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The Gibbs elasticity modulus represents an important tool to predict the foamability for transient and permanent foams like polyurethane flexible systems. Elasticity is related to foamability and so is used as a synonymous for the purpose of this paper. In this article we propose a method and a thermodynamic model to analyze the espumability of silicone surfactants in polyol binary mixtures using surface tension data. The present work describes foamability through the Gibbs elasticity modulus expressed in terms of first and second derivatives of surface pressure vs bulk composition. Furthermore, the Gibbs adsorption equation and the corresponding novel surface equation of state based on a modification of the Langmuir isotherm resulted in an elasticity equation with analytical solution. It is shown that according to foam model systems of surfactant solution in polyol used at commercial processes, optimum concentration level of surfactant obtained at this article by Gibbs adsorption equation and maximus on elasticity modulus finally match.

Polyurethane foams are made by mixing an isocyanate and a polyol component [

Commercially polydimethylsiloxane based surfactants are of a polymeric nature with a natural spread of molecular weight. The exact structure is part of the proprietary knowledge of the surfactant suppliers (

Analytical grade nonyl phenol ethoxylates were purchased from Sigma and used with no further purification.

Silicone Tegostab BF-2370 provided by Goldschmidt AG (

Surfactant-polyol mixtures were prepared at controlled temperature of 30˚C. The samples were placed in a thermostated vessel during the surface tension measurements and the temperature was regulated (within ± 0.1˚C).

Surface tension methods A dynamic method for surface tension was chosen, the maximum bubble pressure method displayes surface tension based on instantaneous bubble formation [6-8]. This is a dynamic method selected for surface tension measurements due to polyurethane foam formulation taking seconds during its production.

Surface tension vs composition was measured using QC3000 SensaDyne Surface Tensiometer (within ±0.1 dyne/cm) fitted with a bath (Haake K20/DC30) allowing to control the temperature (within ±0.1˚C). An inert process gas (nitrogen or dry air) is bubbled slowly through two probes of different radio that are immersed in a test fluid. The bubbling of the nitrogen through the probes produces a differential pressure signal (ΔP) which is directly related to the fluid surface tension (σ).

The Young-Laplace equation rules the relation among curvature, surface energy and pressure difference between two phases; it has been used to describe spherical and non spherical shapes either in the absence or under the influenece of an external field. The general expression for the mechanical equilibrium between phases separated by spherical surface is

where r is the sphere radius and DP is the difference in the pressure between phases. The difference in the pressure P_{2} at the large probe from the pressure P_{1} at the small probe results in a differential pressure equation keeping the two probes at the approximate same immersion depth cancelling the effects of liquid level (

Foam formation test Foam was produced using a glass column of 1000 ml fitted at the bottom with a porous glass disk. Samples of the polydimethylsiloxane dispersions were carefully pured into the column and foam was produced at a constant temperature of 30˚C by passing gas trough the porous glass (pore diameter 0.2 μm) at a controlled flux between 20 and 60 ml/min during ten minutes (

Although surface properties for both kinds of polyols were similar (

Some works have been published describing methods to calculate equilibrium constants of molecular complexes in aqueous solution or to predict activity coefficients at infinite dilution from surface tension data [9, 10]. In this line it has been recently developed a model [

where p is surface pressure, Γ is the excess surface concentration and is equall to the reciprocal area, G_{s} is the maximum surface concentration, R is the universal gas constant, T is the absolute temperature and β is a measure of the lyophobicity, so, based on the Langmuir isotherm, where θ is the surface coverage,;

The Gibbs adsorption isotherm allows us to transform an isotherm to a surface equation of state [

The combination of Equations (4) and (5) and integral form leads to the corresponding surface equation of state SEOS

from the Lagmuir-Frumkin equation;

Equation (7) does not include atractive effects, so is proposed a general function f(θ);

That can be expressed as series expansion;

Obtaining the final Equation (3) to which the experimental data are fitted;

Polidimethylsiloxanes form transient foams in polyols, meaning that the foam is present as long as the gas flux is present and depended on concentration. Pluronic L61 showed lower foamability with bubbles breaking up as soon as the gas flux was interrupted. On the other hand Polypropylene glycol showed better foamability with a maximum that coincides with the critical concentration just as predicted for transient foams.

Gibbs elasticity modulus is closely related to foamability and is expressed as;

Г is the surface concentration and is equal to the reciprocal of the molecular area;

leads to the equation;

Surface tension expressed as surface pressure;

Expressions (12) and (13) can be substituted in Equation (10) providing;

Surface pressure can be used instead of surface tension and the associated area changed for specific area, thus

That can be expressed as;

From Gibbs adsorption equation;

Surface coverage can be used instead of surface tension, so

The first derivative with respect to mol fraction is;

By substituing relationships (18) and (19) into Equation (16) to express Gibbs elasticity modulus as;

The combination of Equation (3) and (20) leads to

Equation (21) can be expressed as follows;

The bulk concentration x is related to the surface concentration θ and the foam goes to maximum (

It has been shown that it is possible to characterize the surface tension of polydimethylsiloxane based surfactants in viscous model systems of a polyurethane foam forming medium. Even though both polyols don’t share same characteristics, the surface tension equilibrium was very similar when PDMS is involved as surfactant. However Pluronic L61 presented poor foaming behaviour explained by the rectification process ocurring along the foam column. PDMS showed transient foam behaviour with maximum foamability around concentration level

used in polyurethane foam formulations. Polypropylene glycol presented better foaming properties compared to Pluronic L61 which had poor foaming behaviour explained by the rectification process ocurring along the foam causing Pluronic a lower capability to re-establish the surface concentration limiting the adsorption to the newly created surfaces. Even though foam is a dispersed system and hence out of equilibrium, the hypothesis of local chemical equilibrium between lamellar solution and foam’s surface shows a proximity towards equilibrium conditions and allows for the thermodynamic description of elasticity; which for the purpose of this work was referred to foamability. The equation presented allows a direct relation to the Gibbs adsorption equation and is applicable for transient foams to predict concentration stability zone for polyurethane flexible systems that involve polidymethylsiloxane-poliol mixtures, where foamability decreases beyond a critical concentration. Foamability, or the foam volume produced by an amphiphile in a viscous medium like a poliol at a given concentration, is satisfactorily described by means of the Gibbs elasticity modulus (ε) presented in terms of the first and second derivatives of surface pressure and the bulk concentration.

This work was supported by Dirección General de Administración del Personal Académico (DGAPA) under the project PAPIIT-IT118711.