthermore, the dissolution rate of the total amount added will decrease with increasing initial particle size. Decrease of Gas Solubility due to the Increase in Concentration of Dissolved Compound. The solubility of gases in liquids, Gsol, decreases with increasing concentrations of dissolved substances. The decrease in gas solubility is compound specific.15−17 Available literature data predominantly concern oxygen solubility; therefore, adjustments have been made to reflect the solubility of air by assuming that the ratio of nitrogen and oxygen solubility’s do not change with increasing concentrations of dissolved solute. It is also known that the nitrogen/oxygen solubility ratio decreases by ∼2% for a salinity of 40 g/L, which corresponds to a concentration of approximately 0.68 M NaCl.18 The oxygen solubility function data are fitted to a polynomial function to facilitate the calculation of its derivative (dGsol/dCvol). Incorporation of this correction has only negligible effects on the simulation results. Increase in Gas Oversaturation Due to Decreased Gas Solubility. According to eq 8, the decrease in gas solubility

Δp =

4·σ d

(12)

where σ is the surface tension of water (7.5 × 10−4 N/cm), and Δp is the increase in pressure. Therefore a sufficiently large degree of gas oversaturation is required to overcome the additional pressure barrier in minute air bubbles which act as bubble nucleation/growth centers. Reduction of the Actual Gas Oversaturation via Gas Exchange at the Liquid Surface of the Solution in the Vessel. Dissolved gases will exchange with the atmosphere above the liquid surface. The reduction of gas oversaturation due to this exchange is also assumed to be a first order kinetic process according to eq 13 dGos = −Gos ·kse dt

(13)

where kse is the rate constant of gas exchange at the surface of the solution. In the methodology used in the BARDS experiments, it is assumed the solution is well-stirred, thereby resulting in a homogeneous distribution of the dissolved gases. The gas exchange at the surface rate constant (kse) will then linearly depend on the cross section of the liquid/air surface and will further be inversely proportional to the volume of the test solution. If the ratio of this cross section to the liquid volume is not constant, then effects on the BARDS response may be expected theoretically. 2207

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Transient Gas Oversaturation Following Decrease in Gas Solubility and Gas Exchange at the Liquid Surface. When combining eqs 9, 11, and 13, the following expression for the gas oversaturation Gos results dGos dGsol dCvol =− · − Gos ·(kbg + kse) dt dCvol dt

gas exchange at gas/liquid interfaces. Ultimately, a new equilibrium concentration of dissolved gas will be attained according to Henry’s Law. In addition to gas exchange at the liquid surface of the solution, this may occur at the gas/liquid interface of minute gas bubbles introduced into the solution. The BARDS profiles resulting from this gas supersaturation dependent bubble growth is evident as a shoulder or U-shaped responses with increasing concentration. However, due to the increased pressure existing in the small bubbles (Young− Laplace) an additional pressure barrier has to be overcome to enable gas transfer into these bubbles in agreement with Henry’s Law constant. Therefore, a minimum threshold level of gas oversaturation (Cthresh) is required to enable bubble growth. Below this threshold existing bubbles will not grow and may shrink and collapse or escape at the surface of the solution. Immediately after addition of the solute only a small amount will have dissolved, and, therefore, no noticeable gas oversaturation will be present. Sufficiently high gas oversaturation will be attained with additions exceeding Cthresh before complete disappearance of the nucleation/growth centers can take place, and growth of these bubble nucleation centers will occur. In the model, the main source of bubble nucleation centers is assumed to consist of minute introduced gas bubbles, loosely adhered to the surface of the added solute particles, or trapped within porous solute particles. It is also assumed that the amount of gas/bubbles introduced by the solute, acting as bubble nucleation/growth centers, is linearly dependent on the concentration of the added solute (Cadd). The BN parameter depends on the chemical and morphological properties of the solute. These properties together determine the bubble nucleation power, which is a quantitative measure of initiation of bubble growth in a gas oversaturated solution, thereby resulting in a U-shaped response in the BARDS spectrum. Furthermore, the particle size will determine the total surface area of the solute and in this way the number of adhered bubbles. The number of bubble nucleation/growth centers is assumed to decrease in time via escape from the solution. This process is described by the term exp(‑kel·t) in eq 16. Decrease of Total Gas Volume Due to Shrinking and Collapse of Bubbles (Redissolution) and via Escape/ Annihilation of Gas Bubbles at the Liquid Surface of the Solution. Three potential sources of gas bubbles can be distinguished, each with their own specific behavior. They include (a) gas bubbles loosely adhered to the solute particles and instantly liberated into the solution after addition, (b) gas trapped within porous solute particles, where gas is gradually liberated into the solution during the dissolution of these particles, and (c) gas bubbles grown by desolvation of gas in a gas oversaturated solution. Gas bubbles originating from sources (a) and (b), considered as entrained gases, may shrink and collapse (redissolve) or continue to exist as bubble nucleation centers. All remaining bubbles will gradually disappear from the solution via escape/ annihilation at the liquid surface of the solution. First order elimination kinetics is assumed for the disappearance of the gas bubbles with, theoretically, specific elimination rate constants for each source of bubbles. The choice for first order kinetics is based on the following assumptions. In the methodology used in the BARDS experiments, it is assumed the solution is well-stirred, therefore resulting in a reasonably homogeneous distribution of gas bubbles. Visual observation during the BARDS experiments

(14)

Origin of Bubble Nucleation and Growth Centers. Reduction of the actual gas oversaturation via gas exchange at the gas/liquid surface of bubbles in the solution requires the presence of bubble nucleation and/or growth centers. Based on observations it is assumed that trace solid contaminations in the solution or irregularities at the inner vessel surface do not play a significant role in our methodology.20 Instead, minute gas bubbles introduced into the solution upon addition of the solute, or generated thereafter, are the dominant source of bubble nucleation or growth centers. The following origins may in this respect be distinguished. First, gas bubbles loosely adhered to the solute particles are instantly liberated into the solution after addition. Second, gas trapped within porous solute particles is also introduced. This gas is gradually liberated into the solution during the dissolution of these particles. The volume fraction of bubbles present in the solution determines the compressibility of the liquid and in this way also the pitch of its fundamental resonance frequency. Therefore, the size of the frequency drop (V-shaped response to f min) upon addition of the solute is a good indication of the trapped and adhered gases that are introduced and equally so the number of bubble nucleation/growth centers. This mechanism is also indirectly indicated by the correlation between the BARDS frequency response strength caused by the induced gas oversaturation (U-shaped response) and the V-shaped response. A comparison of the magnitude of the acoustic profiles of the three potassium phosphates, in Figure 3, suggests a threshold value of the V-shaped response is required before the onset of the U-shaped signal. Growth of Total Gas Volume of the Bubbles Due to Oversaturation. The growth rate of the bubble volume due to gas oversaturation is approximated by a first order kinetic process in eq 15 dVtb = Gos ·kbg ·V ·25 dt

(15)

where Vtb is the total bubble volume generated by gas oversaturation and desolvation, and V is the volume of the solution. Gos is expressed as molar concentration. The volume of a mole of gas is taken as 25 L. Based on a semiempirical approach, a new expression for the bubble growth rate constant kbg can be derived kbg =

n Cadd n + Cn Cadd thresh

·(BN ·Cadd) ·e(−kel·t )

(16)

where Cadd is the concentration of the amount of substance added, Cthresh is the added concentration related to the threshold which must be exceeded for bubble growth, and n is the exponential value, set to 5 in the model, defining steepness of threshold in the kbg step function term. BN is a solute specific factor related to its bubble nucleation power, t is the time after addition of the solute, and kel is the elimination rate constant of bubble nucleation/growth centers. The transient gas supersaturation induced by the dissolution of solutes such as electrolytes or carbohydrates is reduced via 2208

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Analytical Chemistry

Article

U-shaped response. This behavior strongly indicates that there are common underlying processes involved. However, the magnitude (normalized for concentration) of both the V- to Ushaped response characteristics and the transition concentration range are all compound specific. The time point of the frequency minimum and the rate and shape of the return of resonance frequency to preaddition values depends not only on the specific compound and its concentration but also on experimental conditions e.g., stirring/ mixing rate and solvent properties e.g. viscosity. Potassium Phosphates. It is evident in Figure 3A and B that small and intermediate respective amounts of K3PO4 and K2HPO4 results in a ‘V-shaped’ profile, characterized by an instantaneous decrease in the resonance frequency followed by a first order return to preaddition steady state frequency. It is assumed that this drop in frequency is directly caused by gases entrained with the solid into the solvent either trapped between or within particles. It should be noted that it requires about ten times the concentration of the dipotassium salt to elicit this response compared to the tripotassium phosphate salt. Note there is no V-shaped response observed for KH2PO4. The data in Figure 3C, for KH2PO4, suggest this is a compound which generates insufficient nucleation centers to initiate bubble growth in a gas oversaturated solution. This may explain the necessity for significantly larger concentrations to achieve a comparable acoustic response to those in Figure 3A and B. In case of monopotassium phosphate, the bubble-less gas exchange at the liquid/air interface at the solution surface becomes an important route for the reduction of gas-super saturation. The phosphates profiles show an additional ‘U-shaped’ response for all compounds upon concentration increase. This is attributable to the decrease in gas solubility with increasing dissolved mass, thereby resulting in bubble formation and growth. The already present gas bubbles may serve as additional nucleation sites in the gas supersaturated solution. Assuming that this number of nucleation sites is related to the size of the instant (V-shaped) response this may at least partly explain the difference in magnitude of the U-shaped response between the three phosphates. The required concentrations for a frequency drop at 50 s after addition to ∼4 kHz are in the proportion of 1:2.5:15 for tri-, di-, and monopotassium phosphate, respectively. Since KH2PO4 displays almost no instant response, very few bubble nucleation centers are probably present (Figure 3C). This may explain its comparatively low BARDS response. In the virtual absence of bubble nucleation and growth centers, part of the supersaturation will probably dissipate via bubble-less gas exchange at the liquid surface in accordance with Henry’s Law. For a complete interpretation of the observed differences in response, additional factors should also be considered, such as the different number of potassium ions that may affect the gas solubility and the solubility and dissolution rate differences. The potassium phosphates data in Figure 3 suggest the different acoustic behavior of each of the three compounds may be related to the chemical nature of the compound, e.g., a difference in cationic make up. However, investigations have equally shown that physical properties of the compounds, such as particle size, density, and crystal structure are controlling factors of the acoustic response. It is likely that the acoustic responses are a result of the inter-relationship between the chemical and physical properties of the compounds under investigation.

show that the gas bubbles present in the solution have small diameters (