Dependence of emulsion stability on particle size: Relative importance of drop concentration and destabilization rate on the half lifetimes of O/W nanoemulsions

ABSTRACT This study reports the behavior of ionic dodecane-in-water nanoemulsions in distinct salt concentrations. Systems of smaller particle size (74–285 nm) were synthesized by a sudden dilution of an equilibrated mixture. Larger size systems (384–670 nm) were obtained from a set of formerly smaller nanoemulsions that evolved unperturbed for 2 weeks. Characteristic destabilization times for flocculation, coalescence, and Ostwald ripening were evaluated. In general, it was observed that stability increases with drop size. However, this size dependence is largely the consequence of the lower particle concentration of the coarser emulsions. GRAPHICAL ABSTRACT


Concept of a characteristic time
It is known that the kinetic stability of a suspension toward aggregation depends on the repulsive forces between its particles. The electrostatic forces and their dependence on the ionic strength of the solution are the basis of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of the stability of lyophobic colloids. [1,2] The quantitative implementation of the DLVO theory relies on the kinetic description of the process of irreversible aggregation. According to Smoluchowski, [3] the number concentration of aggregates of k primary particles (size k) existing at a given time (n k (t)) results from a balance between the aggregates produced by the collisions of smaller clusters of size i and j (such that i þj ¼ k), and the aggregates of size k lost by the collisions with clusters of any other size: i¼1j¼kÀ i k ij n i ðtÞn j ðtÞ À n k ðtÞ X 1 i¼1 k ik n i ðtÞ ½1� Equation (1) can be solved assuming a constant kernel of flocculation rates (k ij ¼ k f ): Using Equation (2), a simple expression for the time dependence of the total number of aggregates (n) per unit volume results: Here, n 0 stands for the initial number of aggregates and τ c is a characteristic aggregation time: This characteristic time s c (Equation 4) is also equal to the half lifetime of the dispersion since it depends solely on the concentration of aggregates and their flocculation rate (k f ): The product of these two variables defines the collision frequency of the aggregates. This frequency depends on three factors: (1) the number of aggregates susceptible to collision, (2) their cross-section, and (3) their velocity. The last two factors are considered in calculating the flocculation rate.
In the absence of hydrodynamic interactions, and assuming that the movement of the particles is solely the result of their thermal interaction with the solvent, a simple expression for the flocculation rate can be obtained: [3] Here, the radius of collision is equal to R ¼ (R i þ R j )/2 ¼ R (for i ¼ j), and D is the relative diffusion constant. At room temperature, the value of k f ¼ 6 � 10 À 18 m 3 /s. When interaction forces are incorporated (DLVO theory [4][5][6] ), the rate constant is modified by the term W, known as the stability ratio: [6,7] The simplicity of Equation (7) and its apparent autonomy with respect to particle size is the result of a number of approximations; [3,5,[8][9][10][11] the most important of which is the selection of a constant kernel for the resolution of Equation (1). In general, the kernel {k ij } and the stability ratio (W ij ) depend on R i and R j . Therefore, there is no simple analytical solution of Equation (1) that must be evaluated numerically. [12][13][14][15] Moreover, even under the constant kernel approximation, Smoluchowski supposed that the product D ij R ij should be approximately equal to 2 D R, partially neglecting its size dependence, as well as the variation of the hydrodynamic radii with the topology of the aggregates. [9,[16][17][18][19]

Emulsion stability
Although oil-in-water (O/W) emulsions are also kinetically stabilized dispersions, they show significant differences in regard to aggregation with respect to suspensions [20][21][22] . While the surface charges of solid particles are fixed, being a product of their chemical synthesis, those of emulsions are a consequence of the surfactant adsorption to the interface of the drops. [23,24] Adsorption is a reversible process that depends on the physicochemical conditions of the system. Changes in drop size distribution (DSD) of the emulsion induce changes in its total interfacial area that modify the adsorption equilibrium of the surfactant. Hence, the charge of the drops is expected to vary as a function of time even if the actual adsorption mechanism is very fast. [25][26][27] The adsorption of ionic surfactants is enhanced in the presence of salt, which increases the charge of the drops and favors their repulsion. Concurrently, the electrostatic forces between the drops are screened by the increase of the ionic strength of the solution. Unlike that predicted by DLVO, this originates situations in which the stability of the systems may either increase or decrease depending on the concentration of salt. [28][29][30] In a typical situation, the concurrent occurrence of flocculation, coalescence, Ostwald ripening (OR), and creaming limits the quantification of emulsion stability. [31][32][33][34][35][36] It also hinders the identification of the fundamental parameters that determine the observed experimental behavior. Stability prediction is generally reduced to ad hoc explanations supported by the predominance of one particular phenomenon [37,38] such as the appraisal of the ripening rate through the temporal variation of the cube average radius of an emulsion. [39][40][41] It has been shown both experimentally [42] and theoretically [43,44] that this measurement is generally influenced by flocculation and coalescence (FC). Another example is the assessment of emulsion stability by using the creaming rate. Nevertheless, it has been also demonstrated that this measurement is not reliable whenever OR predominates. [45] Even if the composition of the phases is such that Dρ ∼ 0 and aggregation does not occur, a very unstable nanoemulsion (showing a significant evolution of its average radius with time) does not present any sizeable evidence of creaming.
In this study, the stability of nanoemulsions with average radii between 74 and 670 nm was appraised in terms of characteristic half lifetimes and rates of occurrence. The rates were measured by employing different techniques for each phenomenon (see below). A set of characteristic times was calculated using the functional form of Equation (6), which allows decoupling the effect of the number density of particles from the rate of incidence of each process.

Measurement of the absolute aggregation rate: Determination of k FC
With few exceptions, [46][47][48][49] the aggregation rate is generally estimated through the rate of doublet formation. The authors recently published a novel procedure to reproduce the turbidity (τ) of an emulsion during periods of the order of the flocculation half lifetime: [22,50,51] Here, σ 1 , σ k,a , and σ k,s represent the optical cross-sections of the primary drops, aggregates of size k, and spherical drops of size k (R ¼ ffi ffi ffi k 3 p R 0 ). Fitting of Equation (9) to the experimental variation of the turbidity as a function of time allows for the calculation of a mixed FC rate, k FC , and x a . In the case of ionic nanoemulsions, aggregation is induced by injecting a high concentration of salt into the sample vessel. It is noteworthy that the last two terms of Equation (9) can be regrouped in order to define an average cross-section for an aggregate of size k: The magnitude and sign of x a are related to globularity of the formed aggregates. [52,53] The significance of the obtained values of k FC was confirmed by contrasting the experimental observations with the results of emulsion stability simulations (ESSs). [54] Average expressions for k F and k C based on macroscopic models Over the years, several average expressions were proposed for the evaluation of FC. [20,55,56] According to ESSs, the variation in the number of aggregates as a function of time in the presence of an appreciable repulsive barrier conforms to a remarkably simple expression: [26] Here, k A formerly represented the rate of flocculation and k B the rate of coalescence. A and B (with B ¼ 1 -A) are coefficients that measure the extent of the occurrence of these two processes. Equation (11) was obtained by adding the solutions of first-and second-order equations representing the processes of coalescence and flocculation, respectively. However, the proposed mathematical solution is not general and the rates obtained contain information from both FC. [26,27] Recently, Yang et al. [57] solved a similar differential equation to describe the aggregation of particles and the breakage of their flocs: Here, k 1 and k 2 are the mean kinetic constants associated to flocculation and breakage, respectively. It is clear that in order to consider coalescence instead of breakage it is only necessary to change the sign of the second term on the right-hand side and reinterpret k 2 as a coalescence rate. Following Yang et al. [57] , the general solution of the FC problem would be Equation (13) can be evaluated if the variation of the total number of aggregates as a function of time is known. Such information is not available by conventional means. In fact, the average radius of the emulsion provided by common light scattering instruments contains information from both drops and aggregates because these equipments only sample the hydrodynamic radii of the scatterers. Hence, only an order of magnitude calculation can be made by using Equation (13). For this purpose, the total volume of oil is divided by the average radius of the dispersion at each time in order to estimate the average number of "aggregates" (drops).

Creaming
Creaming describes the upward movement of droplets due to the higher density of the surrounding liquid. The drops of any emulsion are subject to a constant buoyancy force resulting from a balance between the earth's field of gravity and Archimedes' law: Here, g is the acceleration due to gravity, ρ i is the density of phase i, and the subscripts 1 and 2 respectively refer to the disperse and the continuous phase. This force causes the formation of cream at the top of the container. Using Stokes' law, a simplified formula for the velocity of an isolated rigid spherical particle that creams in an ideal unbounded liquid (ν) can be obtained: When hydrodynamic corrections are included and the problem of a polydisperse mixture of spheres is considered, a volume-fraction dependence results. [58] In this case, the rate of creaming can be evaluated using [59,60] v R; Most studies of emulsion stability consider the creaming rate as a fundamental stability criterion. Implicitly, it is assumed that any process leading to destabilization will generate larger droplets (or aggregates) subject to the gravitational force. As such, the creaming rate appears to be a reliable measure of stability. Nevertheless, it is important to note that, according to Equations (14) and (16), a homogenous nanoemulsion is not expected to produce a substantial amount of cream over a period of several months [45,52,61] unless its drops aggregate.

OR rate
The theory of Lifshitz, Slyosov and Wagner (LSW) [39][40][41] predicts that the ripening rate can be quantified in terms of the linear increase of the cubic critical/average radius [62] of the emulsion as a function of time: Here, R c , C(∞), and α stand for the critical radius of the dispersion, their bulk solubility in the presence of a planar O/W interface, and the capillary length: Here, R g is the universal gas constant and V m represents the molar volume of the oil.
During experimentation, the linear variation of the cube average radius of an emulsion as a function of time is often associated with the process of OR, especially in the case of nanoemulsions. [31,63] However, the value of the ripening rate is found to be systematically larger than the one predicted by LSW for O/W emulsions. [42,64] Moreover, the prediction of the solubility of a mixture of hydrophobic components is difficult in spite of the fact that it has a decisive influence on the ripening process. [65] According to ESSs, [42,44] the variation of the cube average radius of an emulsion is often influenced by the effect of FC. Consequently, the curves of R 3 versus t -usually employed to characterize the OR rate -probably identify a mixed phenomenon. Moreover, FC prevents the elimination of the drops by dissolution in alkane/water emulsions of low ionic strength [42,43] : FC increases the radius of the drops at a higher rate than the one employed by OR to dissolve them. Otherwise, the radius decreases as a consequence of molecular exchange. [42][43][44]66] A typical curve of R 3 versus t is predicted to show a sawtooth variation around an average slope. Depending on the predominating process, the slope varies between 10 À 22 and 10 À 24 m 3 /s for FC or close to the predictions of Equation (17) in the case of OR. In the former case, the slope of the curve changes with the ionic strength of the aqueous solution. Moreover, the periods of decrease depend on the solubility of the oils. In the case of dodecane, slopes of the order of 10 À 27 m 3 /s are obtained. [44] According to Nazarzadeth et al., [67] the behavior of the DSD helps to identify the predominating process of destabilization. These authors coincide with the view that only OR can explain the falls in the average radius, while the periods of increase can be either attributed to FC or OR. They further observed that at high surfactant concentrations the DSD becomes wider when the average radius falls, extending at both ends due to the generation of small and large drops as a consequence of the molecular exchange. Instead, when the small drops disappear by dissolution, the radius increases and the DSD becomes thinner since its low-end tail progressively disappears. At low surfactant concentration, coalescence predominates. The increase of the average radius due to coalescence masks the decrease induced by OR. The fall periods become shorter, and the DSD widens and becomes positively skewed due to the formation of larger drops. Thus, it might be possible to observe changes of R 3 versus t, which basically correspond to either OR or to the combined process of FC. In any event, the slope of R 3 versus t is the most reliable measure of the overall stability of an emulsion because it summarizes the effect of all destabilization processes. [45] If the increase of the average radius (dR 3 /dt) is due to Oswald ripening, a characteristic time can be defined for this variable. Consequently, we considered the following relationship between the average radius and the total number of particles per volume: Hence, [45,68] This yields a second-order differential equation for the number density of particles: The solution of this equation is similar to Equation (3) but with a different rate: Finally, the expression for a characteristic time deduced from dR 3 /dt is equal to

Formulation-composition map
Bidimensional scans of formulation (wt% NaCl) and composition (weight fraction of water, f w ) at 5 wt% SDS and 3.3 wt% iP were made. The composition was varied from f w ¼ 0.1 to f w ¼ 0.9 with increments of 0.1. The salinity of the aqueous phase was equal to 2, 4, 6, 8, and 10 wt% NaCl. The systems were prepared in vials, subsequently weighing each component until the final composition was obtained. The vials were gently shaken for 2 hours using a Varimix (Barnstead International, USA) in order to facilitate contact between the phases. Finally, the systems were placed in a thermostatic bath at 25°C until equilibrium was reached, that is, until complete phase separation was observed. The number of phases was then visually evaluated. Anisotropic liquid crystalline phases were identified using polarized light.

Emulsion preparation
An equilibrated mixture of water þ liquid crystal þ oil with 10 wt% SDS, 8 wt% NaCl, 6.5 wt% iP, and a weight fraction of oil f w ¼ 0.20 (ϕ ¼ 0.84) was used as a starting composition. [22] This blend was suddenly diluted with water while constantly stirring. The procedure allowed the synthesis of mother nanoemulsions (MN) with an average radius (R) of 72.5 nm, 5 wt% SDS, 3.3 wt% iP, and f w ¼ 0.62 (ϕ ¼ 0.44). These emulsions were then further diluted with an appropriate W/SDS/NaCl/iP solution until the final (formulation-composition map) conditions were attained (f w ¼ 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, and salt concentrations of 2, 4, 6, 8, and 10 wt% NaCl). An additional set of emulsions was prepared leaving a set of MN to evolve in time until their average radius reached R ∼ 500 nm [mother "mini"-emulsions (MM)]. As before, aliquots of MM were then diluted in order to prepare systems with the same physicochemical conditions of the smaller nanoemulsions (SNE), but with a larger particle radius (LNE). For the purpose of identification, the systems were labeled with two numbers separated by a hyphen, which indicate the weight percentage of water (% f w ) and the NaCl concentration (wt% NaCl), respectively.
To avoid the risk of perturbation and/or the contamination of a unique sample, a stock of each emulsion was divided into (1) a volume of 4 mL to measure the variation of the turbidity of the concentrated emulsion as a function of time (gravity effect) and (2) 15 vials stored at 25°C and used to study the evolution of the emulsion over a period of 6 hours. At a predetermined time, the content of each vial was appraised to establish the state of the emulsion. To do so, two aliquots were drawn in order to measure (1) the average radius of the emulsions and (2) the turbidity of a diluted sample. The whole procedure was repeated three times with independently prepared mother emulsions. In all cases, the final concentrations of SDS and iP were fixed to 5 and 3.3 wt%, respectively. In total, 30 SNE and 30 LNE were studied. Tables 1 and 2 show the composition and the initial average radius of each emulsion.

Droplet size measurement
The average size of the dispersions and their DSD were measured using a LS 230 from Beckman-Coulter, USA.

Drop size as a function of time
The overall stability of the systems was characterized by the slope of a cubic average radius as a function of a time curve.

Evaluation of k FC
At specific times, an aliquot was taken from one of the set of vials corresponding to each system. The aliquot was diluted with an aqueous solution of SDS in order to reach a volume fraction of oil equal to ϕ ¼ 10 À 4 ([SDS] ¼ 8 mM). The value of the turbidity was measured using a Turner spectrophotometer (Fisher Scientific, USA) at k ¼ 800 nm. [22,51] This procedure was repeated using different vials for a total of 6 hours. When the whole set of measurements was completed, Equation (9) was fitted to the experimental data of the turbidity (τ ¼ 230Abs) using Mathematica 8.0.1.0, Wolfram, USA. This study was only possible for systems with 2 and 4 wt% NaCl because destabilization occurred too quickly at higher salt concentrations.
The optical cross-sections required in Equation (9) were calculated using the theory of Rayleigh-Gans-Debye. [22,47,48,69,70] The theory is valid whenever Here, k is the wavelength of light in the liquid medium and m the relative refractive index between the particle and the solvent. In the case of a dodecane/water emulsion (m ¼ 1.07), the values of C RGD corresponding to radii, R ¼ 50, 60, 70, 80, 100, and 500 nm are 0.07, 0.09, 0.10, 0.12, 0.15, and 0.75, respectively. However, these values are relatively low and produce errors in the cross-sections lower than 10%. [69] The values of k FC , t 0,teo (theoretical starting time of the aggregation process), and x a were directly obtained from the fitting of Equation (9) to the experimental data. The initial particle radius is a parameter of the calculation that can be systematically varied (R teo ¼ R exp � δ) to optimize the fitting. The errors were calculated using the procedure described in ref. [22,51] . The effect of buoyancy during these measurements is believed to be small. [53] As mentioned above, the evaluation of k FC requires further dilution of the aliquots drawn from the concentrated emulsions. Hence, the number of particles per unit volume used in the determination of k FC corresponds to the dilute (d) system (k FC ; d ) and not to the actual, concentrated (c) emulsion (k FC;c ) under study. Simple arithmetic shows that k FC;d has to be multiplied by the dilution factor in order to obtain k FC;c : If an aliquot of volume V c is removed from a concentrated emulsion with an aggregate density n c , and diluted with aqueous solution until reaching a final volume V d , the new aggregate density n d fulfills the following relationship: This allows the definition of a dilution factor f d : Using the expression of Smoluchowski for a total number of aggregates at time t: the following equality is obtained: But for any time t, and in particular for t ¼ 0: Therefore, This means that the value of k FC obtained from the turbidity measurements of the dilute systems has to be divided by the dilution factor in order to get the rates of FC of the concentrated emulsions:

Independent FC rates
From the data of size variation as a function of time and considering a monodisperse emulsion, the change of the total number of drops (∼n) versus time was determined using n ¼ ϕ/V 1 , where V 1 is the volume of a spherical drop with radius R(t). ) were fitted to this data in order to obtain the corresponding rates.

Influence of gravity
The change of turbidity as a function of the height (H) of each emulsion was evaluated over 24 hours using a Quickscan (Beckman-Coulter, USA). Transmittance (T r ) profiles were obtained and analyzed in order to determine (1) the time required for the initial transmittance to increase up to 50% at half the height of the container (τ c (T r ¼ 50%)), and (2) the creaming rate: ν(R 0 , ϕ) (Equation 16) determined from the evolution of the creaming layer (ΔH) as a function of time (Migration software v1.3, Formulaction, France). This rate was calculated from the T r profiles corresponding to H < H max /2 (where H max is the total height of the emulsion in the container) using 50% of transmission as threshold. From these data, the software automatically calculated the creaming rate (ν(R 0 , ϕ)) and its corresponding correlation coefficient (r 2 ). This creaming rate was also used to compute the time needed for the drops at the bottom of the container to reach H max /2: τ c (ν(R 0 , ϕ)).

Stability regions
Stability regions were plotted on an formulation-composition map using a function of the Origin program. The program creates XYZ contours maps following four steps [71] : (1) Thiessen triangles are drawn to connect all xy data points.
(2) Characteristic points are found choosing a particular level z c , and identifying those vertexes of the triangles whose z-coordinates lay above and below z c . The x c , y c coordinates of the characteristic points result from linear interpolation. In the present case, a z c level is defined by the values of the stability measurements (log τ c in seconds). Once the lines that define the regions according to these values are traced, an appropriate scale of colors was selected in order to observe the trends of destabilization of the systems. Each level represents an increment of 0.5 log τ c . Figure 1 shows the formulation-composition map of a dodecane/water system with 5 wt% SDS and 3.3 wt% iP. The map is divided by a dash curve that crosses the diagram from the upper-right to the lower-left corner in a "step" fashion, separating the regions of low and high conductivity corresponding to W/O and O/W emulsions, respectively. At the extremes, (1) low f w < 0.2, NaCl <6 wt% and (2) high f w > 0.8, NaCl >13 wt%, abrupt changes in conductivity are observed. This behavior has been associated with the occurrence of multiple emulsions. [73,74] At the center of the map is an ample threephase region (a microemulsion phase in equilibrium with excess water and oil). Finally, an unexpected one-phase O/W microemulsion region was observed (probably Winsor-WIV [72] ) between 0.85 < f w < 0.90, and 8 wt% < NaCl < 10 wt%. This behavior has been previously reported at high values of f w and near the ternary zone. [75] The nature of the phase behavior of the systems is also described in Figure 1. Below the three-phase region and between 0.2 < f w < 0.9, a two-phase region consisting of a direct micellar solution (W m ) with an excess oil phase (O) is observed. Around 12 wt% SDS, the three-phase region is divided into two zones: (1) a lower subregion containing an O/ W microemulsion (W m ) phase, a bicontinuous microemulsion (D) and an excess oil phase (O); and (2) an upper subregion

General stability: Slope of R 3 versus t
According to Table 3, the slopes of dR 3 /dt corresponding to SNE (6.0�10 À 25 -6.6�10 À 22 m 3 /s) and the ones of LNE (2.0�10 À 25 -6.1�10 À 22 m 3 /s) are comparable: the order of magnitude is the same and the absolute value of one is within four times the absolute value of the other (under similar physicochemical conditions). There are only three exceptions to this rule, two of which correspond to the lowest regression coefficients found. Values between 10 À 22 -10 À 24 m 3 /s had been previously predicted by ESS for similar emulsions, subject to the predominance of FC. [42,44] In several cases, the slope increases with the salt concentration (Table 3) The curves on the right correspond to LNE and the ones on the left to SNE. Within the range of times studied, either the behavior is not linear (SNE 70-10,  or the data are rather disperse (r 2 < 0.90). The correlation coefficients of SNE are slightly higher than the ones of LNE and roughly increase for f w < 0.7.
Several emulsions (SNE: 70-10, 80-8, 80-10, 85-2, and 90-4; LNE: 65-2, 65-10, 70-6, 80-2, 80-6, and 90-2) show a slope of R 3 versus t that decreases progressively with time, producing a concave-downwards curve. This behavior has been associated in the past to FC: as the aggregates grow, the diffusion coefficient is expected to decrease and the electrostatic repulsion is anticipated to increase. [42,44] Hence, the instability of the system toward aggregation should progressively decrease. ESS of alkane/water emulsions typically produce concave-downward plots of R 3 versus t whenever the repulsive interaction between the drops is not negligible. [43,44] Figure 2 shows a contour diagram of overall stability, calculated with the slopes of Table 3. Unlike most contour diagrams shown in this paper, this particular data corresponds to a rate of destabilization and not to a characteristic time. Again, the results for both SNE and LNE are qualitatively similar. The diagrams show similar colors and trends. Based on the theory of aggregation, the stability is expected to decrease with the increase of the salinity and (less pronouncedly) with the increase of the volume fraction of internal phase (lower f w values). However, the most unstable region lies in the middle of the diagram along the diagonal direction. Unstable dilute systems are observed even at 4 wt% NaCl (Figure 2a). Remarkably, there is a fringe of  relative stability nearby the three-phase region (which lays above 10% NaCl; see Figure 1). Such behavior could be related to maximum surfactant adsorption. [76] Ostwald ripening According to LSW theory, the OR rate for a dodecane/water emulsion without surfactant is 1.3 � 10 À 26 m 3 /s. In the presence of SDS, values between 1 � 10 À 25 and 3 � 10 À 26 m 3 /s have been found (Kabalnov et al. [77] (ϕ ¼ 0.1, SDS 0.1 M); Taylor et al. [34] (ϕ ¼ 3.5 � 10 À 5 , SDS ¼ 3.5 � 10 À 5 M); Weiss et al. [35] (ϕ ¼ 0.4, SDS ¼ 3.3 M)). Surfactants are expected to decrease the OR rate due to the direct dependence of the capillary length on the interfacial tension. In any event, the referred values are significantly different from the ones shown in Table 3. The literature generally argues that whenever this occurs the system has not reached the stationary state for which Equations (17) and (18) are valid. [40,41] If despite the inconsistencies outlined above Equation (23) is used to appraise alternative OR rates, the contour diagrams shown in Figure 3 are obtained. In this case, the logarithm of the characteristic time of the process (log τ c (k p )) is shown. It is observed that the characteristic times calculated for SNE (0.7 < log τ c (k p ) < 3.6) are smaller than the ones of LNE (1.8 < log τ c (k p ) < 5.8) and do not show any monotonous trend.

Mixed FC rate
The methodology previously developed for the appraisal of k FC in dilute systems during a period of 60 seconds was modified to evaluate the rate of concentrated systems during a long period of time (see . Due to the high instability of the dispersions at NaCl > 6 wt%, the k FC rates could only be evaluated for the systems of lower ionic strength. Some emulsions required longer periods of measurement than others (16 hours for SNE 65-2 and 5 days for LNE 65-2, 70-2, and 75-2). In these cases, the use of shorter times of measurement produces negative values of x a . For most emulsions, a 6-hour period was sufficient. Figure 4 illustrates the quality of the fittings. The fittings were obtained using a symbolic algebra code (Mathematica, Wolfram, USA). The software does not provide regression coefficients for this type of nonlinear fitting, but does report standard deviations (see Table 4). Although most theoretical curves  approximate the experimental trend (Figure 4), the errors suggest that two SNE and most LNE do not reliably reproduce the theoretical model. However, the magnitude of the deviations could be partially the result of the relatively low density of the data collected.
As expected, most rates are equal or higher for 4wt% NaCl than for 2wt% NaCl. This is more significant in the case of LNE (see Table 4). When systems of similar composition but different sizes are compared, the differences are often larger than the ones previously found for the case of dR 3 /dt. The values of x a for SNE changed between 0.97 and 0.15. Higher values were exhibited by LNE (1.99 < x a < 0.71). This suggests a higher proportion of globular (spherical) aggregates in the case of SNE.
From the values of k FC , their corresponding characteristic times (τ c (k FC )) were computed using Equation (6) ( Figure 5). In general, the characteristic times of LNE are considerably longer than the ones of SNE. This is basically due to their lower number of particles per unit volume, which enhances the differences found between the rates of both sets of systems (see Equation 6). Despite the limitations outlined above, the characteristic times deduced from k FC ( Figure 5

Average FC rates
Whereas the formalism that supports the evaluation of k FC is sound, but the precise meaning of x a is uncertain, [52] the models of Smoluchowski, [3] Yang et al., [57] and Urbina-Villalba et al. [26] require the change of the number of aggregates as a function of time, which cannot be determined by ordinary techniques. Hence, the value of the average radii of the emulsions determined by light scattering was assumed to represent the average hydrodynamic radius of the actual aggregates, and hence, used to estimate the total number of aggregates at each time. Fitting of Equations. (3), (11), and (13) to the data of n versus t provided an average flocculation rate k S , [3] a pair of kinetic rates for the mixed aggregation-coalescence process, [26] and approximate values for the separate rates of FC (k F , k C ). [57] . According to previous ESS results [21,26] and analytical developments, [20] Smoluchowski's model can only reproduce the mixed process of FC in the absence of a repulsive barrier. Hence, it is not surprising that most regression coefficients shown in Table S-1 are appreciably lower than the ones of Table S-2. Notice that the range of values of k F (1.0 � 10 À 23 < k F < 6.6 � 10 À 20 m 3 /s) is similar to the one of k S (1.0 � 10 À 23 < k S < 5.6 � 10 À 20 m 3 /s) and overlaps with the range of experimental rates deduced from dR 3 /dt for OR (2.0 � 10 À 25 < k p < 6.6 � 10 À 22 m 3 /s). On the other hand, the values of k C between 3.0 � 10 À 20 and 5.9 � 10 À 17 s À 1 (10 14 À 10 11 days) indicate that coalescence does not occur during the time of study (6 hours). Although the small order of magnitude of these values suggests that a multiplication by n 0 is absent, [55,56] we did not find any mistakes either in the formula of Yang et al. or in the fitting of the theoretical expression. In addition, the original equation of Yang et al. [57] was formulated for the process of reversible aggregation. If coalescence does not occur, reversible aggregation is certainly a reasonable possibility. [54] The similarities in the distribution of the different stability regions between τ c (k FC ) (  Figure 3. According to the characteristic times, LNE shows higher mean lifetimes than SNE, that is, the stability increases as a function of size. The moderate differences between the rates of occurrence of each process are enhanced by the effect of the number density of particles in Equation (6). Figure 6 shows the isostability contours produced by τ c (T r ¼ 50 %). Some systems are not represented in the figure because they either reached the stability criteria in less than a minute (log τ c (T r ¼ 50%) < 1.8) or they did not reach the criteria during the time of the measurement (log τ c (T r ¼ 50%) > 5).

Creaming
In apparent contradiction to the behavior of the characteristic times resulting from the OR and the FC processes, no substantial differences are observed between SNE and LNE. This is rather remarkable since the creaming velocity depends on the second power of the average radius of the dispersion (Equation 15). Moreover, there are significant differences between the values of the creaming velocity calculated with Equation (15) and the ones experimentally observed (see Table 5). This is partially due to the erratic behavior of the thickness of the creaming layer as a function of time. Examples of these behaviors are shown in Supplementary Figure S-4, which illustrates the typical variations of the thickness observed. In some cases, the change of H versus t is smooth but not linear, and in others it is totally erratic. As a result, it is difficult to appraise the velocity of creaming from the slope of H versus t. However, these plots allow for determining the approximate time elapsed before the creaming process causes a sensible change in the transmittance of the dispersion. This time, referred to here as the "starting creaming time" (SCT) is the interval required for each system to attain a particle size distribution that can be sensibly affected by the effect of gravity.
The average radius of each emulsion at SCT can be computed from linear fittings of the experimental variation of the average radius as a function of time. These results are shown in Table 6. Notice that the increase in the average radius of the SNE systems is huge in comparison to the one of LNE systems. At the SCT, and except for the 90-4 dispersion, the average radius of the SNE systems varies between 228 and 1071nm,  while the one of the LNE systems spans between 340 and 1099nm. The radii are of comparable magnitude, and as a result, the rates of creaming at SCT are similar (see Supplementary Material, Table S-3), and the isostability contours produced by τ c (T r ¼ 50%) for SNE and LNE are also alike. The results of the previous paragraphs might mistakenly lead one to believe that the rate of destabilization of the SNE systems is higher than the one of the LNE systems. However, the SCT of the smaller emulsions is substantially longer than the one of the larger systems. As a result, if the increase of the average radius during SCT is taken as a measurement of stability (see Table S-3), contours plots similar to the one of Figure 2 are reproduced.
It should be remarked that due to the definition of the SCT any increase of the radius of an emulsion previous to this time does not include a substantial influence of creaming. In the present case, neither the creaming rate nor the characteristic times (defined in terms of the creaming rate) allow to discriminate between the stability of SNE and LNE.

Conclusions
Medical applications of nanoparticles are strongly dependent on particle size. [78][79][80][81][82] In most cases, this dependence is related to the requirements and characteristics of the biological systems under study. This paper is concerned with the inherent stability of the emulsions, which might also be related to the effectiveness of these systems in biological applications. The size dependence of emulsion stability is a matter of constant discrepancies. [83,[84][85][86] In the usual scenario, the deformability of large (micron-metric) drops plays a crucial role due to the time required for the drainage of the intervening film between flocculated particles. But the small drop sizes studied in this article disfavor the process of deformation due to the high internal Laplace pressure of the drops. Hence, a more basic dependence on size could be evidenced. However, as it is illustrated in Figure 2, the overall rate of destabilization of most nanoemulsions is similar. No general trend is observed when the rates of the different destabilization processes corresponding to SNE and LNE are compared. It is likely that the lack of a monotonous trend might be partially caused by the high instability of the systems under consideration, which foster large errors in the appraisal of some rates. However, large systematic differences in the characteristic times are observed between nanoemulsions of equal volume fraction of oil but distinct particle size. This fact strongly suggests that it is the number density of the particles that determines the characteristic time of destabilization. In liquid dispersions, the number density decreases as the particle size increases. Hence, the characteristic times of systems with larger average radius are longer.
While the hydrophilic-lipophilic deviation (HLD [87] ) concept is linked to the overall free energy of the ternary systems through its connection to the chemical potentials of the surfactant, the DLVO theory concerns the free energy of interaction between two particles. Hence, it is not surprising that the HLD concept is more closely related to the underlying phase behavior of emulsions. What is surprising is that the rates of destabilization are grossly connected with the HLD since there are multiple mechanisms with distinct kinetics that lead to phase separation. But again, it should be noticed that the isostability contours span several values of HLD (% NaCl in this case) as a function of the volume fraction of oil.
Bottle tests are traditionally used for the assessment of emulsion stability. They measure the time required for the separation of a certain amount of disperse phase in the path to equilibrium. As Smoluchowski's characteristic time, the lapse required for any destabilization process not only depends on the intrinsic rate of occurrence of the process, but also on the number density of the intervening particles (Equation 6). Comparison of the rates disregards the effect of the particle concentration and vice versa. According to the present results, nanoemulsions with larger particles sizes (LNE) are in general more stable than equivalent systems with smaller particle sizes (SNE). This difference in stability is caused by differences in both destabilization rates and number density of particles. However, it is the number density of particles that predominates. This parameter enhances the differences in the size dependence of the destabilization rates that appear to be relatively small for the range of particles studied (74-670 nm).