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A unified nonlinear fractional equation of the diffusion-controlled surfactant adsorption: Reappraisal and new solution of the Ward–Tordai problem
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Received: ,
Accepted: ,
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.
Peer review under responsibility of King Saud University.
Abstract
The article addresses a reappraisal of the famous Ward–Tordai equation describing the equilibrium of surfactants at air/liquid interfaces under diffusion control. The new derivation is entirely developed in the light of fractional calculus. The unified approach demonstrates that this equation can be clearly reformulated as a nonlinear ordinary time-fractional equation of order 1/2. The work formulates versions with different isotherms. A simple solution of the case with the Henry’s isotherm and a discussion of a Cauchy problem involving the Freundlich isotherm are provided.
Keywords
Ward–Tordai equation
Time-fractional equation
Fractional semiderivative
Henry isotherm
Cauchy problem
Introduction
Surfactants are broadly encountered in aqueous systems used as cleaning and wetting agents, dispersants stabilizers, lubricants, foam stabilizers, catalysts, as well as to stabilize pharmaceutical, cosmetic and agrochemical formulations, etc. (Davies and Rideal, 1963; Lenzi et al., 2005; Gaines, 1966; Gosh, 2009). In such systems molecules attempt to be at a position in the fluid where there are forces of attraction in as many directions as possible thus attaining local dynamic equilibria. For the molecules located at the surface, however, there exist forces directed inwards to the fluid which are not balanced outwards. As a result, the fluid attempts to minimize the free area due to resisting expansion. This cohesive feature of the fluid can be measured as a force per unit of length of the interface, and it is known as surface tension. The dynamics of the surface tension depends on the amount of molecules of surfactant accumulated (adsorbed) at the interface. The classical equation of Ward and Tordai (1946) describes the transient in the surface adsorption of surfactant when the supply of surfactant molecules is under diffusion control from the fluid to the fluid-air interface. This work stresses the attention on a unified re-formulation of the Ward–Tordai equation in a straightforward manner entirely in the light of the fractional calculus.
Physical background
The amount of surfactant adsorbed at an air/water interface is usually calculated indirectly from interfacial tension measurements (Dudnik and Lunkenheimer, 2000; Dannov et al., 2000). The surfactant concentration of the liquid bulk C0 and its equilibrium surface tension
corresponds to the equilibrium amount at the interface
are interrelated by the Gibbs equation (Gaines, 1966).
Isotherm
Equation of state
Henry
Langmuir
Frumkin
Freundlich
Volmer
Initially the surface is cleaned so that the initially adsorbed amount at and the surface tension is that of the solvent . The system is out of equilibrium and will return to the equilibrium state. In this context, let us consider an aqueous surfactant solution in equilibrium with its air–water interface. The surfactant concentration is and its equilibrium surface tension corresponds to the equilibrium surfactant concentration denoted as , interrelated by the Gibbs Eq. (2). The system is out of equilibrium and will return to the equilibrium state. Hence, the surfactant molecules will be transported to the surface by diffusion (Ward and Tordai, 1946; Baret, 1968; Mysels, 1982; Li et al., 1994; Campanelli and Wang, 1998; Liu and Messow, 2000; Liu et al., 2009).
Ward–Tordai equation: the common approach at a glance
Consider a process entirely controlled by the diffusion transport through the stagnant fluid (Ward and Tordai, 1946; Baret, 1968; Mysels, 1982; Borwankar and Wasan, 1983; Li et al., 1994, 2010; Campanelli and Wang, 1998; Liu and Messow, 2000; Liu et al., 2009) and instantaneous adsorption of the surfactant molecules at the interface. When the diffusion through the bulk of the liquid is linear, then the Fick’s second law describes the transfer of the surfactant to the surface, with initial and boundary conditions presented by the model.
The final solution of the model (3) with help of (4) and the imposed boundary and initial conditions at
(short times) is (Ward and Tordai, 1946; Baret, 1968; Liu and Messow, 2000; Liu et al., 2009; Li et al., 2010).
The development of the Ward–Tordai equation (5a) is frequently referred to a solution of the model (3) by the Laplace transform (Hansen, 1960; Borwankar and Wasan, 1983; Chang and Frances, 1995; Kralchevsky et al., 2008) which differs from the original approach of Ward and Tordai (1946) and underlying solution of Carslaw (1921) using Green functions. Some specific point will be commented next.
In order to be correct, it is worthy to mention that Ward and Tordai (1946) have started the solution by directly applying the result developed by Carslaw (1921) for a problem in heat transfer analogous to the model (3) and involving a convolution integral (like that in (5a). For the readers familiar with the book of Carslaw and Jaeger (1959), Ward and Tordai used the solution of problems 2.5 and 14.2, exactly equation (2) in section 14.2. Further, looking for an expression of the gradient of the subsurface concentration
from this solution Ward and Tordai applied Maclaurin’s theorem. Consequently the subsurface gradient was expressed as:
Ward and Tordai represented the right-hand side of (6a) as
involving differentiation and integration of fractional order (sic!), without a reference source. Because the solution needed to find the time-derivative of
, they applied the operator
(sic!) to
that led to:
The first independent solution of the model (3) has been developed by Sutherland (1952) with a linear relationship
where
is average defined as
and considered independent of concentration. Then the boundary equation becomes:
Alternatively, a solution based on the reflection and linear superposition has been developed independently by Mysels (1982). Many attempts have been applied to develop the Ward–Tordai equation (Petrov and Miller, 1977) and to solve it for various adsorption isotherms (as well as to solve the original model (3)) among them: analytical solutions by series presentation of the convolution integral (Hansen, 1960; Petrov and Miller, 1977; Ziller and Miller, 1986), orthogonal collocation (Ziller and Miller, 1986) and numerically by an implicit difference method (Miller, 1981; Borwankar and Wasan, 1983; Chang et al., 2006; Li et al., 2010), finite element method (Fenandez and Muniz, 2011; Fenandez et al., 2012a,b). A comprehensive review of the existing models and possible analytical solutions is provided by Chang and Frances (1995).
The equation of Ward and Tordai is not enough to describe the adsorption process at the air–water interface because it relates two unknown functions and . In fact, is a function of at the relationship depending on the equilibrium isotherm describing the process at the air–water interface (see Table 1).
Besides, Eq. (5b) cannot be considered as a simple Abel equation (Linz, 1985), because of the nonlinearity imposed by the function . The main problem in the solution of the Ward–Tordai equation and the proper evaluation of comes from the fact that is in a convolution integral with a weakly singular kernel and simultaneously depends on . If the subsurface concentration is known independently through the adsorption process, then can be calculated immediately. In this context, Johansen et al. (1991), for instance, have suggested empirical forms of the subsurface concentration expressed by exponential functions: and . The parameters and are overall measures of the diffusion, adsorption, and desorption rates and are determined from the transients in equilibrium adsorption experiments.
Aim and article structure
This article presents a reappraisal of the model (3) in light of the fractional calculus that finally yields a nonlinear time-fractional ordinary equation of order 1/2. The article demonstrates an alternative derivation of the Ward–Tordai equation by using time fractional semi derivatives of Riemann–Liouville (Section 2). Further, the analysis in Section 3 allows developing a time-fractional nonlinear ODE analogous to the Ward–Tordai equation with a non-linear term depending on the type of the adsorption isotherm describing the equilibrium at the surface. Section 4 deals with the derived time-fractional equation in case of the Henry isotherm and analyses possible solutions. Section 5 addresses the general Cauchy problem pertinent to the formulated fractional ODE and the case of the Freundlich isotherm. The discussion section analyzes the results developed and formulates new problems.
New development of the Ward–Tordai equation by time-fractional semiderivatives
Now, we present an alterative solution of the model (3) by the tools of the fractional calculus only, directly leading to the Ward–Tordai equation. The transport of the surfactant from the bulk to the surface is described by Eq. (3a) which can be represented as (Babenko, 1984):
The long-time solutions of (14a, b) as well as of (17a) with various non-linear relationships
are special, not straightforward resolvable tasks, and some of them will be discussed next. However, a simplification for
can be expressed as (Hansen, 1960; Daniel and Berg, 2001; Kralchevsky et al., 2008):
Certainly, the new approach to derive the Ward–Tordai equation presented in this section is straightforward, starts from the basic model (3) and does not use underlying solutions of similar problems taken from other sources. Moreover, it is entirely developed by the tools of the fractional calculus.
Formulation of a unified nonlinear fractional equation
Even though we have developed the Ward–Tordai equation in a simple manner, the equation of the adsorption isotherm has to be accounted for in order to accomplish the solution of the problem. Moreover, since we stress the attention on application of fractional calculus, this section demonstrates that it is possible to create a unified time-fractional equation describing the time evolution of the surfactant adsorbed at the interface. This equation is equivalent to the Ward–Tordai equation but now it is in a form which is “readable” by people solving fractional calculus models.
Now, starting from Eq. (17a) and taking into account that the relationship
is the adsorption isotherm, we read:
Equations
Coefficients
Henry
,
Langmuir
, for
Frumkin
,
Freundlich
,
,
Volmer
Solution examples
The Henry isotherm
Even though this is the simplest case we will use it to demonstrate how the new developed time-fractional ordinary equation (20a) relates to existing solutions. With
and
we have
and
equation (20a) reads:
Zero initial condition
With initially clean interface, that is
and applying the Laplace transform to Eq. (21) we get:
Non-zero initial condition
When an amount of surfactant
exists at the interface, then we have
. In that case, the semiderivative of surface excess of surfactant is
(Oldham and Spanier, 1974) and we have:
Short time solution
The widely used short-time approximation of the Ward–Tordai equation is presented by (17b). Now, using the Laplace transform solution of the problem with zero initial condition (22a) we may develop an asymptotic series (restricted to 3 terms only for seek of simplicity of the analysis), namely:
Formulation as a Cauchy problem and the case of the Freundlich isotherm
Certainly, the original Ward–Tordai equation (5a) is a Cauchy problem determining locally and uniquely the solution of the model (3). As commented by Baret (1968), the solution of (3a) and the Ward–Tordai equation (5a) are, in fact, the compatibility relations between the Cauchy’s condition and the condition (3d).
Let us consider the linear fractional differential Eq. (20a) with the Freundlich isotherm expressed in the form:
It was proved by Kilbas et al. (2006) that if the condition:
In general, it is proved (Kilbas et al., 2006) that the condition (32) is equivalent to:
Equation excludes the case of as it is stated by (37b) due to a singularity in , but this case corresponds to Henry’s isotherm and the straightforward solution is presented by (25) and (28).
Moreover, for
, for instance, that is
representing a convex Freundlich isotherm, we have from (38b) that
and then the solution (38a) is a simple square-root law of the time, namely
Conclusions
The article performed a reappraisal of the famous Ward–Tordai equation entirely developed in terms of fractional calculus. The unified approach demonstrates that Ward–Tordai equation can be clearly reformulated as a nonlinear ordinary time-fractional equation of order 1/2. In addition, the approach used allowed to formulate versions with different isotherms. The simple solution of the case with the Henry’s isotherms is provided. The Cauchy problem involving an example with the Freundlich isotherm is discussed.
References
- Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dyn.. 2004;38:191-206.
- [Google Scholar]
- Heat–Mass Transfer: Methods for Calculation of Thermal and Diffusional Fluxes. Moscow: Khimia Publ.; 1984. (in Russian)
- Kinetics of adsorption from solution. Role of diffusion and the adsorption–desorption antagonism. J. Phys. Chem.. 1968;78:2755-2758.
- [Google Scholar]
- The kinetics of adsorption of active surface agents at gas–liquid surfaces. Chem. Eng. Sci.. 1983;38:1637-1649.
- [Google Scholar]
- Comments on modelling the diffusion controlled adsorption of surfactants. Can J. Chem. Eng.. 1998;76(2):51-57.
- [Google Scholar]
- Mathematical Theory of Heat conduction. London: MacMillan; 1921.
- Conduction of Heart in Solids (second ed.). Oxford: Oxford University Press; 1959.
- Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data and mechanisms. Colloids Surf., A. 1995;100:1-45.
- [Google Scholar]
- A model for simulating the dynamic surface tension behaviour of aqueous surfactant dispersions. Colloid Polym. Sci.. 2006;285:57-63.
- [Google Scholar]
- Maximum bubble pressure method: universal surface age and transport mechanism in surfactant solutions. Langmuir. 2006;22:7528-7542.
- [Google Scholar]
- Diffusion-controlled adsorption at liquid–air interface: the long-time limit. J. Colloid Interface Sci.. 2001;237:294-296.
- [Google Scholar]
- Adsorption kinetics of ionic surfactants after a large initial perturbations, effect of surface elasticity. Langmuir. 2000;16:2942-2956.
- [Google Scholar]
- Interfacial Phenomena. New York: Academic Press; 1963. (p. 284)
- Dynamic surface tension and adsorption kinetics of nonionic surfactants at the air–water interface. Langmuir. 2000;16:2802-2807.
- [Google Scholar]
- Dynamic surface tension and adsorption mechanism of surfactants at air–water interface. Adv. Colloid Interface Sci.. 2000;85:103-144.
- [Google Scholar]
- Numerical analysis of surfactant dynamics at air–water interface using the Henry isotherm. J. Math. Chem.. 2011;49:1624-1645.
- [Google Scholar]
- Numerical behaviour of a linear mixed kinetic-diffusion model for surfactant adsorption at the air–water interface. J. Math. Chem.. 2012;50:429-438.
- [Google Scholar]
- A mixed diffusion surfactant model for the Henry isotherm. J. Math. Anal. Appl.. 2012;389:670-684.
- [Google Scholar]
- Techniques to measure dynamic surface tension. Curr. Opin. Colloid Interface Sci.. 1996;1:296-303.
- [Google Scholar]
- Insoluble Monolayers at Liquid–Gas Interfaces. New York: Wiley; 1966.
- Colloid and Interface Science. New Delhi: PHI Learning; 2009.
- The theory of diffusion-controlled adsorption kinetics with accompanying evaporation. J. Phys. Chem.. 1960;64:637-641.
- [Google Scholar]
- Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006.
- Chemical physics of colloid system and interfaces. In: Birdi K.S., ed. Handbook of Surface and Colloid Chemistry (third ed.). Boca Raton: CRC Press; 2008. p. :199-377.
- [Google Scholar]
- Contact angle measurement and contact angle interpretation. Adv. Colloid Interface Sci.. 1999;81:167-249.
- [Google Scholar]
- Producing bimodal molecular weight distribution polymer resins using living and conventional free-radical polymerization. Ind. Eng. Chem. Res.. 2005;44:2568-2578.
- [Google Scholar]
- Kinetic equations for transfer-controlled adsorption kinetics. Colloids Surf., A. 1994;88:251-266.
- [Google Scholar]
- A simple solution of the Ward–Tordai equation for the adsorption of non-ionic surfactants. Comput. Chem. Eng.. 2010;34:146-153.
- [Google Scholar]
- Analytical and Numerical Methods for Volterra Equations. Philadelphia: SIAM; 1985.
- Diffusion-controlled adsorption kinetics at the air/solution interface. Colloid Polym. Sci.. 2000;278:124-129.
- [Google Scholar]
- Diffusion-controlled kinetics of aqueous micellar solution at air/solution interface. Colloid Polym. Sci.. 2009;287:1083-1088.
- [Google Scholar]
- On the solution of diffusion controlled adsorption kinetics for any adsorption isotherms. Colloid Polym. Sci.. 1981;259:375-381.
- [Google Scholar]
- Diffusion-controlled adsorption kinetics. General solution and some applications. J. Phys. Chem.. 1982;86:4648-4651.
- [Google Scholar]
- Fast adsorption at the liquid–gas interface. Adv. Colloid Interface Sci.. 1996;69:63-129.
- [Google Scholar]
- The Fractional Calculus. New York: Academic press; 1974.
- On the solution of the diffusional problem in adsorption kinetics. Colloid Polym. Sci.. 1977;255:669-674.
- [Google Scholar]
- Design and accuracy of pendant drop methods for surface tension measurement. Colloids Surf., A. 2011;384:442-452.
- [Google Scholar]
- The kinetics of adsorption at liquid interfaces. Aust. J. Sci. Res.. 1952;A5:683-696.
- [Google Scholar]
- Time-dependence of boundary tensions in solutions. I. Role of diffusion in time effects. J. Chem. Phys.. 1946;14:453-461.
- [Google Scholar]
- On the solution of diffusion controlled adsorption kinetics by means of orthogonal collocations. Colloid Polym. Sci.. 1986;264:611-615.
- [Google Scholar]