This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (

The first algorithm for Emulsion Stability Simulations (ESS) was presented at the V Conferencia Iberoamericana sobre Equilibrio de Fases y Diseño de Procesos [Luis, J.; García-Sucre, M.; Urbina-Villalba, G. Brownian Dynamics Simulation of Emulsion Stability In: ^{st} Ed., Tojo J., Arce, A., Eds.; Solucion’s: Vigo, Spain, 1999; Volume 2, pp. 364–369]. The former version of the program consisted on a minor modification of the Brownian Dynamics algorithm to account for the coalescence of drops. The present version of the program contains elaborate routines for time-dependent surfactant adsorption, average diffusion constants, and Ostwald ripening.

Emulsions are dispersions of two immiscible liquids, kinetically stabilized by the action of a surface-active substance known as a surfactant. The role of the surfactant in emulsion stability is decisive. Even the external phase of the emulsion resulting from the mixing of oil (O) and water (W) depends on the surfactant solubility [

In order to understand the role of the surfactant in the stability of emulsions, simulations can be a valuable tool. However, there are severe computational restrictions for the simulation of emulsions apart from the large number of processes involved. First, the number of surfactant molecules in a typical system is very high even at dilute concentrations. Hence, it is not possible to simulate the movement of surfactant molecules explicitly along with the movement of the drops. Second, the time step of the simulation has to be very small in order to sample appropriately the potential of interaction between the drops. Third, drops of small size exhibit Brownian motion due to their thermal interaction with the solvent [

In this review we concentrate on the evolution of oil-in-water (O/W) emulsions composed of non-deformable drops. The next section introduces the technique of Brownian Dynamics simulations and the most relevant aspects of the classical theory of Derjaguin-Landau-Verwey-Overbeek (DLVO). Section 3 describes the algorithm of Emulsion Stability Simulations (ESS) in detail. In section 4 some illustrative results of ESS are discussed. Following, the modifications of the former algorithm required for the simulation of deformable drops are outlined. The paper finishes with a brief conclusion and the bibliography.

The movement of one small particle (1 nm – 10 μm) diffusing in a quiescent media is the result of external and random forces. Random forces represent the effect of millions of collisions occurring between the solvent molecules and the particle surface. In the absence of external forces the energy for the displacement of a spherical particle is provided by the solvent, which at the same time takes away some energy from the particle in the form of friction. The whole process is a manifestation of the Fluctuation-Dissipation theorem that is expressed concisely in the form of the Stokes-Einstein equation [

In this equation, _{B}

The diffusion tensor can be evaluated from the resistance tensor: ^{−1}_{B}

The specific form of the diffusion constant of a sphere depends on the velocity of the liquid at the particle surface. If the surface is rigid and smooth, the velocity of the fluid becomes zero at the surface. In this case the diffusion constant comes out to _{0} =_{B}

In the case of liquid drops, porous spheres, and bitumen drops, the diffusion constant can be expressed as:

The addition of more particles to the system increases the complexity of the problem considerably. First, the random movement of the particles must be connected in such a way that they fulfil the Stokes-Einstein relation. Second, the movement of each particle generates fluxes (disturbances) in the solvent which affect the movement of the surrounding particles and its own. Thus, it is necessary to account for hydrodynamic interactions between the particles. Third, the particles interact with deterministic forces other than external forces. Thus, their movement is a combination of deterministic hydrodynamic and random forces.

One of the most widely used algorithms for Brownian Dynamics simulations is the one of Ermak and McCammon [_{i}_{j}_{ij}

Here _{j}_{i}_{i}_{j}_{ij}_{ij}

According to the Ermak and McCammon

When an average diffusion constant is used instead of a tensor, its value is de-coupled from the random deviates. Using this approximation and _{0} _{B}^{2}_{B}_{B}^{2}

The above considerations are profoundly related to the outcome of ESS calculations and its discussion is not a mere technicality. The most famous theory of colloidal stability, Derjaguin-Landau-Verwey-Overbeek’s DLVO theory [

The presence of a strong repulsive force between the particles generates a potential barrier and two minima at each side of the barrier (

On the one hand, primary minimum flocculation occurs at very small distances of separation and is assumed to be irreversible. Irreversible meaning that the aggregates formed do not separate if one lowers the ionic strength of the solution, increasing the repulsive force. This behaviour is due to the strong van der Waals force experienced by the particles at short distances of separation. On the other hand, secondary-minimum flocculation is usually reversible for small particles and could be “irreversible” for micron size drops [

The occurrence of primary minimum flocculation depends on the diffusive passage of the particles over the potential barrier. According to Chandrasekhar [_{f}_{A}_{R}_{T}_{A}_{R}

Fuchs demonstrated that the theory of Smoluchowski [_{0}

According to _{1/2}_{0}

In the case of non-deformable drops, primary minimum flocculation

Emulsion Stability Simulations start from a cubic box that contains

In _{eff,i}

In the case of non-deformable particles _{eff,i}_{ij}_{int} (where:
_{int} < _{ij}_{ext}_{eff,i}_{ij}_{ext}_{eff,i}_{eff,i}_{0}

The program includes several forms for
_{eff,i}_{0}

In _{i}_{0} a radius of a reference. The third term on the right hand side of

In ESS, non-deformable drops coalesce if the distance between their centres of mass gets smaller than the sum of their radii:

When this occurs a new drop is created at the centre of mass of the coalescing drops. The radius of the new drop results from the conservation of volume:

In ESS the attractive force between the drops is determined by the effective Hamaker constant of two oil drops separated by water [_{H}^{−21} J for hydrocarbons and lattices. In the case of Bitumen some old experimental evidence indicated a value of _{H}^{−19} J, but recent evaluations suggest a much lower value (_{H}^{−21} J).

The van der Waals potential for two spherical drops of different radius (_{1}_{2}_{1}, _{2}/_{1}, and _{ij}_{1}−_{2}.

There is always an attractive force between two drops of similar composition suspended in aqueous media. However, the repulsive force depends on the characteristics of the O/W interface and the type of surfactant adsorbed. Oil drops often exhibit an electrostatic surface potential between −10 mV (octacosane [^{−} ions to the oil/water interface. If this is the case, an initial value of the surface charge per unit area (σ_{0}) can be introduced as input in order to reproduce the total value of the surface charge _{T}

When ionic surfactants are present, they add their charge to the initial surface charge of the drop (σ_{0}

Here _{i}_{i}_{s}_{i}^{−19} Coul.). Γ stands for number of surfactant molecules per unit area at the oil/water interface. The interfacial area of one surfactant molecule at maximum packing (_{s}^{2} [

The effective charge of an ionic surfactant molecule (_{s}_{s}_{0}_{0}_{s}_{s}_{s}

The estimation of

The current version of the program contains four analytical expressions for the calculation of the electrostatic force [

In _{0} is the permittivity of vacumm, ^{−1} the Debye length, and Φ_{P}_{0} _{B}

Notice that knowledge of _{T}_{T}

Surfactants can be ionic, non-ionic, zwitterionic, and have a complex molecular structure. Hence, the interaction potential between the drops can vary amply. The program of ESS includes expressions for van der Waals, electrostatic, oscillatory, depletion, hydration, and steric forces.

In the case of non-ionic surfactants the stabilization force between emulsion drops is assumed to be steric [

The ESS code has several expressions for the calculation of the steric interactions [

In _{w}_{i}_{i}_{i}_{2}

As in the case of the electrostatic potential, the steric potential depends on some parameters and is a function of the surface excess.

The value of the surface excess of a surfactant at the interface of emulsion drops cannot be measured directly. It is extrapolated from the variation of the interfacial tension in systems with a macroscopic O/W interface. Unfortunately, emulsions are constantly evolving, continuously changing their drop size distribution and total interfacial area, _{T}

As a result, the surface excess is not constant and the interaction potential between the drops changes as a function of time.

Ionic surfactants are not soluble in the oil phase. They can migrate to the oil phase in the form of inverse micelles if the salt concentration of the system is unusually high. In the typical case, ionic surfactants adsorb to the O/W interface before they form micelles. The amount of surfactant required for the complete coverage of the interface can be larger or equal to the critical micelle concentration (CMC). It varies with the volume fraction of oil (φ) and _{T}_{T}

In the case of non-ionic surfactants the situation is more unpredictable. The partition of non-ionic molecules depends on the affinity of its lipophilic and hydrophilic moieties for the oil and water phases. For example, the partition coefficient of alkylphenol oligomers between water and n-alkanes is equal to [

The routines of surfactant distribution attempt to recreate the most common experimental situations. The strategy of ESS is to apportion the surfactants to the interfaces of the drops in such a way that it reproduces the variation of the surface excess in the experimental system. Consequently, only the movement of the drops is considered explicitly in

Some of the routines for surfactant distribution have a formal theoretical background [

This is the simplest methodology. It consists in ascribing to each drop a number of surfactant molecules (_{s}_{,}_{i}

Here _{s,T}

If the number of drops decreases as a consequence of coalescence, _{s}_{s,i}_{i}_{s}_{s,i}

In the case that the adsorption is basically controlled by the diffusion of surfactant molecules from the bulk to the subsurface [_{T}_{s}

The mechanism of surfactant adsorption can be very involved, including barriers of adsorption, reorientation at the surface, etc. However, Liggieri et al. [_{s}_{app}_{app}

The routine of time dependent surfactant adsorption uses _{app}_{T}_{T}_{app}^{1/2}. The number of surfactant molecules of drop

Here _{s}_{s}

Equilibrium isotherms are only attained after long periods of time. In this routine we assume that: a) the adsorption of surfactant is very fast, and b) equilibrium adsorption is obtained instantaneously.

Gibbs isotherm only applies to that range of surfactant concentration between the beginning of the decrease of the interfacial tension _{T}_{1}_{T}_{cmc} the maximum number of surfactants per drop is already adsorbed: _{s,i}_{T}_{1}_{T}_{T}_{1}_{T}_{0}.

In terms of finite differentials the equation of Gibbs is equal to:

_{1} stands for the value of the interfacial tension at _{1}_{i}_{s,i}_{i}

Although a large number of numerical techniques are available for the simulation of Ostwald ripening [_{i}

Here, _{o,i}_{oil}_{c}

Here

In _{m}

A simple equation for the exchange of oil molecules is obtained:

At any step of the simulation there exists a critical radius _{c}_{i}_{c}_{i}_{c}_{i}_{=} _{c}_{i}

In ESS the value of

Recently we implemented a new procedure to avoid a substantial decrease in the number of particles [_{0}_{1}_{0}_{i}_{i}_{i}_{i}

Use of _{0}

The algorithm for ESS is shown in

At the beginning of the simulation the code reads or generates a drop size distribution and a set of co-ordinates for each particle. Additionally, the time step(s) of the simulation (single or double) must be specified. A combination of a small time step and a large one is used for the calculation of dilute dispersions [_{min}) must also be selected. The maximum range of the interaction potential usually approximates this distance (50 nm ≤ d_{min} ≤ 100 nm). A double time-step calculation implies the use of a longer time step (Δ_{L}_{ij}_{min}. If the distance between two particles becomes equal or lower than d_{min}, all particles are returned to their previous positions, and a small time step (Δ_{S}_{S}_{L}_{ij}_{min}. This causes the continuous use of Δ_{S}

After the creation of the DSD, the program distributes the surfactant molecules among the drops. There are 11 routines to cover the most common experimental situations.

At this point the interfacial tension of each drop can be determined. The calculation of the diffusion constants is also executed here, because some of its expressions depend on interfacial parameters. First, the program assigns the diffusion constant of Stokes to all particles according to their radius. Second it makes the corrections necessary to account for the effect of the interfacial properties and hydrodynamic interactions on the diffusion constant (

Following, the forces are calculated and the drops are moved according to

If coalescence occurs, the total interfacial area of the emulsion changes along with the volume of one (or several) drop(s). Therefore the surfactant population had to be redistributed among the drops. The same occurs if Ostwald ripening happens. However, in the latter case a minimum number of drops must be maintained. Therefore, the program checks if the remaining number of drops is higher than a fixed value (_{1}_{0}

If the Ostwald ripening process is not selected, the program neither exchange oil molecules nor does it re-builds the drop size distribution when _{1}

It is important to remark at this point that the change of the number of particles during the simulations is only equal to the variation of the number of aggregates

Whenever a repulsive barrier is present, the change in the number of aggregates as a function of time has to be calculated at the end of the simulation using a different program [

The general results of the simulations can be classified into three categories depending on the total surfactant concentration of the system [

If the surfactant concentration is not enough to stabilize the initial drop size distribution, drops coalesce as soon as they collide with each other. Hence, the change in the number of particles is equal to the change in the number of aggregates [

The value of
^{−5} ≤ _{H}^{−21} J [

In the case of _{H}^{−19} J:

Notice that the value of
_{S}_{B}^{−18} m^{3}/s) based on the Brownian motion of the particles only.

The attractive force between the drops increases the value of
_{H}^{−21} J the effect of the attractive potential is small and the hydrodynamic interactions dominate (see _{H}^{−19} J (polar oils, metal salts) the attractive force dominates. In either case, the effect of the volume fraction remains. As

Notice that

The fact that

Inhomogeneous surfactant distributions as well as reversible adsorption, also lead to fast aggregation and to the coalescence of drops, favouring the validity of _{app}^{−12} m^{2}/s) [

The data of _{s}^{−5} M) is not enough to prevent the coalescence of drops during the course of the simulation. Hence, the number of single particles is equal to the _{a}_{agg}_{in agg}).

In general, the initial interfacial area of an emulsion is higher than the one that can be stabilised with the surfactant concentration available. In this case a Smoluchowskian drop in the number of particles occurs at short times. In the preparation of emulsions, chemical methods usually employ a larger surfactant concentration than mechanical ones. This favours the formation of a repulsive barrier as soon as the drops are formed. In this case, the dynamic of aggregation is not expected to follow

According to ESS of non-deformable droplets, the variation of the number of aggregates in the presence of an appreciable repulsive barrier conforms to a remarkably simple expression [_{1}_{2}_{1}_{2}_{c}_{a}_{a}

We have also compared the prediction of

Series expansions of the exponential term and the flocculation function of _{1}_{2}

It was also found that the interfacial area of the emulsion after ∼200 seconds was proportional to the number of surfactant molecules. The slope of this curve provides a measurement of the area per surfactant molecule that is required (∼ 236 Å^{2}) in order to avoid an initial pronounced drop in the number of particles. Using the effective charge of a surfactant molecule (0.21e), the corresponding value of the surface charge can be obtained (σ = 13.95 mCoul/m^{2}). The electrostatic surface potential can also be calculated (Ψ_{0} ∼44 mV) using the relation between σ and Ψ (_{i}

From the variation of _{a}_{a}_{a}

We believe that the drastic change in the slope of _{a}_{i}

A more conventional effect of the repulsive barrier was observed in the simulation of hexadecane-in-water nanoparticles (_{i}

This equation is very similar to _{a}

_{a}

_{B}

According to Verwey and Overbeek [_{B}_{0}^{14} m^{−3}). This corresponds to ^{9}. Much higher values of the stability ratio (10^{10} – 10^{17}) are found in the literature published between 1917 and 1940 (see Refs. [

The wide range of stability ratios reported during the past century is partially caused by the inadequate definition of _{S}_{B}

We studied this problem in Ref [

The simulations indicated that the systems with barrier heights higher than 20 _{B}_{a}_{a}

Recently Kuznar and Elimelech studied the deposition of micrometer-sized particles on a single layer of packed glass beads using a flow-cell [_{B}_{B}_{B}_{B}

The experimental evidence described above strongly supports the predictions of Ref. [

In

For the evaluation of

These calculations are the simulation analogue of the numerical evaluation of

As shown in _{B}_{B}

In our view, the agreement between theory and experiment shown in

On the one hand a 0.1 g/L solution of sodium dodecyl sulfate (SDS) causes a drop of 15 mN/m in the interfacial tension of a dodecane/water system in a period of 100 s [_{T} = 0.014 g/L) achieves the same fall in only 6 s in the presence of 0.1 g/L NaCl. Which chemical condition is more effective for stabilising a dodecane-in-water emulsion considering that the electrostatic interaction is screened in the latter case?

In order to throw some light into this problem, we computed the evolution of a set of oil-in-water emulsions with volume fractions between 0.10 ≤ ^{−12} m^{2}/s ≤ _{app}^{−9} m^{2}/s, and two surfactant concentrations _{T}^{−4} M, and _{T}^{−4} M [^{−10} m^{2}/s, unless their molecular weight is very large. Ionic surfactants show kinetically limited adsorption [^{−10} m^{2}/s. Furthermore, according to Rosen et al. [^{−4} M is the lowest surfactant concentration required to achieve a 1-second surface tension reduction that does not change much if the surfactant concentration is increased. In these simulations the surfactant was assumed to be ionic. Therefore, the interaction potential between the drops was supposed to be DLVO. Up to our knowledge this was the first off-lattice simulation that took into account the effect of time-dependent surfactant adsorption along with the explicit movement of the drops. The question we addressed in that article was: Under what conditions of _{T}_{app}

The stability of an emulsion depends on the balance between: a) the mean time between the collisions of the drops, and b) the time necessary for a

A close look at _{f}_{f}_{H}^{−19} J, and _{f}

The time required for an effective surfactant adsorption can be roughly estimated from _{s}_{s}^{2}):

_{msa}_{app}_{T}_{app}

Comparison of the values of _{f}_{msa}_{app}^{−10} m^{2}/s) can stabilize emulsions with ^{−4} M is used. The lower surfactant concentration (_{T}^{−4} M) can only stabilize emulsions with _{app}^{−12} m^{2}/s, a concentration of 5 × 10^{−4} M can only stabilize an emulsion with

If the surfactant concentration is high _{T}^{−4} M, and its diffusion constant fast _{app}^{−9} m^{2}/s, the initial drop size distribution of the emulsion is preserved [_{app}^{−12} m^{2}/s, _{T}^{−4} M, complete destabilisation of the system is obtained. The most common cases happen between these extreme situations. They show an initial decrease in the number of drops due to the early destabilisation of the former DSD. Destabilisation occurs faster when the mean free path between the drops is small (high volume fraction, inhomogeneous spatial distribution of drops, etc.). As time passes, surfactant adsorption progressively increases until it reaches a point in which the repulsive potential generated by the surfactant is enough to preserve the current DSD. At this point the kinetic rates of coalescence and flocculation markedly change. The variation of number of aggregates as a function of time follows the prediction of _{i}

In our calculations, _{T}^{−4} M is the lowest surfactant concentration able to generate a substantial repulsive potential for _{app}^{−10} m^{2}/s [^{−4} M.

As _{app}_{T}^{−4} M and

We also observed a marked effect of the hydrodynamic interaction in the most concentrated system (

It should be remarked that the effect of the ionic strength was not considered in Ref. [_{T}^{2} _{H}^{−21} J,

Furthermore, it should be noticed that the value of the Hamaker constant used in the simulations of Ref. [_{H}^{−21} J) should decrease the flocculation rate, diminishing the effect of the time dependent adsorption even more.

Taking into account the results of the simulations [_{T}^{−4} M, mostly between 0.20 ≤_{T}_{app}

The Liftshitz-Slezov-Wagner (LSW) theory of Oswald ripening, predicts a linear variation of the cube of the average radius of a dispersion as a function of time (
_{c}

The particles are fixed in space.

The system is infinitely dilute (implying the absence of interactions).

The molecules of the internal phase are transported from one particle to another by molecular diffusion.

The concentration of oil is the same through the whole system except in a direct neighbourhood of the particles, where it is given by the Kelvin equation.

LSW does not provide analytical expressions for the variation of _{c}_{OR}

Sakai ^{−4}) a _{OR}^{−26} m^{3}/s was found. This rate is three times larger than the theoretical estimation (1.3 × 10^{−26} m^{3}/s). Moreover, the DSDs of all measured systems were skewed to the right, in contradiction with the predictions of LSW. Using freeze-fracture electron microscopy, Sakai et al. [

In Ref. [

The first set of simulations followed the original algorithm of De Smet [_{c}_{c}_{c}

We carried out similar calculations with the ESS program including Ostwald Ripening (ESS+OR simulations). For these simulations 1000 non-deformable drops were used. The potential was completely attractive, characterized with a Hamaker constant, _{H}^{−21} J. When the movement of the particles is incorporated, the simulations showed that ^{3}^{3}_{FC}^{−22} m^{3}/s (r^{2} = 0.9979). This rate is four orders of magnitude higher than the theoretical value of _{OR}

This finding motivated the measurement of _{c}^{−4}) were prepared using an aqueous solution of NaCl (0.5 M) to screen the electrostatic charge produced by the preferential adsorption of hydroxyl ions. Immediately after sonication, the light scattered by the emulsion was measured at 90° using a BI-200SM goniometer (Brookhaven Instruments). The mean diffusion coefficient was derived from the intensity autocorrelation function using a cumulant analysis [

^{−23} m^{3}/s was found for ^{3}_{OR}^{−26} m^{3}/s). Moreover, if the emulsion is measured for a longer period of time (∼200 s), a small temporary plateau is reached, and the typical order of magnitude of OR (10^{−26} m^{3}/s) is recovered.

The above result demonstrated that it is possible to obtain a linear relation of

In Section

The average size of the particles in the system can be calculated substituting

Smoluchowski supposed that the dispersion was initially composed by particles of equal radius. Hence, some prescription was necessary to connect the size of the aggregates with the initial particle radius (_{0}). For example:

For coalescing emulsions _{i agg}_{0} produces a very simple formula:

If the aggregates formed are not linear and/or coalescence occurs, ^{3}_{F}_{F}_{f}

Notice that a real exponent _{0} (1 + _{f}_{0} ^{p}_{0} (1 + _{f}_{0} ^{1/}^{Df}_{f}^{3} =
_{f}_{0} _{f}

The data shown in _{F}_{f}_{F}_{f}_{0}
^{3}^{3}

The problem of drop deformation is formidably complex. Deformation results from the interplay between hydrodynamic and interaction forces. Hence it depends on all the parameters that affect these forces. As soon as deformation occurs, the geometrical part of the interaction potentials changes. Analytical forms for the potentials of truncated spheres are available in the literature, but they are function of the radius of the film between flocculated drops and its width. Moreover, the process of deformation itself involves several stages including the formation of a dimple, its evolution to a plane parallel film and the destabilisation of the film either by surface oscillation or by the formation of holes [

The present version of the ESS code has routines for non-deformable and deformable drops, but cannot simulate both types of drops during the same calculation. Thus, if the mode of deformable droplets is selected, it is assumed that the deformation of the drops occurs independently of the energy required for this process. Both types of simulation use the same equation of motion (

In the mode of deformable droplets, the drops move as spheres until they reach the initial distance of deformation _{0}_{ij}_{i}_{j}_{0} the code calculates the dimensions of truncated spheres that are compatible with _{0}_{ij}_{crit}

In order to calculate the forces and diffusion constants three regions of approach are defined:

Region I: The distance of separation between the centres of mass of the drops _{ij}_{,} is greater than _{i}_{j}_{0}. In this case the drops maintain their spherical shape. Consequently:

Region II: It covers the range of distances between the beginning of the deformation _{film}_{film}_{f}_{max} =
_{i}_{j}_{0}

Region III: The film already attained its maximum radius, _{film}_{f}_{max} =

Notice that the drops behave as spheres in Region I (_{ij}_{i}_{j}_{0}) even in the case in which the mode of deformable drops is selected. This means that the potential of interaction and diffusion constant in Region I correspond to spherical particles. When the distance of separation between the surfaces of the drops is less than _{0}

As soon as deformation occurs, the analytical forms of the interaction potentials change. Analytical forms of the potentials for truncated spheres are available in the literature. However, they are expressed as a function of the radius of the film between flocculated drops and its width [

where: _{1}=_{1}+
_{2}=_{2}+
_{film}

Different types of interaction potentials for truncated spheres are found in the literature [_{film}_{ij}_{ij}

Additionally, two new contributions to the free energy appear. The first one is called dilatational or extensional energy and is caused by the increase of the interfacial area of the particle as it looses its spherical shape. The analytical form for the extensional potential corresponding to the change between a sphere and a truncated spheroid is [_{film}_{,}_{i}

The variation of the interfacial curvature requires an additional amount of energy. This contribution can be positive or negative depending on the value of the spontaneous curvature of the interface:
_{c,0}

where

and _{0}

Constant _{0}_{b}^{−11} N for _{0}_{B}_{0}_{B}_{0}^{−12} N.

The potentials of surface deformation occur at close separation distances, within or nearby the same region of influence of the other potentials of interaction like electrostatic, steric, etc. In the absence of other repulsive potentials, deformation (dilatational + bending) generates two minima in the interaction potential separated by a repulsive barrier, similar to the ones exhibit by the DLVO potential (see

It is not yet possible to deduce a general analytical expression to relate _{film}_{ij}

It should be noticed, that _{film}_{,}_{i}

Accurate estimation of _{0}_{crit}_{film}_{0}_{0}_{0}_{i}

_{0}_{f}_{max} cannot be larger than the radius of the smallest drop (_{f}_{max} =
_{0}_{f}_{max}.

In regard to _{crit}_{crit}_{film}

The form of the diffusion constant employed at every distance of separation corresponds to the shape of the approaching drops. Within Region I the same formalism used for non-deformable drops is employed [_{T}_{film}_{B}_{T}

Most equations compiled by Gurkov and Basheva [

Thus, the analytical form of
_{ij}_{int} is modified as soon as the drops change their spherical shape to form truncated spheres with a plane parallel film. Hence, the additional friction generated by the creation of two planar disks between flocculated drops, decreases the diffusion constant beyond the estimation of Honig et al.

The process of coalescence of deformable droplets is involved. It is known that thin liquid films not necessarily drain until reaching _{crit}

Depending on the properties of its interfacial layers, films can collapse through the formation of holes [

By default, the mechanism of coalescence of deformable drops is the one of film drainage. As the drops approach, a plane parallel film forms and thins until _{crit}

It is known that thin liquid films of macroscopic radii experience surface oscillations. In some circumstances the interfacial waves at each O/W interface may grow until they touch. In this case, a channel between the oil phases of the drops could form causing their coalescence. This mechanism is additional to the one of film drainage. Other possible process like the condensation of holes in common black films are also known to promote film rupture [

If the additional mechanism of surface oscillations is selected, a random number is assigned to the surface of each drop every time a pair of particles enters Regions II or III above. The random number varies between −1.0 and 1.0 units.

The amplitude (height) of each capillary wave (_{crit}

The time of existence of a doublet (_{ij}

_{ij}_{crit}_{ij}_{i}_{j}_{0}. If the doublet separates _{ij}

_{ij}_{ij}

At each time step, the value of _{ij}

_{Vrij}

The value of the tension in _{i}_{j}_{i}

Here, _{0}, and _{cmc}_{s,i}_{i}_{0}.

Coalescence occurs whenever the total height of the surface oscillations is greater than _{crit}

It is clear from above, that all coalescence mechanisms are defined in terms of doublets. If aggregation of multiple particles occurs,

The force routine is divided into two parts to account for non-deformable and deformable droplets. In the case of deformable droplets, the program estimates the value of _{0}_{ij}

The incorporation of deformable droplets is very recent [_{0}_{0} ≤15 nm). The particle radius was changed between 100 nm and 100 μm, for γ = 1 mN/m. Within these limits, the diffusion constant used (

For the 96-nm particles of ^{−4} s and 4.81 × 10^{−3} s (_{H}^{−21} J, _{H}^{−20} J, _{0}

The coalescence time between two _{i}_{i}_{i}_{i}_{c}_{i}_{c}_{i}_{0}_{f max}_{c}_{i}

In order to compare the effect of deformation on the rates of aggregation and coalescence, we evaluated an average (mixed) rate based on the variation of the number of

For _{FC}^{−17} m^{3}/s) is two orders of magnitude higher than the one of deformable droplets (5 × 10^{−19} m^{3}/s). For ^{−16} m^{3}/s and 4 × 10^{−17} m^{3}/s, respectively).

This review describes in detail the algorithm of Emulsion Stability Simulations developed by our group. It was not until very recently that we completed the routines for deformable droplets and Ostwald ripening. Consequently, it is now that we are ready to study the role of the surfactant molecule in systems exhibiting the concurrent occurrence of most destabilization processes. This is why we are confident that the most outstanding results from ESS are yet to be produced. However, we had shown that the previous studies related to the behaviour of non-deformable drops are very insightful. In particular they had remarked the role of the secondary minimum in the flocculation of drops, suggested a new threshold for the coalescence of drops, and demonstrated that the cube of the average radius of an emulsion can change linearly with time as a consequence of flocculation and coalescence.

The technical assistance of Ms. K. Rahn is gratefully acknowledged.

^{3}vs

Classical DLVO potential. The curve corresponds to the interaction potential between two 3.9-μm drops of dodecane (_{H}^{−21} J) suspended in water. The drops are partially covered with sodium dodecyl sulfate (_{s}^{−4} M, _{s}

Flowchart of Emulsion Stability Simulations.

Smoluchowskian decrease of the number of aggregates as a function of time. Subscripts “_{a}_{in agg}), and the number of flocs (_{agg}), respectively (_{H}^{−19} J).

Behavior of system #13 from Refs. [_{s}^{−5} M, _{H}^{−19} J).

Variation of the number of particles as a function of time for a dodecane in water nano-emulsion stabilized with Brij 30 (

DLVO potential (red line) between two spherical drops of bitumen (_{i}^{2} (Ψ_{0} ∼44 mV, _{H}^{−19} J). The ionic strength of the solution is equal to 1.4 × 10^{−2} M. The van der Waals potential is shown in blue.

Potential energy between two spherical drops of bitumen (_{i}^{2} (Ψ_{0} ∼44 mV, _{H}^{−19} J). The ionic strength of the solution is equal to 1.4 × 10^{−2} M.

Behavior of system #12 from Refs. [_{s} = 4.10 × 10^{−5} M, _{H}^{−19} J).

Experimental variation of the cube average radius as function of time. The data corresponds to a dodecane-in-water nanoemulsion produced by sonication.

The curves correspond to the interaction potential between two deformable drops of dodecane suspended in water (_{i}_{H}^{−21} J, _{s}^{−4} M, γ = 48.5 mN/m, _{0}^{−12} N). (a) Sum of Extensional, Bending and van der Waals potentials. (b) Sum of Electrostatic and van der Waals potentials.

Time required for maximum surfactant adsorption (_{msa}

_{app}^{2}/s) |
_{T} |
_{msa} |
---|---|---|

1 × 10^{−9} |
5 × 10^{−4} |
0.03 |

1 × 10^{−9} |
1 × 10^{−4} |
0.87 |

1 × 10^{−10} |
5 × 10^{−4} |
0.35 |

1 × 10^{−10} |
1 × 10^{−4} |
8.67 |

1 × 10^{−12} |
5 × 10^{−4} |
34.7 |

1 × 10^{−12} |
1 × 10^{−4} |
867 |

Relationship between _{f}_{F}_{0}_{0}^{18} m^{−3}).

_{f}^{3}/s) |
_{F}^{3}/s) |
||
---|---|---|---|

7.37 × 10^{−18} |
3.11 × 10^{−22} |
0.78 | |

6.62 × 10^{−18} |
2.58 × 10^{−22} |
0.72 |