Langmuir 1999, 15, 6775-6780
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Emulsions: A Time-Saving Evaluation of the Droplets’ Polydispersity and of the Dispersed Phase Self-Diffusion Coefficient† L. Ambrosone* and A. Ceglie Consorzio per lo sviluppo dei Sistemi a Grande Interfase (CSGI) c/o Department of Food Technology (DISTAAM), Universita` del Molise, via De Sanctis 86100 Campobasso, Italy
G. Colafemmina and G. Palazzo Dipartimento di Chimica, Universita` di Bari, via Orabona 4, 70126 Bari, Italy Received March 4, 1999. In Final Form: May 19, 1999 The NMR pulsed magnetic field-gradient spin-echo (PGSE) technique is used to study restricted diffusion in emulsion droplets. Experimental and theoretical difficulties are discussed. A limiting law for evaluating the self-diffusion coefficient directly from the data is derived. For the polydispersed system, the methods allow us also to obtain information about the droplet size distribution. The reliability of the polynomial representation of simulated and experimental data is investigated, with emphasis on the application to emulsion. Experimental results are given for the water-in-oil system formed by water-swollen Sephadex G-50 and cyclohexane at 25 °C. The agreement between the self-diffusion calculated and that measured in water-Sephadex is quite good.
Introduction Nonequilibrium, heterogeneous systems of two immiscible liquids, dispersed one into another in the form of droplets having radii of the order of magnitude of micrometers, are known as emulsions. The two liquids’ dispersion, usually an aqueous and an organic solution, is stabilized by surfactant molecules. Aqueous droplets dispersed in an organic continuous phase are called waterin-oil (w/o) emulsion, while the term oil-in-water (o/w) indicates the opposite case. Emulsions play a ubiquitous role in everyday life: a large portion of foodstuff1, paints, and pharmacological preparations are included in the above definition, as are several intermediate and final products in industrial processing.2,3 For this reason, at the dawn of colloidal science a notable effort was devoted to this topic.4,5 However, recognition of the thermodynamic instability of the emulsions and the lack of sophisticated investigation tools rapidly moved researchers’ main interest toward areas where the system behavior can be accounted for by a simpler equilibrium treatment. Notwithstanding the difficulties of the investigation, the applicative interest in emulsions is still considerably high. Thus, the renewed interest in emulsions, manifested by the scientific community in recent years, is not surprising. Starting from a consolidated background in the equilibrium properties of colloidal systems, several models have been proposed to rationalize the kinetic stability of
emulsions.6,7 At the same time there is an increased interest in the development of techniques able to extract the basic information concerning the emulsion structure, i.e., the size distribution function. Very often emulsions are opaque and present multiple (light) scattering. This is a considerable obstacle to the use of optical techniques such as light scattering and optical microscopy (although recently, efforts have been made to overcome this problem8). On the other hand, the optical appearance of the sample is totally insignificant for NMR measurements. Actually, several methods have been proposed to obtain the droplet size from NMR data, mainly obtained by means of pulsed magnetic field gradient (PGSE) experiment.9 Recently, we proposed some methods that are able to work without any “a priori” assumption about the distribution function.10,11 Obviously, the numerical results depend on the value of the self-diffusion coefficient, D, inside the droplets. The aim of this paper is to show how it is possible to extract the value of the “local” (inside the droplet) selfdiffusion coefficient of the dispersed phase, and the second and fourth moments, µ2 and µ4, of the size distribution function directly from the echo attenuation data. Furthermore, it is also possible to obtain an approximate value of the mean radius of the droplets as well as the error associated with this procedure. If the distribution’s form is known or if one is interested only in the moments of a given size distribution (as usually happens in emulsifica-
* To whom correspondence should be addressed. Phone: +39.0874.404647. Fax: +39.0874.404652. E-mail: ambroson@ unimol.it. † Dedicated to the memory of our dear friend and colleague, professor Americo Inglese (1946-1998).
(6) Vaessen, G. E. Ph.D. Thesis, Eindhoven University, and references therein. (7) Wennestrom, H.; Morris, J.; Olsson, U. Langmuir 1997, 13, 6972; and Dickinson, E. Curr. Op. Colloid Interface Sci. 1998, 6, 633. (8) For recent reviews see: Chestnut, M. H. Curr. Op. Colloid Interface Sci. 1997, 2, 158. Klein, R.; Nagele, G. Curr. Op. Colloid Interface Sci. 1996, 1, 4. (9) Callaghan, P. T. Aust. J. Phys. 1984, 37, 359. Balinov, B.; So¨derman, O.; Warnheim, T. J. Am. Oil Chem. Soc. 1994, 71, 513. Balinov, B.; Urdahl, O.; So¨derman, O.; Sjo¨blom, J. Colloid Surf. A 1994, 82, 173. (10) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. J. Chem. Phys. 1997, 107, 10756. (11) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. J. Chem. Phys. 1999, 110, 797.
(1) Larsson, K.; Friberg, S. E. Food Emulsion; Marcel Dekker Inc.: New York, 1994. (2) Orr, C. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1988; Vol. 3, p 369. (3) McKay, R. B. Technological Applications of Dispersions; Marcel Dekker: New York, 1990. (4) Ostwald, W. Kolloid Z. 1910, 8, 103. Ostwald, W. Kolloid-Z. 1910, 7, 64. (5) Bancroft, W. D. J. Phys. Chem. 1913, 17, 501.
10.1021/la990257e CCC: $15.00 © 1999 American Chemical Society Published on Web 08/03/1999
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tion or demulsification kinetics), the treatment here discussed supplies a time-saving method to monitor the evolution of the emulsion microstructure and to calculate the distribution parameters. Theory Restricted Diffusion in Spherical Cavities. For the sake of clarity, in the following we will discuss the PGSENMR experiments performed on emulsions in terms of w/o systems. In principle, the framework and the conclusions that will be drawn hold also for o/w systems by simply exchanging the words oil and water. Generally, the emulsion droplets are spherical because this is the shape that minimizes the surface for a given volume and, hence, the surface free energy. This is valid up to a volume fraction of 0.74, which is the value for the closest packing of spheres. The emulsions with volume fraction well below 0.74 are called dilute emulsions. We have already showed elsewhere10 that for such systems it is possible to get information about the droplet size polydispersity by analyzing the experimental data in terms of a continuum of isolated spheres. If the molecules’ rootmean-square displacement during the measurement timescale is large enough (in comparison with the droplet size), the diffusion is restricted by the presence of the droplet walls. The echo attenuation for diffusion within spherical boundaries does not have an exact solution. However, if one assumes a Gaussian distribution of the phase displacement,12 the echo attenuation for a single sphere of radius R will depend on the pulse gradient strength (g), the interpulses interval (∆), the pulse duration (δ) and R according to:13
(
E(q,R) ) exp -
2γ2g2 D
∞
∑ m)1
1 R4m(λ2m
)
fm(q,R) - 2)
(1)
with
fm(q,R) ) 2 + e-RmD(∆-δ) - 2e-RmDδ - 2e-RmD∆ + e-RmD(∆+δ) R2mD (2) 2
2δ -
2
2
2
where λm ) RmR is the mth root of the equation
(1/2)J3/2(λ) - λJ′3/2(λ) ) 0
(3)
Jn/2(ϑ) being the Bessel coefficient of a generic argument ϑ and of the order n/2, D the self-diffusion coefficient of the dispersed liquid inside the droplet (which can be different from that of the neat liquid), and γ the magnetogyric ratio of the observed nucleus. In eq 1, q ) γδg/2π and the transverse relaxation times are assumed to be independent of R. Equation 1 describes the restricted diffusion in an isolated sphere which can be can considered fixed in the space because the droplets’ size is, at least, 4 orders of magnitude larger than a water molecule; therefore, the droplet motion can be neglected (furthermore, in most emulsions of practical interest, the continuous phase is highly viscous). In polydisperse systems, the molecules within droplets sharing the same size diffuse (12) The conditions under which such an approximation holds are discussed in detail in: Neuman, C. H. J. Chem. Phys. 1974, 60, 4508. For a comparison between simulated data and the approximation, see Balinov, B.; Jo¨nsson, B.; Linse, P.; So¨derman, O. J. Magn. Reson., Ser. B 1993, 104, 17. (13) Murday, J. S.; Cotts, R. M. J. Chem. Phys. 1968, 48, 4938.
according to eq 1. Taking into account that the 1H NMR experiment provides information on proton spin density in the sphere (i.e., is a measure of the volume of the sphere), the observed echo attenuation can be written as:
I)
∫0∞E(q,R)Φ(R)dR
(4)
where E(q,R) is given by eq 1 and the volume fraction distribution function is defined by
Φ(R)dR )
R3P(R)dR
∫0∞R3P(R)dR
(5)
P(R)dR being the fraction of droplets sharing radius values lying between R and R + dR. Recently we reexamined the nature of the restricted diffusion in polydisperse systems and found that, when the volume fraction distribution function is within the range of diffusion length, L ) x2D∆, the experimental data contain enough information to properly describe the distribution function.10 Analysis of Self-Diffusion Coefficient and Moments of the Distribution. The diffusion length, L, is determined by the parameters of the PGSE experiment, and its value is strictly dependent on the D value. Since it is not possible to solve eq 1 using D as an adjustable parameter, D is usually taken as equal to the self-diffusion coefficient of the neat liquid (pure water for w/o emulsions). This choice is questionable. Indeed, the interactions with the surfactant polar head could, in principle, affect the diffusional properties. More important, real emulsions are composed of several chemical constituents whose partition between hydrophilic and lipophilic domains is not, generally, exactly known. Therefore, the dispersed phase is an aqueous solution of unknown composition rather than pure water. Furthermore, in several cases of applicative interest (e.g., food emulsions) the quantitative composition of the system is not easily accessible, being that the emulsion is formed by an enormous number of different chemical species (phospholipids, ions, amino acids, proteins, fatty acids, and so on). Computational Procedure for Small δ. A limiting law for evaluating the second moment of Φ(R), and the self-diffusion coefficient, D, from NMR measurements is derived by observing that when (Dδ)/R2 < 1, eq 2 reduces to form
fm(q,R) ) R2mDδ2 - (1/3)R4mD2δ3 + ...
(6)
and, furthermore, if δ f 0, eq 1 transforms into
E(q,R) ) 1 - 2γ2g2R2S2δ2 + (2/3)γ2g2DS0δ3 + ...
(7)
where Sn stands for ∞
Sn )
∑ m)1
1 λnm(λ2m
- 2)
(8)
The Sn calculation deserves some particular attention, so it will be discussed in detail in the appendix. Now, applying eqs 6 and 7, the observed echo attenuation (eq 1) results in
I(q) ) 1 - γ2g2
µ2 2 1 2 2 3 δ + γ g Dδ + ... 5 3
(9)
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It should be noticed that the term δ3 in eq 9 is sizedistribution independent and therefore allows the determination of the self-diffusion coefficient, while the second term directly gives the second moment. It is convenient to use a complete polynomial; therefore, we define the function f (δ) as
f(δ) ) (1 - I(q))/δ2 ) γ2g2
µ2 1 2 2 - γ g Dδ + ... 5 3
(10)
In this form, the function f (δ) can be treated as the virial equation for the real gas.14,15 Once µ2 is obtained, the mean radius of the volume fraction distribution can be estimated. Obviously, it is important to calculate the error in radius which arises if one assumes that the mean h . It is well-known radius is xµ2, whereas actually is R from the statistics that the standard deviation, σ, is related to the 2n moment through the relationship
h2 σ2 ) µ2 - R
(11)
Generally σ < R h ; consequently, we are allowed to write
xµ2 ) Rh [1 + ( /2)p 1
2
- (1/8)p4 + ...]
(12)
where, in accordance with the Hunter definition,16 p ) σ/R h is the polydispersity degree. Defining the relative error, , we have
) (xµ2 - R h )/R h ) (1/2)p2 - (1/8)p4 + ...
(13)
From eq 13 one sees that for a polydispersity degree of 40%, the relative error does not exceed 8%, indicating h in most that xµ2 can be used as a good estimate of R applications. Computational Procedure for Large δ. In this case, large δ means that (Dδ)/R2 > 1. Simple considerations can greatly simplify eq 1 and eq 2. Again we will refer to w/o emulsion and into consideration PGSE experiments performed on water protons in the droplets. Taking as typical droplets’ representative parameters: R ) 2-10 µm, ∆ ) 0.1 s, D ) 2.3 × 10-9 m2 s-1 for water at 25 °C, the term R2mD(∆ + iδ) (with i ) -1, 0, 1) varies 50 λ2m to 2 λ2m, where λm is defined by eq 3. Since the smallest λm value is greater than 2, the exponential terms in eq 2 are practically negligible, and thus eq 2 is well approximated by
(
fm(q,R) ) 2δ 1 -
R2 λ2mDδ
)
+ ...
(14)
Because the mean radius of emulsion is of the order of micrometers (see introduction), under the usual experimental conditions the relationship γ2g2R4/D < 1 is valid. Then, the echo attenuation can be expanded in Taylor series
E(q,R) ) 1 + 4 (γ2g2/D2)R6S6 + 4 (γ2g2/D)R4S4δ + ... (15) and applying eq 15, the observed echo attenuation (eq 4) is (14) Michels, A.; Abels, J. C.; Ten Seldam, C. A.; De Graaff, W. Physica 1960, 26, 381. (15) Michels, A.; De Graaff, W.; Ten Seldam, J. C. Physica 1960, 26, 393. (16) Hunter, R. J. Foundation of Colloid Science; Clarendon: Oxford, 1987.
I(q,R) ) 1 +
166 γ2g2 16 γ2g2 4 µ δ + ... (16a) µ6 2 7875 D 175 D
Once more, it is convenient to define the function
g(δ) ) 1 - I(q,R)
(16b)
so that a polynomial least-squares procedure can be used to directly achieve µ4 and µ6. Experimental Section Materials. Sephadex G-50 superfine from Sigma Chemical Co. was used without any further purification. Cyclohexane from Carlo Erba was of special reagent grade. Double-distilled water was always used in preparing emulsions. The samples were prepared as follows. The dry Sephadex was swollen overnight with water at room temperature. The gel was then transferred in vials, and an excess of cyclohexane was added. After several hours of gentle stirring, the sample was transferred into the NMR tube, where the gel sedimented on the bottom (well within the measuring coils). NMR Self-Diffusion Measurements. The diffusion behavior of the components in the emulsion samples was measured with the FT PGSE technique monitoring the water peak decay.17 Measurements were performed in 5-mm NMR tubes on a spectrometer TESLA BS-587A operating at 80 MHz for proton and equipped with a pulsed-gradient unit Autodif 504, Stelar SNC. The fieldgradient strength was calibrated using the dimethyl sulfoxides because its self-diffusion coefficient is known.18 The temperature was fixed at 25.0 ( 0.1 °C. Simulated Measurements. The procedure for the simulation of an NMR measurement and the details of the superimposed pseudorandom noise have been described elsewhere.10 Here the polydispersity is generated using a log-normal distribution function:
Φ(R) ) (1/x2πσR) exp[-(ln R - ln R0)2/2σ2] (17) where R is the droplet radius, R0 its geometric mean, and σ the geometric standard deviation of the R-ratio distribution around R0. This distribution form was selected as it is a good model for polydispersity in food emulsions. Results and Discussion Simulated Measurements. To recognize the potentialities and the limitations of the above-reported theoretical considerations, these considerations have first been tested on simulated measurements. Because the functions f(δ) and g(δ) were meaningful for small and large values of δ and R2/D, respectively, it is important to know which is the value δb that divides the experimental interval into the two subintervals implicit in the power series representation. At the beginning we have no information about the system under consideration, so δb can only be estimated. To estimate such a value, we treat the emulsion as a monodisperse system and define δb through the relation Dδb ) R2. It follows that, according to Callaghan,19 R2 is calculated from the limit form of eq 1, reporting ln(I) vs δ2, and D is assumed to be the self-diffusion coefficient of the pure water (or pure oil for o/w emulsions). Of course, R2/D gives only an estimate of δb, so it is safer to not use the closer points, being that δb is really a discriminating (17) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (18) Bruker Almanac, 1990. (19) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon: Oxford, 1991, Chapters 6 and 7.
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Ambrosone et al. Table 1. Comparison of Theoretical Self-Diffusion Coefficient and Moments, with the Calculated Values for a Simulated Measurement D × 109/ m2 s-1 µ2 × 1011/m2 µ4 × 1021/m4 µ6 × 1032/m6 a
Figure 1. (a) Simulated echo attenuation for a model emulsion. The dotted line is the boundary separating the small and the large δs regions (see text for details). (b) Polynomial interpolation to calculate µ2, D, µ4, and µ6, according to eqs 10 and 16b.
interval and not a single value. This is particularly true for the points in the right interval, because most emulsions show, in their size distribution, a tail toward larger radii; moreover, in the presence of solutes, the true self-diffusion coefficient is expected to be lower in emulsion droplets than in neat liquid. Figure 1a shows the δb value calculated for a simulated measurement. For this case we selected R0 ) 5 × 10-6 m and σ ) 0.3 in eq 17 as typical polydispersity parameters. The time scale of the experiment is ∆ ) 0.140s and the strength of the applied magnetic field gradient is g ) 0.1 Tm-1. It is evident from Figure 1b that δ ) δb divides the experimental interval into two subintervals which can be thought of as independent measurements. Among the experimental (simulated) points on the left interval (δ < δb), those with δ < 0.008 s were used to compute the coefficients of a thirddegree polynomial, f(δ). To the contrary, among the points on the right interval (δ > δb), those with δ > 0.012 s were used to evaluate the coefficients of another third-degree polynomial, g(δ). The choice of the polynomial degree deserves some comment. When the experimental results are presented by a polynomial of the form Pm(δ) ) m ∑k)0 bkδk (model function) and the coefficients bk are calculated by a least-squares method, the theoretical interpretation of them is realistic only when the deviations show a Gaussian distribution, which may require an additional power of δ. Because the experimental points scatter around the model function with accuracy of the order of magnitude of the experimental accuracy, with the procedure of least squares we calculate a polynomial of sufficiently high degree to obtain such a scattering of the residuals. Only this polynomial is accepted as model function. In Table 1, the results obtained by means of this procedure are compared with the theoretical ones. The good agreement indicates that the method can be used to calculate the value of D in emulsion systems. The value of µ6 is determined with poor precision, as expected, but
theoretical
calculated
2.30 2.99 1.28 7.89
2.2( 0.2a 2.97( 0.07 1.1( 0.5 3.1( 0.3
Uncertainties represent standard deviations of the fit.
fortunately, this value is not essential for determining the distribution. Moreover, an estimate of the mean size of the droplets may be obtained using eq 12 truncated to the first term. For the case shown in Figure 1, we found xµ2 ) 5.5 × 10-6 m, which draws away from the true value for less than 6%. It is important to underline that if the distribution’s form may be supposed known, the method also gives a powerful tool for obtaining its parameters. To fully characterize any unimodal, distribution functions are necessary for only two parameters. Once the relationships between these parameters and the moments of the distribution are known, the distribution function can be figured out. Indeed, one can easily verify that, for a lognormal distribution, we have µn ) Rn0 exp(n2σ2/2), where R0 and σ are defined in eq 17. With the aid of both this relationship and the data in Table 1, we found R0 ) (5.1 ( 0.4) × 10-6 m and σ ) 0.23 ( 0.09. Although the above-described physical situation is generally met under the usual experimental conditions, it is important to point out the disadvantages that might arise in some particular cases. When R is sufficiently high, the value of δb is so large (for fixed experimental conditions) that in the right interval (δ > δb), there are very few points (or there are none). This circumstance would usually be met in o/w emulsions made of highly viscous oil, D being much smaller than that of pure water. In Figure 2 an example of such a case is reported. A log-normal distribution was simulated using R0 ) 7.5 × 10-6 m and σ ) 0.30 (solid line in Figure 2a), and the corresponding echo decay for the same values of D, ∆, and g of the previous case was simulated (dots in Figure 2a). It is easy to verify that δb ) 0.023 s; thus, all the experimental points are in the left interval. It follows that all the data points may be used for computing D and µ2. The data have been interpolated with a sixth-degree polynomial (solid line in Figure 2a), thus obtaining D ) (2.4 ( 0.9) × 10-9 m2 s-1 and µ2 ) (6.25 ( 0.2) × 10-11 m-2. Two ways can be followed to get information about the distribution function. The first consists of changing the experimental conditions (increasing ∆, for example) in order to have more points in the interval δ > δb, and calculating µ4 and µ6, as described above. The second way, when the increase of ∆ is not feasible, uses the calculated self-diffusion coefficient to generate the distribution function through the more exhaustive direct method described in a previous paper.11 The distribution function calculated with the last procedure is shown in Figure 2b. Experimental Measurements. To test our approach, we need a system with a known size distribution or at least with an estimate of the droplet size. Furthermore, in such a system the diffusion coefficient of the dispersed phase should be known. A third constraint is related to the droplet size: When the mean radius is small, most of the experimental points lie in the region δ > δb. If R is sufficiently small, it is not possible to overcome this problem by varying the experimental conditions, and thus the evaluation of D becomes complicated. For D of the
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Figure 3. Fitting of the experimental data for the emulsion system Sephadex G-50-cyclohexane-water at 25 °C. The insert shows the distribution calculated with the direct method.
Figure 2. (a) Polynomial fitting to calculate D and µ2 using the points in the left interval for a simulated measurement. (b) Comparison between the calculated distribution using the direct method (O) and the theoretical distribution (b).
10-9
m2
s-1,
order of magnitude of we cannot apply the above procedure for droplets’ radii smaller than 3 × 10-6 m, and it is necessary to use an iterative procedure which will be the subject of a forthcoming paper.20 Unfortunately, most of the emulsion systems formed by a neat liquid dispersed as droplets of well-known dimensions also have small radii and high diffusion coefficients, and thus they cannot be used to test the proposed procedure. In the attempt to obtain large water droplets, waterswollen Sephadex in cyclohexane dispersion was prepared. Sephadex is a bead-formed gel prepared by cross-linking dextran with epichlorohydrin. It swells in aqueous solutions, and the swollen beads are widely used as media for gel filtration.21 The beads of Sephadex G-50 superfine have a large dimension (R ) 5 - 20 × 10-6 m for dry beads), and when dispersed in an inorganic medium they should provide a collection of closed water domains with a mean radius of about 8 × 10-6 m. Under these conditions, δb ) 0.03 s, which means that it is easy to have several experimental points in the left interval. Experiments were carried out at ∆ ) 0.1 s and g ) 0.18 T/m. Notwithstanding the high concentration of dispersed phase, the echo-decay is clearly not exponential (Figure 3), indicating that there (20) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. submitted 1999. (21) Janson, J. C. Chromatographia 1987, 23, 361.
is no water exchange between beads. We find that a polynomial of the seventh order can represent f(δ) within the experimental accuracy. This approach leads to D ) (2.4 ( 0.4) × 10-9 m2 s-1 (to compare with D ) 2.1 × 10-9 m2 s-1 of water-swollen Sephadex without cyclohexane) and µ2 ) (6.87 ( 0.08) × 10-11 m2. From µ2 a value of R ) (8.3 ( 0.9) × 10-6 m is obtained, which sounds reasonable considering the size of the dry beads.22 It should be stressed that in these experiments the moments of the distribution come from an average on a volume fraction basis. In other words, we are dealing with the moments of the volume fraction distribution described by eq 5. If one needs the averages calculated on the number of particles, the approach is the following. The evaluated diffusion coefficient (D) is used to fully evaluate the volume distribution function (Φ(R)) using the direct method previously described.11 The resulting distribution is displayed in the insert of Figure 3 (the direct computation of the second moment gives µ2 ) 6.87 10-11 m2). The subsequent evaluation of the droplet size distribution, P(R), from eq 5 gives a number average diameter of 15 × 10-6 m. Conclusions In a PGSE-NMR experiment the echo attenuation contains information on the droplet size distribution, and the distribution function can be calculated, provided that the local self-diffusion coefficient is known. Because of the mathematical complexity of eq 1, such coefficient has never been calculated directly from experimental data. Generally, information about the local self-diffusion coefficient of the dispersed phase is obtained either assuming an ideal behavior of the liquid inside the droplets or directly by measuring the diffusion coefficient after macroscopic demixing. If we are working with model emulsions (water/oil/surfactant) we have prepared for a pedagogical purpose, both the assumptions are sound. But, using “real life” emulsions, the drawbacks of the assumptions become evident. It is often difficult to obtain the chemical composition of the dispersed phase of a complex multicomponent emulsion (e.g., butter, margarine, mayonnaise, and so on), and sometimes they are patent protected. On the other hand, to induce a demixing it is not a trivial task (in particular for commercial products where a considerable effort was made to form a stable emulsion), and it can require a considerable amount of time. In most cases, such a time-consuming procedure could preclude or vitiate a correct study of the emulsion (22) Gel Filtration Principle and Methods, Pharmacia LKB Biotechnology, 5th ed., Lund, 1991.
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aging processes during industrial treatment. Therefore, it is important to have a time-saving tool to compute D directly from the PGSE-NMR data. A careful investigation of eq 1 reveals that a dimensionless parameter (Dδ/R2) exists which permits us to divide the experimental interval into two subintervals. On each of such sub-intervals, a polynomial least-squares procedure can be used to draw valuable information: the translational self-diffusion coefficient and the second moment of distribution in the left interval and the fourth moment in the right interval. If the distribution’s form is known, this approach gives a rapid method for completely determining the distribution function directly by the PGSE-NMR data. In any case, when it is used together with the direct method,11 it becomes a powerful tool to investigate the emulsion systems. Acknowledgment. We gratefully acknowledge constructive criticism by Dr. Mauro Giustini. This work was supported by the MURST of Italy (Prog. Cofinanziati 1998).
d v+1 (ϑ Jv+1(λiϑ)) ) ϑv+1Jv(λiϑ) dϑ
by ϑ2n, integrating by parts, and applying the properties of Bessel functions, we have
λ2i
∫01 ϑv+2n+1Jv(λiϑ)dϑ ) (v + 2n + p/q)Jv(λi) 1 4n(v + n)∫0 ϑv+2n-1Jv(λiϑ)dϑ
b(n) i )
b(0) i )
Using the Bessel functions theory it is easy to obtain vI2n-1
{
0 for i ) j p2 2 1 2 2 λ - v + 2 Jv(λi) for i * j (A.2) 2 i 2λi q
(
)
which indicates that the Bessel functions Jv(λiϑ) and Jv(λjϑ) are orthogonal with respect to the weight function ϑ. As a consequence, it is possible to use the Fourier method to expand a generic function in a series of Bessel functions on the interval [0, 1]. We choose to expand the function f(ϑ) ) ϑv+2n (n being a positive integer); that is
vRn vI2n-1
b(n) ∑ i Jv(λiϑ) i)1
)
2λ2i (λ2i
2
2
2
- v + p /q
∫1ϑv+2n+1Jv(λiϑ)dϑ
)J2v(λi) 0
(A.8)
- v2 + p2/q2)Jv(λi)
∫01 ϑv+2n-1Jv(λiϑ)dϑ, vRn )
(
)
(
)
4n(v + n) v + 2n + p/q Jv(λi), vβn ) (A.9) 2 λi λ2i
To extract the value of the sums defined in eq 8, we note that when n + p > 0, the series A.3 is uniformly convergent on the entire interval [0,1]; therefore, putting ϑ ) 1 in eq A.3, for different n values the sums Sn are obtained. Applying this procedure to the case under investigation, from eq 3 we have ν ) 3/2, p ) -1/2, q ) 1, and, consequently,
b(0) i )
(λ2i
2 - 2)Jv(λi)
(A.10)
(A.3) thus, putting ϑ ) 1 in eq A.3 (for n ) 0), we have
Multiplying both the sides of this equation by ϑJv(λjϑ) and integrating, by virtue of eq A.2, we get
b(n) i )
(A.7)
2(v + p/q) (λ2i
∞
ϑv+2n )
)
- vβn b(n-1) i
where we have used the following abbreviations
23
∫01ϑJv(λiϑ)Jv(λjϑ)dϑ )
(
with
Let λ1, λ2, ..., λi, ... be the distinct roots of the equation
(A.1)
(A.6)
can be calculated Consequently, the coefficients b(n) i recursively by
Appendix
pJv(ξ) + qξJv(ξ) ) 0
(A.5)
(A.4)
This equation allows us to calculate the Fourier coefficients in eq A.3; however, the computation is easier if one has at one’s disposal a recursion formula for the integral. To obtain such a formula, we proceed as follows. Multiplying both the sides of the identity (23) Watson, G. N. A Treatise on the Theory of Bessel Function; Cambridge University Press: Cambridge, 1966, Chapter XVIII.
∞
S0 )
(λ2i - 2)-1 ) 1/2 ∑ i)1
(A.11)
2 (0) for n ) 1, b(1) i ) (3 - 10/λi )bi and, therefore,
∞
S1 )
-1 2 λ-2 ) 1/10 ∑ i (λi - 2) i)1
(A.12)
When we take λi ) RiR, varying n, eq A.4 can generate all the sums used in the text. LA990257E
