Large-Scale Supramolecular Structure in Solutions of Low Molar Mass

J. Phys. Chem. B 2006, 110, 4329-4338

4329

Large-Scale Supramolecular Structure in Solutions of Low Molar Mass Compounds and Mixtures of Liquids: I. Light Scattering Characterization Maria´ n Sedla´ k* Institute of Experimental Physics, SloVak Academy of Sciences, WatsonoVa 47, 043 53 Kosˇice, SloVakia ReceiVed: NoVember 29, 2005

Static and dynamic laser light scattering were used to bring evidence of large-scale supramolecular structure in solutions of low molar mass electrolytes, nonelectrolytes, and mixtures of liquids. It was shown that solutes are distributed inhomogeneously on large length scales. Regions of higher and lower solute concentration exist in solution and give sufficient scattering contrast for experimental observation. A detailed light scattering study showed that these regions can be characterized as close-to-spherical discrete domains of higher solute density in a less dense rest of solution. These domains do contain solvent inside and can be therefore characterized as loose associates (giant clusters, aggregates). Their size distributions are significantly broad, ranging up to several hundreds of nanometers. Characteristic sizes of these inhomogenities thus exceed angstrom dimensions of individual molecules by several orders of magnitude. The number of solute molecules per domain varies approximately in the range 103 - 108. Phenomena described were observed in a very broad range of solutes and solvents. Among others, selected data on most common substances of great chemical and biological importance such as sodium chloride, citric acid, glucose, urea, acetic acid, and ethanol are presented.

Introduction The structure of solutions of low molar mass compounds including mixtures of liquids is given by various types of solute-solute, solute-solvent, and solvent-solvent interactions. In most common cases, i.e., solutions of common organic molecules and inorganic salts (electrolytes), the known microscopic structure due to such interactions is formed on the angstrom/nano scale and extends in maximum up to several intermolecular distances between solute molecules. These solutions are assumed to be structureless on larger length scales up to macroscopic, which means that the structure (distribution of solute) on such length scales is considered homogeneous. Here we report on the basis of extensive static and dynamic laser light scattering data existence of supramolecular organization (inhomogeneous distribution of solute) in solutions of electrolytes, nonelectrolytes, and mixtures of liquids. Characteristic range of this large-scale supramolecular structuring exceeds dimensions of individual molecules or intermolecular distances by several orders of magnitude. The first signatures that the common assumption about homogeneous structure on larger length scales does not have to be necessarily true, were two-fold. In 2000, Georgalis, Kierzek, and Saenger published a short communication1 regarding the observation of submicron size clusters in undersaturated solutions of some aqueous electrolyte solutions. On the other hand, numerous works were devoted to the effect of association in polyelectrolyte solutions during the last three decades, reported in tens of various ionic polymers as well as in fully hydrophilic inorganic large anions.2-29 In the course of these studies, which were performed also in our laboratory,22-27 we repeatedly noticed that some kind of association was present also in solutions of ionic monomers (precursors of ionic * E-mail: [email protected].

polymers), i.e., low molar mass electrolytes. This was the motivation for the current detailed work. Naturally, the character of attractive forces between solute particles leading to their association can be different in solutions of low- and high molar mass compounds, respectively. Our approach was therefore to make an extensive and detailed study without any a priori assumptions. It appeared anyway that the association of low molar mass compounds is not restricted to electrolytes but occurs also in solutions of certain classes of nonelectrolytes and mixtures of liquids. The complexity of the reported effect of supramolecular structuring in solutions of low molar mass compounds required a very detailed and timeconsuming investigation, which is now presented in the current series of two papers. The organization of the work is as follows. The first paper deals with a detailed characterization of the solution structure due to inhomogeneous distribution of solute. The second paper30 deals with the kinetics of the structure formation as well as with its long-time stability. Under preparation is another work giving a detailed classification of systems with respect to their ability to form this supramolecular organization, with the aim to shed light on the molecular mechanism of the formation of supramolecular structures by correlation of supramolecular structuring with molecular properties and interactions between involved constituents. Static and dynamic laser light scattering, the main techniques used in this work, probe the structure and dynamics of the scattering medium on scales given predominantly by laser wavelength and current technical limitations in photon correlation. The length scales probed in static light scattering experiments by angular distribution of scattering intensity range typically from ca. 20 nm to several microns, while time scales probed in dynamic light scattering experiments by photon correlation range from ca. 100 ns to seconds (what enables in turn to yield also structural information from ca. 1 nm to several microns). Both techniques are especially powerful in the studies

10.1021/jp0569335 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/03/2006

4330 J. Phys. Chem. B, Vol. 110, No. 9, 2006

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of aggregation and association phenomena, including macromolecules as well as small molecules.2,31 The suitability of these techniques for the study of association phenomena comes from the fact that intensity of scattered light as well as its other static and dynamic characteristics change very strongly upon association.

correlation measurements. Characteristic decay times of dynamic modes τi and their relative amplitudes Ai(τi) were evaluated through the moments of distribution functions of decay times A(τ) obtained by fitting correlation curves using CONTIN32 and GENDIST33,34 programs as

g(1) (t) ) Experimental Section Experiments were performed on analytical grade chemicals (mostly from Merck, Darmstadt). The following substances were used in this work: D-glucose, urea, acetic acid, citric acid, NaCl, CdSO4, sodium para-toluene-sulfonate, ethanol, and benzene. Water was freshly double-distilled in a quartz apparatus and subsequently deionized by analytical grade mixed-bed ionexchange resins (Bio-Rad, Richmond, CA). The resistivity of water was above 15 MΩ cm. The static light scattering (SLS) and dynamic light scattering (DLS) measurements were made using a Stabilite 2017-04S argon laser (Spectra Physics, Mountain View, CA) with 514.5 nm vertically polarized beam. Laser power was limited to 300 mW. No change of data with laser power was observed in this range. A laboratory-made goniometer with angular range from 30° to 150° was used to collect data for both static and dynamic light scattering experiments. The scattering cell was thermostated at 25 °C with a precision of ( 0.1 °C. Scattering intensities were measured by photon counting. Solvent scattering I0 was subtracted from total solution scattering It to obtain excess scattering intensity I

I ) It - I0

(1)

The subtraction of solvent scattering was done separately for each angle. In the case of mixtures of liquids, the excess scattering was taken as a compositional fluctuation contribution to total scattering. Anisotropic scattering and scattering due to density fluctuations were subtracted from the total scattering according to the relation

I ) It - Ian - Id

(2)

where Ian is anisotropic scattering and Id is scattering due to density fluctuations. Excess scattering intensities were normalized using a doubly distilled and filtered benzene as a standard and expressed as ratios I/IB, where IB is benzene total scattering. Great attention was paid to the purity of samples necessary for work with weakly scattering systems.22-27 Scattering cells were thoroughly cleaned from dust. All solvents were filtered through 0.2 µm filters for the same reason. After dissolving solute, solutions were additionally centrifuged 10 min at 5000 g. This very gentle centrifugation had no effect on samples but was able to eliminate some remaining dust residuals. Since supramolecular structures developed in solutions have large dimensions (large compared to 0.2 µm filter pore size used for solvent filtration), solutions were not filtered again since the aim was not to disturb the supramolecular structures but rather to study them in their native state (as well as kinetics of their growth). In the case of mixtures of liquids, it was possible to filter separately each liquid directly into the scattering cell and hence to mix perfectly clean liquids. Experiments on blank samples (nonstructured mixtures or pure liquids) showed a complete dust removal (correct absolute values of scattering intensity without upturns in scattering intensity tracks, flat angular dependencies, as well as flat correlation curve baselines). An ALV5000 correlator with a fast correlation board option and an ALV800 transputer board (ALV, Langen, Germany) were used for photon

∫0∞ A(τ)e-t/τ dτ

(3)

Diffusion coefficients were calculated as Di ) (1/τi)q-2, where q is the scattering vector defined as q ) (4πn/λ0)sin(θ/2), with n the solution refractive index, λ0 the laser wavelength, and θ the scattering angle. Two diffusive modes were detected. They were characterized by diffusion coefficients Df, Ds, and amplitudes Af, As (subscripts f and s refer to faster and slower, respectively). Correlation curves at various angles were recorded concurrently with integral scattering intensities I(θ) (solution excess scattering) and IB(θ) (scattering of a benzene standard). Normalized excess scattering amplitudes of the two dynamic modes Af(θ) and As(θ) were calculated as

As (θ) ) Af(θ) )

I(θ)/IB(θ) 1 + Af (θ)/As(θ) I(θ)/IB(θ) 1 + As(θ)/Af(θ)

(4)

(5)

assuming that I(θ)/IB(θ) ) Af(θ) + As(θ). Dimensionless ratios As(θ)/Af(θ) and Af(θ)/As(θ) were taken from DLS spectra of relaxation times. Blank DLS experiments were performed with pure solvents to ensure that no autocorrelation is found in the time window where both modes occur. Angular dependencies of scattering intensity were usually measured several times on one sample and subsequently averaged. An optimized regularization technique ORT35,36 was used for the calculation of domain size distributions from static light scattering data. The inversion of Fredholm integral equations of the first kind in optical sizing, which is an illposed problem from the mathematical point of view, is overcome in this case by the regularization technique. The size distribution function is searched as a linear combination of cubic B splines φi(R) in a window delimited by Rmin and Rmax n

D(R) )

ciφi(R) ∑ i)1

(6)

where the natural limits 2Rmin ) π/qmax and 2Rmax ) π/qmin may be exceeded to a certain extent35,36 (qmax and qmin are the maximum and minimum experimentally available scattering vectors). Scattering vector is defined as q ) |q b| ) (4πn/λ0)sin (θ/2), with n the refractive index of the medium, λ0 the light wavelength in vacuum, and θ the scattering angle. Scattering intensity is expressed via coefficients ci, which are the unknowns that can be determined by a constrained least-squares condition. In our particular case the amplitude of the slower mode As calculated by eq 4 (scattering contribution from domains) was used as the input scattering intensity for the ORT calculation of domain size distributions. The relative refractive index of particles with respect to solvent, which comes into calculation, is in our case an unknown parameter since the scattering contrast of domains is not exactly known. However, very similar results are obtained irrespective of concrete values of refractive index used in the calculation. Samples were centrifuged in a microchip-controlled centrifuge Jouan KR22i (Jouan, France). This centrifuge enables very

Large-Scale Supramolecular Structure Characterization I

Figure 1. Angular dependencies of light scattering from solutions of electrolytes, nonelectrolytes, and mixtures of liquids. Selected in this plot are aqueous solutions and mixtures of some very common substances: (1) D-glucose, c ) 4.1 mass %, (O) urea, c ) 4.5 mass %, (b) acetic acid, c ) 6.0 mass %, (0) NaCl, c ) 15.8 mass %, (*) citric acid, c ) 18.5 mass %, and (4) ethanol, c ) 20 mass %. Scattering intensities I are excess intensities and are normalized to benzene scattering IB.

precisely to set not only the relative centrifugal force (RCF) and centrifugation time (CT), but also the acceleration and braking by definition of the integral of the dependence of RCF on time. This allows to exactly and reproducibly apply gravitational forces. Temperature was controlled by cooling and was set to T ) 25 °C. Samples were centrifuged 3 h at 22 000 g for this purpose. A swing-out rotor was used in which the scattering cell is positioned perpendicular to the axis of rotation during centrifugation so that the gravitational force acts along the cell axis toward its bottom. Measurements of osmotic coefficient φ defined as φ ) π/πid, where π is osmotic pressure and πid is ideal osmotic pressure, were done by K-7000 vapor pressure osmometer (Knauer, Berlin). Measurements were done at 30 °C. This temperature was chosen in order to reach an optimum quality of the signal. It was checked by light scattering that the small difference between temperatures where light scattering was performed (T ) 25 °C) and where osmometry was performed (T ) 30 °C) did not play any significant role with respect to the structure of solutions and mixtures reported. Results and Discussion Figure 1 gives an introductory brief summary of main results. It shows angular dependencies of scattering intensities from several solutions and mixtures of low molar mass compounds. Two main nontrivial features can be seen immediately: (i) scattering intensities are strongly angularly dependent, although no dependence is expected because dimensions of constituent molecules are much smaller compared to the length scales probed (given by q-1, where q is the scattering vector), and (ii) scattering intensities are much higher than expected. Systems in which these features are found can be divided into three classes: solutions of electrolytes, solutions of nonelectrolytes, and mixtures of liquids. Examples from each class have been selected in Figure 1. These are aqueous solutions and mixtures

J. Phys. Chem. B, Vol. 110, No. 9, 2006 4331

Figure 2. Typical example of bimodal dynamic light scattering from solutions of low molar mass compounds and mixtures of liquids. Data were obtained on a 4.1 mass % aqueous solution of D-glucose. Autocorrelation function of the electric field of scattered light g(1)(t) is bimodal and consequently the spectrum of relaxation times A(t) obtained by Laplace inversion is composed of two modes, which are diffusive in nature. The two modes are referred to as the faster and slower, respectively. The faster mode corresponds to “ordinary diffusion of solute”. The slower mode appears due to the presence of inhomogeneous solute distribution (solute association) at large length scales. Scattering angle θ ) 90°.

of some very common substances: sodium chloride, citric acid, glucose, urea, acetic acid, and ethanol. A detailed classification - an exact range of substances in which described phenomena occur (and in which not) - is currently under investigation and will be published in a separate paper. Figures 2-5 show more detailed light scattering data. Results on a 4.1 mass % aqueous solution of D-glucose are shown, however, features described apply to the whole range of solutes and solvents and have therefore universal character. We would like to specify at this point that only selected representative data sets are used in the presentation of particular aspects of the problem (solution of each partial issue) in this paper. It means in practice that usually several experiments of the same type were performed on different systems, but only the result of one is presented here for the purpose of clarity of the presentation and in order to keep the size of this presentation in acceptable limits. Therefore each ”experiment” means here rather a set of experiments of certain kind on various systems (solute/solvent pairs). Autocorrelation function of the electric field of scattered light g(1)(t) in Figure 2 is bimodal, and consequently the spectrum of relaxation times A(t) obtained by Laplace inversion contains two distinct peaks. One peak is located in the microsecond range and corresponds to “ordinary diffusion” of the solute. As expected, the characteristic decay frequency Γ scales as Γ ∼ q2. Here we will not discuss more details regarding this wellunderstood mode. Instead we will focus on the second mode in the millisecond range, which is unexpected. The total scattering intensity I can be decomposed into contributions (amplitudes) from the two modes, i.e., I ) Af + As. Subscripts f and s refer to faster and slower, respectively. These amplitudes correspond to the area under the peaks in the spectrum. Figure 3 shows angular dependencies of scattering. The angular independence of Af is expected since concentration fluctuations due to ordinary diffusion of small molecules occur on length scales much smaller than q-1. On the other hand, As is strongly angularly dependent. The angular dependence of the total scattering is thus fully due to the angular dependence of the scattering


Figure 3. Typical example of static light scattering data obtained on solutions of low molar mass compounds and mixtures of liquids. The data were obtained on the same sample as shown in Figure 2 and contain absolute values and angular dependencies of the total scattering (O) and scattering contributions from the faster (0) and slower (b) modes, respectively. Intensities shown are excess intensities (solvent contribution subtracted) and are normalized to benzene scattering IB.

contribution from the slower mode. The observed angular dependence of As indicates the presence of large inhomogenities in the refractive index in solution (comparable to q-1), where “large” is here understood in comparison with the size of individual solvent or solute molecules. The slower mode reflects the dynamics of these inhomogenities in solution. The main aim of this paper is to give a detailed analysis of this inhomogeneous structure and related dynamics. Since in our case nothing is a priori known about the source of this slow-dynamics scattering, we try not to stick with a certain a priori model and rather focus to a maximum extent on obtaining as much model-independent information as possible. Observed inhomogenities in the refractive index can be, in principle, due to inhomogenities in the local concentration of solute and/or due to inhomogenities in the local arrangement of asymmetric solute molecules. Since the effect is observed equally well for both spherically symmetric and asymmetric solutes, the former mechanism must be dominant. It means that regions of relatively higher and lower solute concentration exist in solution. The next question concerns the architecture of these regions, which should be reflected in shapes of angular dependencies As(θ). To discuss As(θ) shapes in detail, we first analyze them by the Debye-Bueche method, in which linearity is expected in the plot I -1/2 vs q2 (or As-1/2 vs q2 in our specific case). The Debye-Bueche model37 describes scattering from a random two-phase system where the spatial correlation function of the refractive index, which is a Fourier transform of scattering intensity, is in the form 〈δn(0)δn(r)〉 = -r/a, where br is the position vector and r ) |rb|. Refractive index n is strongly correlated for r < a and weakly correlated for r > a. Randomly chosen points in the scattering medium separated by distance less than a can be then characterized as belonging to one phase. However, plots As-1/2 vs sin2(θ/2) are not linear as shown in Figure 4. A clear departure from linearity is seen. We have analyzed tens of angular dependencies As(θ) obtained on various systems by the Debye-Bueche method. In very few cases DebyeBueche plots are linear, but this is only accidentally. Linearity in plots As-1/2 vs sin2(θ/2) is not a universal feature describing data. Similar results were obtained when trying to analyze data with the Ornstein-Zernike function38 I(q) ) I(0)/(1+ ξ2q2), where ξ is the correlation length. Strong departures from such dependence were found. Therefore, we proceeded in the analysis

Sedla´k

Figure 4. Debye-Bueche plot of the scattering amplitude of the slower diffusive mode (the same sample as in Figure 2). A clear departure from linearity demonstrates that the solution structure cannot be described as a random two-phase system. Similar departures are found in other solutions and mixtures.

TABLE 1: Results Obtained by Depolarized Light Scattering on Aqueous Salt Solution (0.133 M 3CdSO4 × 8H2O)a experiment

I/IB

As

Af

Ds, 10-9cm2/s

Df, 10-6 cm2/s

VU VV VH

2.62 2.57 0.051

2.38 2.34 0.051

0.24 0.23 ∼0

16.5 16.4 18.8

4.32 4.38

a

VU means a classical experimental arrangement where the initial light is vertically polarized while scattered light is unpolarized (no polarization filter used). VV means that both initial and scattered light are vertically polarized. VH means that initial light is vertically polarized while scattered light is horizontally polarized. Scattering amplitudes Af, As of the faster and slower mode, respectively, represent excess scattering (solvent contribution subtracted) and are expressed in units of the scattering intensity of a benzene standard IB. I is the total excess scattering. Df, Ds are the faster and slower diffusion coefficients, respectively.

of As(q) dependencies by assuming that inhomogenities can be considered as discrete objects. They can be referred to as domains, clusters, aggregates, or associates.39 It was further found that As(q) does not obey any power dependence of the type As(q) ∼ q-d, which would indicate that scattering reflects internal structure of fractal objects with fractal dimension d and radius of gyration Rg . q-1. The next issue considered was the asymmetry of scattering objects. Substantially asymmetric objects give strong depolarized scattering (initial polarization of laser light is changed after the scattering process). An example of results obtained by depolarized light scattering is shown in Table 1. Depolarized scattering is fairly weak (IVH/IVV ) 0.02), and the analysis of the depolarized dynamic light scattering data shows that the depolarized scattering is due to the presence of large domains with slow dynamics giving apparent diffusion coefficient roughly equal to the slow diffusion coefficient from conventional scattering. The slightly higher value of diffusion coefficient in the depolarized scattering experiment is due to the influence of very slow rotational diffusion which superimposes on the translational diffusion. As expected, no contribution from the fast mode (coupled diffusion of Cd2+ cations and SO42- anions in our particular case) is seen in depolarized scattering. Since the depolarized scattering from domains is very weak, it can be concluded that domains can be considered as close-tospherical objects. Therefore an optimized regularization tech-


Figure 5. Volume size distribution of clusters obtained by ORT analysis35,36 of the angular dependence of scattering from the slower mode. Clusters mean domains with different solute concentration with respect to the rest of solution and are considered as spherical or closeto-spherical objects with radius R. The same sample as in Figure 2.

nique (ORT),35,36 which proved to be successful in the computation of size distributions of polydisperse spherical and closeto spherical objects from static light scattering data, was used. Experiments on nonspherical particles showed35 that reasonable results can be achieved by assuming the spherical shape in the calculation. The more the shape deviates from the sphere, the more the distribution is skewed and broadened, but peaks of the distribution function do appear at correct positions.35 Another parameter which comes into the calculation of size distributions is the relative refractive index of particles with respect to solvent m ) n1/n2, where n1 is the complex refractive index of the particle and n2 is the complex refractive index of the solvent. Since the imaginary part of the complex refractive index (absorption coefficient) is small, it is assumed that the real part m′ of the complex relative refractive index obeys m′ = m. The m value is in our case an unknown parameter since the scattering contrast of domains (the difference between solute concentration inside and outside the domain) is not known. Fortunately, very similar results are obtained irrespective of concrete realistic m values used in the calculation. “Realistic” means here a range of values from 1.0 to the upper limit given by an assumption of completely dense solute aggregates without solvent inside. Figure 5 shows domain volume size distribution Dv(R) calculated from angular dependence As(θ) obtained on aqueous solution of D-glucose described in Figures 2-4. Volume size distribution of particle sizes Dv(R) means that Dv(R) dR represents the volume fraction of particles with radius from the interval (R, R+dR). This representation was preferred over the number distribution based on the number fraction of particles with radius from the interval (R, R+dR). Nevertheless, number distributions can also be calculated from measured data if necessary. A relatively high degree of polydispersity can be seen in the size distribution. Domain sizes span from ca. 30 nm to almost 300 nm. The volume fraction decreases with increasing R. The distribution function for R < 30 nm cannot be calculated from light scattering data due to insufficient q range. The resolution limit in the Fourier transform is Rmin ) π/2qmax and Rmax ) π/2qmin. These limits may be exceeded in favorable cases, but only slightly.35 In addition, the distribution function in our case is very broad and hence also the contribution of small particles (e 30 nm) to overall scattering intensity is very low. Therefore, it is impossible to make a conclusion from light scattering data whether the distribution function goes steeply to zero below R ) 30 nm or persists at higher Dv(R) values until very low R.

J. Phys. Chem. B, Vol. 110, No. 9, 2006 4333 The latter case would mean that there is also a large number of very small domains in solution which could in principle approach even solute dimers, trimers, etc. It is necessary to note that static light scattering is in general a very good tool for obtaining broad size distributions in this range of particle sizes, better than dynamic light scattering, which is not able to provide a fully correct information in this situation.35,36 This is not due to insufficient range of q values, but instead due to the nature of the DLS method. DLS works on a log scale, and hence the distribution of sizes over one decade cannot be in the case of DLS considered as broad. The second factor is that large particles from the overall distribution scatter much more light than smaller ones, and therefore the correlation function carries information mostly about the largesize end of the distribution. In SLS experiments, larger particles also scatter much more light than smaller ones, but very differently at different angles, which is a main point why the static method works satisfactorily. Model experiments on solid particles with similarly broad size distributions show that peaks obtained from DLS are located at the large-size end of the size distribution and the shape of the peak carries almost no information about the real size distribution.35,36 Therefore, we used SLS in this work to obtain size distributions. In the case of the diffusion coefficient Ds from DLS experiments, it is uncertain whether conditions for the application of the StokesEinstein relation (calculation of hydrodynamic radius) are fulfilled. Therefore, the hydrodynamic radius is denoted here as Rh,app (apparent). Another complicating factor comes from the fact that particles (domains) may not be solid particles, and therefore a hydrodynamic flow of solvent through diffusing particles cannot be excluded. This effect is widely known, for instance in the case of diffusion of polymer coils, and is referred to as draining. Hydrodynamic radii Rh,app obtained from diffusion coefficients via the Stokes-Einstein relation then do not necessarily correspond to geometrical sizes. Therefore, we restrict ourselves to a qualitative statement that the dynamic process observed in DLS as the slower mode corresponds to diffusion of domains. The characteristic decay frequency Γ of this mode does not scale exactly as required for a true diffusive mode (Γ ∼ q2), but has a stronger q dependence. Therefore, the apparent slow diffusion coefficient Ds ) Γs/q2 is qdependent. This dependence may be due to two factors: (i) as a consequence of the size polydispersity due to the relation Ds = Rh-1 and (ii) due to the influence of internal dynamics. In the presence of internal dynamics of scatterers, the apparent diffusion coefficient can be expanded at low q as40

D(q) ) D(0)(1 + CRg2q2 - ...) for qRg < 2

(7)

where C is a constant dependent on the internal architecture of the scattering particle (the C parameter is determined by the slowest internal mode of motion in the object) and values of C were reported in the range from 0 to 0.2 (ref 40). Thus, for Rg comparable or slightly larger than q-1 (which is the case in this work), a measurable q-dependence of D can be obtained even for monodisperse particles. The contribution of the two effects (polydispersity and internal dynamics) cannot be easily separated in some cases. The influence of polydispersity of course depends on the extent of polydispersity (width of the size distribution). In general, appreciably broad size distributions DV(R) occur in cases investigated in this work. Figure 6A shows an example of the q2-dependence of the apparent slow diffusion coefficient Ds. Data obtained on aqueous solution of urea, c ) 4.5 mass %, is shown (urea solutions show very pronounced angular dependence of both Ds and As - see Figure 1 for corresponding angular dependence of As). The q2 dependence of Ds clearly


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Figure 7. Angular dependencies of scattering from supramolecular domains in aqueous solution of sodium para-toluene-sulfonate, c ) 4.5 mass %. Solution was first gently centrifuged at low centrifugation speeds and times which practically do not affect the sample (10 min at 5000 g) (4). Afterward the following operations were performed: solution was centrifuged 3 h at 22 000 g (O), the upper part of the solution in the cell at the meniscus was stirred (b), solution was stirred at the bottom of the cell (0), solution was repeatedly stirred at the bottom of the cell (9). Experiment illustrates that domains have higher density than the rest of solution and move in the direction of the gravitational field toward the bottom of the cell, not vice versa.

Figure 6. (A) Dependence of the apparent slow diffusion coefficient Ds ) Γs/q2 on scattering vector q. The slow dynamic mode is associated with the presence of large supramolecular domains in solution. (B) Dependence of dimensionless reduced decay rate Γs/(q3KT/η) on scattering vector q (K is Boltzmann’s constant, T is temperature, and η is viscosity of the medium). The solid line corresponds to slope -1. Data were obtained on aqueous solution of urea, c ) 4.5 mass % (see Figure 1 for corresponding angular dependence of the scattering amplitude of this mode As).

deviates from the prediction of eq 7. This is due to a strong influence of polydispersity. The smaller the q the stronger the contribution of the largest objects from the size distribution to the scattering signal (and hence the stronger the influence on Ds). Figure 6B shows a double logarithmic plot of Γs/q3 vs q (Γs/q3 is normalized by KT/η in order to receive a dimensionless ratio; K is Boltzmann’s constant, T is temperature, and η is viscosity of the medium). This reduced decay rate exhibits different scalings with q according to the nature of scattering objects and their internal dynamics.40-43 In our case it does not obey any universal scaling in the whole q range of light scattering. In the limit of large q values, however, it seems that the experimentally measured dependence slowly approaches the prediction for homogeneous spheres Γs/q3 ∼ q-1. This is reasonable since at sufficiently large q values (where q-1 becomes smaller than the smallest Rg from the size distribution), the influence of size polydispersity vanishes. DLS data in the limited available q-range does not allow for more detailed information on the internal structure of domains. Values of diffusion coefficients Ds extrapolated to zero angle yield apparent hydrodynamic radii (calculated via StokesEinstein relation taking solution viscosity into the calculation of the friction factor), which do fall into the range of sizes from ORT size distributions. For instance we can compare obtained apparent hydrodynamic radius with the size distribution shown

in Figure 5. The Rh,app was calculated from Ds extrapolated to zero angle as Rh,app ) 134 nm. In conclusion, size distributions of domains are obtained from SLS data by ORT analysis. DLS data semiquantitatively support these calculations. In general, domain size distributions for various solutions or mixtures possess qualitatively the shape of DV(R) as shown in Figure 5, but quantitatively differ in the distribution width and the mean size Rm. Very narrow size distributions are obtained quite rarely (see for instance acetic acid in Figure 1). In this case the angular dependence of As matches prediction for the sphere form factor. In a vast majority of cases As(θ) cannot be directly compared with known form factors for specific architectures due to the significant influence of polydispersity. The next fundamental question that we would like to answer is whether domains have higher or lower density compared to the rest of solution. In general, clustering due to some attractive mechanism results more frequently in the formation of denser clusters in a less dense continuum matrix. An opposite case, however, is also possible.44 Both above-mentioned cases can be considered in our current work. However, as is known from the Babinet principle, they are not distinguishable in a simple scattering experiment. Therefore, we used a trick to answer this question. A strong gravitational field was applied to sample via centrifugation. In the centrifugation process, in general, objects with higher density than the surrounding continuum move in the direction of the gravitational field while objects with lower density move in the opposite direction. With respect to the concrete arrangement in our experiment, it would mean movement of domains toward the bottom of the scattering cell or toward the solution meniscus, respectively. A swing-out rotor was used in which the scattering cell is positioned perpendicular to the axis of rotation during centrifugation so that the gravitational force acts directly along the cell axis toward its bottom. Results of this experiment are shown in Figure 7. Solution was first gently centrifuged (10 min at 5000 g). Domains are practically unaffected at such low centrifugation force and time. Afterward the solution was strongly centrifuged (3 h at 22 000 g). A significant decrease in scattering intensity and slope of the scattering curve is observed. This is due to the



fact that a certain portion of domains from the initial size distribution was drawn out of the scattering volume. By calculating size distributions from each angular dependence (not shown), it can be seen that the large-size end of the initial distribution is cut, which is in agreement with the commonly known fact that the larger the particle the faster the movement in a gravitational field (large particles are dislocated first). The scattering volume in our experiment was located in the middle of the cell. The question of interest was therefore whether domains move toward the meniscus (against the direction of the gravitational field) or toward the bottom of the cell (in the direction of the gravitational field). The upper part of the solution in the cell at the meniscus was stirred by a thoroughly cleaned stirring bar to check whether domains accumulate at the meniscus. The scattering after stirring was practically unaffected. Then the solution was stirred at the bottom of the cell, and this stirring resulted in a release of domains from the bottom of the cell back in the scattering volume. A similar effect was observed after repeated stirring at the cell bottom. It can be concluded that domains moved in the direction of the gravitational field and therefore had higher density than the surrounding matrix (rest of solution). Because solute had higher density than solvent, we can also conclude that domains had higher solute concentration than the rest of solution. Another result from this particular experiment is that domains cannot be characterized as equilibrium large-scale concentration fluctuations, but rather as long-lived associates of solute (can migrate on macroscopic length- and time scales). The next question, which is again not accessible from a classical light scattering experiment, is whether these are tight associates (solute molecules are in contact and there is little or no solvent inside the associate) or loose associates (solute molecules are not in contact and there is solvent inside the associate). Two limiting cases of possible domain structures are schematically shown in Figure 9. Scattering from an ensemble of particles (such as domains shown in Figure 9) can be approximated as

I(θ) ) kNM2P(θ)S(θ)

(8)

where k is an experimental constant covering the square of the scattering contrast, N is the number of particles in the scattering volume, and M is the particle mass. S(θ) is the solution structure factor reflecting interparticle correlations, and P(θ) is the particle form factor reflecting intraparticle correlations. Upon assumption of negligible interparticle interactions (S(θ) ) 1), the scattering at zero angle (where P(0) ≡ 1) can be written as

I(0) ) kNM2 = kNF2V2 = NFo2R6

(9)

where F is the particle density, V is the particle volume, Fo is the overall scattering contrast including density, and R is the particle radius assuming a spherical shape. The quantity of interest related to the question how tight or loose are the associates is Fo. The radius of the particle is known in our case; however, two other independent variables (N, Fo) are unknown. In other words, it is not possible to estimate Fo without knowledge of N, and vice versa. An interesting qualitative information can be, however, obtained from kinetic measurements on our samples. A typical feature of many kinetic experiments which we have performed30 is that changes in intensity of scattering from domains are observed over time while the size (or size distribution) does not change at all. An example is shown in Figure 8. Data were obtained on an aqueous solution of D-glucose (c ) 0.9 mass %). A significant increase of scattering intensity is observed over time while the size

Figure 8. Angular dependencies of scattering from supramolecular domains in aqueous solution of D-glucose (c ) 0.9 mass %) measured immediately after the preparation (*) and 60 days after the preparation (O). Diffusion coefficient of domains measured at θ ) 90° was Ds ) 26.4 × 10-9 cm2 s-1 (immediately after the preparation) and Ds ) 12.6 × 10-9 cm2 s-1 (60 days after the preparation). Data demonstrate an increase of domain density over time, which means that (at least initially after the preparation) domains are loose structures.

distribution does not change. This increase can be due to an increase of the number of domains (N) or due to increase of the domain density (Fo). It is physically quite unrealistic that the number of domains N would increase with time without any changes in the size distribution. Even more unrealistic would be an opposite effect: a decrease of the number of domains over time in other experimental cases where a decrease of scattering intensity was seen over time with the size distribution unchanged. Therefore, we assume that changes of intensity in such cases are due to changes of Fo. A more important supporting argument follows from measured values of diffusion coefficient Ds. Ds decreased from Ds ) 26.4 × 10-9 cm2 s-1 to Ds ) 12.6 × 10-9 cm2 s-1 in the experiment shown in Figure 8. We interpret this decrease as due to increased domain density and consequently weaker hydrodynamic draining (passage of solvent through domains becomes more difficult). The increase of scattering from domains in Figure 8 means unambiguously that the solute concentration in the rest of the solution (matrix) decreases, and hence microviscosity in the space surrounding the domain decreases, which should act in an opposite direction, i.e., to accelerate diffusion. Decrease of Ds therefore must be interpreted as due to increase of the domain density. In conclusion, domain density changes with time which means that domains cannot be completely tight aggregates without solvent inside. Solute molecules replace solvent molecules in clusters and vice versa. Another argument confirming that domains are not tight aggregates without solvent inside follows from the following consideration. Let us suppose for a moment that domains are completely tight aggregates. This is schematically shown in Figure 9b. The idea of the procedure below is that we define Fo in eq 9 and calculate the remaining unknown parameter N. Under above-mentioned circumstances, the scattering from domains at zero angle can be given as As(0°) ) kMd2Nd, where index d refers to domains (P(θ) ) 1 at zero angle by definition and S(θ) can be considered as S(θ) ) 1 since the concentration of domains is low). The scattering from homogeneously dispersed solute molecules outside domains can be given at zero angle as Af(0°) ) kMm2Nm, where index m refers to individual molecules outside domains. (S(θ) can be approximated as S(θ) ) 1 since interactions between solute molecules at concentra-


Sedla´k

Figure 9. Schematic illustration of two examples of possible domain structures in solutions and mixtures of liquids. Limiting cases of loose (A) and tight (B) structures are shown. Solute molecules are shown as closed circles. Solvent molecules are assumed to occupy the rest. Domains are discrete regions with a higher concentration of solute inside domains than outside domains. An exact calculation of this concentration difference is not possible; however, it can be shown that domains are not completely tight associates without solvent molecules inside as shown in part (B).

tions used are weak). The constant k covering contrast (refractive index increment) is the same in both cases (As and Af) because both scattering species are composed of the same material and differ only in molecular weight (this situation is analogous to monomers and polymers with different molecular weights). Then we can write As(0°) /Af(0°) ) (Md /Mm)2 (Nd/Nm) and evaluate Nd as

Nd ) (As(0°) /Af(0°)) Nm (Mm/Md)2 ) (As(0°) /Af(0°)) Nm/nd2 (10) where nd is the number of solute molecules per domain. Since usually As(0°)/Af(0°) , nd, it can be easily shown that Nm can be very accurately approximated by taking into account analytical (weighing) solute concentration without assumption of formation of any domains (Nm0) and hence Nd ) (As(0°)/Af(0°)) Nm0/nd2. Let us now calculate Nd for the sample described in Figures 2-5 (4.1 mass % aqueous solution of D-glucose). We estimate nd by assuming a close-packing of glucose molecules in a sphere with radius R ) 140 nm as nd ) 3.84 × 108. Nm0 can be calculated based on the known solute concentration and known scattering volume in our experimental setup V ) 3.12 × 105 µm3 as Nm0 ) 4.7 × 1013. By inserting As(0°)/Af(0°) ) 70.0 we obtain Nd ) 0.022. The final result of the calculation is the following: if we would assume completely tight domains, the number of domains per scattering volume would have to be Nd ) 0.022. This is, however, impossible. The number of domains per scattering volume must be orders of magnitude higher in order to yield standard correlation curves, such as those shown in Figure 2. By “standard” we understand here correlation curves without traces of so-called number fluctuations. The effect of number fluctuations means that if the number of scattering particles in the scattering volume is small, then the autocorrelation function contains an additional contribution from these fluctuations (in addition to regular phase fluctuations). Since δN/N ∼ N -1/2 holds for Gaussian fluctuations, the number of scattering particles in the scattering volume must be N > 104 to have relative fluctuations suppressed below the 1% level.45 We have shown that the internal structure of domains is quite far from a model of a tight aggregate of solute molecules without solvent inside. Oppositely, domains are rather loose structures with solvent inside (Figure 9a). Similar calculations were performed for several other samples with the same result. A precise quantitative estimate of the domain density is currently not possible because of reasons outlined earlier. Upper

TABLE 2: Osmotic Coefficients for Aqueous Solutions of D-Glucose and Urea in the Concentration Range Where Light Scattering Experiments Were Performeda D-glucose

a

urea

c, g/kg H2O

φ

c, g/kg H2O

φ

4.90 13.6 40.8 85.0

1.007 1.011 1.007 1.014

4.04 14.6 47.2 58.2

1.01 0.99 1.01 0.98

Osmotic coefficients were obtained by vapor pressure osmometry.

and lower estimates of number of solute molecules per domain can be, however, calculated in the following way. The upper estimate is obtained assuming a close-packing of solute molecules without solvent. The nd value estimated in such a way for the sample shown in Figures 2-5 spans from nd ) 3.8 × 106 in domains with R ) 30 nm up to nd ) 3.8 × 109 in domains with R ) 300 nm. The lower estimate is obtained assuming that the solute concentration in domains is only infinitesimally higher than the concentration outside domains. In this case nd spans from nd ) 1.7 × 104 in domains with R ) 30 nm up to nd ) 1.7 × 107 in domains with R ) 300 nm. Another experimental technique, which is sensitive to solute association, is osmometry. Each completely free solute molecule contributes by KT to osmotic pressure π. Upon association of solute molecules, osmotic pressure decreases. Neglecting the effect of the second and higher virial coefficients (usually possible for systems investigated in this work at given concentrations), the ratio φ ) π/πid (so-called osmotic coefficient) can be taken as a measure of the aggregation (association). πid is the ideal osmotic pressure based on the assumption of no association, i.e., πid ) KT NA cM ) RTcM, where cM is molar solute concentration and NA is Avogadro’s number. π ) RTcM,eff, where cM,eff is effective molar concentration of free species in solution. Upon association, of course, cM,eff < cM. Table 2 shows results obtained by vapor pressure osmometry on aqueous solutions of D-glucose and urea, two components which exhibit a pronounced domain structure. The osmotic coefficient is within experimental precision equal to 1.0 and is independent of concentration in the concentration range investigated. Several conclusions can be drawn from this result. First, the association of solute into domains is practically not reflected in the osmotic pressure data. The reason is the following. If we assume the model of domains as tight aggregates (Figure 9b), then we can derive from eq 10 that Ndnd /Nm ) As(0°) /Af(0°) /nd, where Ndnd /Nm is the ratio of the number of solute molecules bound in domains (osmotically not active) to the



Figure 11. Angular dependencies of scattering from supramolecular domains as a function of solute concentration. Data from Figure 10 are vertically shifted. Dependencies do not fall on a mastercurve. A gradual decrease of angular dependence upon dilution is evident. The meaning of symbols is the same as in Figure 10. Figure 10. Angular dependencies of scattering from supramolecular domains as a function of solute concentration. Data were obtained on aqueous solutions of urea. Scattering intensities As are excess intensities and are expressed in units of benzene scattering. Concentrations c are in g/L: (O) c ) 47.1, (b) c ) 15.2, (0) c ) 5.03, (2) c ) 1.73, (4) c ) 0.605, ([) c ) 0.202, (]) c ) 0.070.

number of osmotically active solute molecules outside domains. For the D-glucose solution described in Figures 2-5 we obtain Ndnd /Nm = 10-7, which is far below the sensitivity of osmometric measurements (5 orders of magnitude below). In the model of loose associates shown in Figure 9a (which seems closer to reality although we do not exactly know domain densities), the ratio Ndnd /Nm can be higher than 10-7. However, the higher this ratio, the looser the domains and consequently the more osmotically free are the solute molecules in domains. We also remind that the difference between concentrations inside and outside domains may be quite low and, as is seen from Table 2, the osmotic coefficient is constant in a broader concentration range around the concentration investigated. It can be understood why osmotic pressure measurements performed in the past on solutions of various solutes could not reveal the domain structure. The sensitivity of osmometry toward this kind of association is much lower compared to scattering methods, which are oppositely extremely sensitive. Osmotic pressure measurements are, however, a good practical tool for checking the dissolution of solutes. Osmotic coefficients such as those reported in Table 2 give information that the solute is completely dissolved. Osmotic pressure measurements can also discriminate between the domain association described in this work and other types of association which can possibly occur in solutions (for instance ion pairing in salt solutions; more details will be given in the forthcoming paper). Figure 10 shows results on the concentration dependence. Angular dependencies of the scattering from domains As are plotted for varying concentrations of solute (aqueous solutions of urea). These data demonstrate that supramolecular domains are present in a very broad range of concentrations. Almost three decades of concentrations were investigated. Upon decreasing concentration, the angular dependence of As becomes continuously slightly weaker while still preserving a significant curvature, which means that domain size distributions are still quite broad but shift toward smaller sizes. Angular dependencies from Figure 10 are replotted in Figure 11 in such a way that

Figure 12. Dependence of reduced scattering intensities As(0°)/c and As(30°)/c on concentration c. Scattering intensities As are excess intensities and are expressed in units of benzene scattering. Concentrations c are in g/L. The same sample as in Figure 10.

they are vertically shifted. It is clear from this representation that curves do not fall on a master curve, but instead a gradual weakening of the angular dependence is observed upon dilution. Figure 12 shows plots of normalized intensities As(0°)/c and As(30°)/c versus concentration c. Scattering from domains at zero angle can be considered as a true measure of the clustering effect. Since no extrapolation of data is needed for As(30°) (which is directly measured), the plot of As(30°)/c is shown, too. Both plots show qualitatively the same effect. A decrease of normalized intensity is seen upon dilution with some leveling

4338 J. Phys. Chem. B, Vol. 110, No. 9, 2006 at low concentrations. This leveling of the normalized intensity means that As decreases further roughly in proportion to the concentration. It cannot be therefore unambiguously concluded where the effect of solute organization into domains may disappear completely upon lowering concentration. The measurements would have to be done on extremely diluted samples, which is far outside the possibilities of light scattering (especially taking into account that we are dealing here with low molecular weight compounds). The trends seen in Figures 10-12 suggest that the effect persists to significantly lower concentrations than measured, however, with a gradual decrease of scattering intensity and a gradual decrease of domain sizes. In principle, the possibility of the existence of a sudden transition occurring somewhere at very low concentrations and leading to a sudden disappearance of the effect cannot be also excluded. The observed concentration dependence is in favor of the association concept rather than a nonequilibrium aggregation concept. It should also be stated that the same results were obtained when particular concentrations were prepared directly and when they were prepared by dilution of the stock solution with the highest concentration. Conclusions A detailed light scattering characterization of large-scale supramolecular structure in solutions of electrolytes, nonelectrolytes, and mixtures of liquids was given. It was shown that solutes are distributed not homogeneously on large length scales. Regions of higher and lower solute concentration exist in solutions and mixtures and give sufficient scattering contrast for experimental observation. Large-scale inhomogenities in concentration are reflected in static light scattering data by the presence of pronounced angular dependencies of scattering intensity as well as by much higher absolute values of scattering intensities compared to intensities expected on assumption of homogeneous mixing at molecular level. The dynamics of these inhomogenities is reflected in dynamic light scattering data as a slow diffusive mode. Results from static and dynamic scattering correlate well and supplement each other. It was shown that this inhomogeneous arrangement of solute cannot be characterized as a random two-phase system. Therefore, discrete objects with solute concentration different than that in the surrounding matrix (rest of solution) were considered further. They were referred to as domains (giant clusters, associates). It was found out that angular dependencies of scattering do not reflect any internal fractal structure of domains and that very low depolarized scattering indicates their closeto-spherical shape. Optimized regularization technique ORT35,36 which proved to be successful in the computation of size distributions of polydisperse spherical and close-to-spherical objects from static light scattering data, was therefore used. Resulting size distributions of domains are significantly broad, ranging up to several hundreds of nanometers. Characteristic sizes of domains thus exceed angstrom dimensions of individual molecules by several orders of magnitude. Number of solute molecules per domain varies approximately in the range 103108. By applying a strong gravitational field on sample, it was found that domains have higher solute concentration than the rest of solution (not vice versa). This is not possible to determine on the basis of a regular light scattering experiment. It was further shown that the internal structure of domains is quite far from the model of a tight aggregate of solute molecules without solvent inside. Oppositely, domains are rather loose structures with solvent inside. The supramolecular organization was found in a very broad interval of concentrations ranging from 0.007 mass % to 20 mass %. A gradual decrease of domain size is, however, seen upon dilution. The phenomena described apply

Sedla´k to a very broad range of solutes and solvents. Among others, selected data on most common substances of great chemical and biological importance such as sodium chloride, citric acid, glucose, urea, acetic acid, and ethanol were presented. Acknowledgment. Support from the Presidium of the Slovak Academy of Sciences (grant No. 2/8001/22) and from the Slovak grant agency VEGA (grant No. 2/2085/2002) is acknowledged. Author is also thankful to E. Gyo¨ngyo¨siova´ for excellent technical assistance in experiments. References and Notes (1) Georgalis, Y.; Kierzek, A. M.; Saenger, W. J. Phys. Chem. B 2000, 104, 3405. (2) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH: New York, 1993. (3) Schmitz, K. S.; Lu, M.; Gauntt, J. J. Chem. Phys. 1983, 78, 5059. (4) Schmitz, K. S.; Lu, M.; Singh, N.; Ramsay, D. J. Biopolymers 1984, 23, 1637. (5) Schmitz, K. S.; Ramsay, D. J. Biopolymers 1985, 24, 1247. (6) Schmitz, K. S.; Ramsay, D. J. Macromolecules 1985, 18, 933. (7) Lee, W. I.; Schurr, J. M. J. Polym. Sci. 1975, 13, 873. (8) Lin, S. C.; Lee, W. I.; Schurr, J. M. Biopolymers 1978, 17, 1041. (9) Fulmer, A. W.; Benbasat, J. A.; Bloomfield, V. A. Biopolymers 1981, 20, 1147. (10) Ermi, B. D.; Amis, E. J. Macromolecules 1996, 29, 2701. (11) Ermi, B. D.; Amis, E. J. Macromolecules 1997, 30, 6937. (12) Ermi, B. D.; Amis, E. J. Macromolecules 1998, 31, 7378. (13) Zhang, Y.; Douglas, J. F.; Ermi, B. D.; Amis, E. J. J. Chem. Phys. 2001, 114, 3299. (14) Tanahatoe, J. J.; Kuil, M. E. Macromolecules 1997, 30, 6102. (15) Tanahatoe, J. J.; Kuil, M. E. J. Phys. Chem. B 1997, 101, 5905. (16) Tanahatoe, J. J.; Kuil, M. E. J. Phys. Chem. B 1997, 101, 10839. (17) Matsuoka, H.; Schwahn, D.; Ise, N. Macromolecules 1991, 24, 4227. (18) Matsuoka, H.; Ogura, Y.; Yamaoka, H. J. Chem. Phys. 1998, 109, 122. (19) Matsuoka, H.; Hattori, N.; Ishida, K.; Tomita, H. Polym. Prepr. Jpn. 1992, 41, 4400. (20) Prabhu, V. M.; Muthukumar, M.; Wignall, G. D.; Melnichenko, Y. B. J. Chem Phys. 2003, 119, 4085. (21) Prabhu, V. M. Curr. Opin. Colloid Interface Sci. 2005, 10, 2. (22) Sedla´k, M. In Light Scattering. Principles and DeVelopment; Clarendon: Oxford, 1996; Ch. 4, pp 120-165. (23) Sedla´k, M. In Physical Chemistry of Polyelectrolytes; Marcel Dekker: New York, 2001; Ch. 1, pp 1-58. (24) Sedla´k, M. J. Chem. Phys. 1994, 101, 10140. (25) Sedla´k, M. J. Chem. Phys. 1996, 105, 10123. (26) Sedla´k, M. J. Chem. Phys. 2002, 116, 5256. (27) Sedla´k, M. J. Chem. Phys. 2005, 122, 151102. (28) Liu, T. J. Am. Chem. Soc. 2003, 125, 312. (29) Liu, T. J. Am. Chem. Soc. 2005, 127, 6942. (30) Sedla´k, M. J. Phys. Chem. B 2006, 110, 4339. (31) Brown, W. Static Light Scattering. Principles and DeVelopment; Clarendon: Oxford, 1996. (32) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213. (33) Jakesˇ, J. Czech. J. Phys. 1988, 38, 1305. (34) Sˇ teˇpa´nek P. In Dynamic Light Scattering. The Method and Some Applications; Clarendon: Oxford, 1993; Ch. 4, pp 177-240. (35) Schnablegger, H.; Glatter, O. Appl. Optics 1991, 30, 4889. (36) Glatter, O.; Hofer, M. J. Colloid Interface Sci. 1988, 122, 496. (37) Debye, P.; Bueche, A. M. J. Appl. Phys. 1949, 20, 518. (38) Ornstein, L. S.; Zernike, F. Phys. Z. 1926, 27, 761. (39) All of these terms are used in the literature in various contexts, and it is difficult to unambiguously choose the most appropriate one. For instance, the term cluster is used frequently in the field in such a way that “large cluster” means association of a few (less than ten) molecules. It should be used therefore with caution in this case where we deal with association of an incomparable number of molecules (many orders of magnitude higher). We consider the term “domain” as sufficiently general, referring to “domain of different local concentration of solute.” (40) Burchard, W.; Schmidt, M.; Stockmayer, W. H. Macromolecules 1980, 13, 580. (41) Akcasu, A. Z.; Benmouna, H.; Han, C. C. Polymer 1980, 21, 866. (42) Tsunashima, Y.; Hirata, N.; Nemoto, N.; Kurata, M. Macromolecules 1987, 20, 2862. (43) Harnau, L.; Winkler, R. G.; Reineker, P. J. Chem. Phys. 1996, 104, 6355. (44) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (45) Bloomfield, V. In Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy; Plenum: New York, 1985; pp 363415.

Large-Scale Supramolecular Structure in Solutions of Low Molar Mass

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