Journal of Mining Science, Vol. 46, No. 4, 2010

DIFFUSION MECHANISM OF THE SURFACTANT-FORMED ADSORPTION LAYER AT THE SOLUTION AND AIR INTERFACE

V. V. Kudryashov

UDC 622.7

The author describes mechanism of arrival of a surfactant at the solution and air interface based on the Brownian diffusion of molecules in a medium with increasing viscosity nearer to the interface. The article explains features of the adsorption layer formation times with molecules of ionizable and non-ionizable wetters, and estimates the adsorption layer formation time versus temperature of the solution and addition of electrolyte. Adsorption, molecules, surfactant, surface, air, solution, diffusion, viscosity

Mineral particles adhere to air bubbles in flotation solutions of surfactants or to drops in sprinkling dusts owing to the formation of an adsorption layer of the surfactant molecules at interfaces liquid – gas or liquid – solid, and due to the liquid spreading on solid surface. Should an adsorption layer form instantly at a liquid – gas or a liquid – solid interface, the liquid spreading on the surface would be the basic process in wet-out of mineral particles. By the research [1], the formation of the surfactant adsorption layer at interfaces of the surfactant solution and air or a solid is governed by concentration and time. Namely, the higher is concentration of the solution, the quicker the adsorption layer forms at the liquid and solid interface. The time for the particles to adhere to a newly formed liquid surface is comparable to the time of the adsorption layer formation. Hence, when particles and bubbles contact in a surfactant solution under dynamic conditions, e.g. in flotation or dust sprinkling, a deterministic factor is the kinetics of the adsorption layer formation by the surfactant molecules at the solid and gas interface. The same is valid for adsorption of ions from solutions at the solid – gas interface [2]. Attempts to explaining the surfactant-generated adsorption layer at the liquid and solid or gas interfaces are known [3 – 5]; for instance, Gauden attributed incoming of surfactant molecules to an interface to diffusion and an activity gradient based on Fick’s law [4]. Chanturia and Shafeev gave the same reason for outgo of ions onto a solid and gas interface [2]. Adamson thought stabilization of equilibrium interfacial tension to be an inverse process of the surfactant film dissolution, based on the Fick law as well [3]. The calculated time of an adsorption layer formation is orders less than in experiments. On this basis, it is taken that outgo of molecules to an interface is a complex phenomenon, including orientation of molecules in facial layer, diffusion of molecules and micells (as well as of ions of ionizable surfactants), destruction of micelle structures (in micelle forming agents), replacement of low active molecules in facial layer by higher active molecules (when chemically impure substances are used) and other phenomena (diffusion of surfactant molecules inward liquid, dehydration of polar groups, depolarization of ions, etc.). Research Institute of Comprehensive Exploitation of Mineral Resources, Russian Academy of Sciences, Moscow, Russia. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 4, pp. 99-103, July-August, 2010. Original article submitted February 11, 2010. 1062-7391/10/4604-0453 ©2010 Springer Science + Business Media, Inc. 453

One of the listed phenomena, namely, orientation of molecules at an interface must take almost no time, commensurable with the time of “sedentary life” of liquid molecules: 10–23 s. The process of replacement of low active surfactant molecules by higher active in a near-interface layer is longer. The transit time of molecules between the surface and near-surface layers was estimated by Adamson with the use of Einstein’s formula [3]: t = x2 / 2D , (1)

where х is near-surface layer thickness in 10–8 m; D is diffusion coefficient, most liquids have D approximately 10-9 m2/s. Thus the transit time is 10-6 s. Even longer is destruction of micelle structure. By [3], dissociation rate constant for micelles of colloid electrolytes equals 100 s-1, which implies an acceptable structural damage time as 0.01 s. The discussed phenomena, solely, as well as dehydration of polar groups and depolarization of ions do not explain the concentration-time character of the wet-out effect produced by surfactant solutions and the adsorption layer formation at liquid and gas or solid interfaces. For this reason, another phenomenon is analyzed below in this paper: Brownian diffusion handling of surfactant molecules and micelles toward an interface in a variable viscosity medium. It is known that viscosity of a solution increases at getting nearer to the solution surface. By the current views [3, 6], a solution with the formed adsorption layer has its structure composed of: — a facial layer made of surfactant molecules oriented in perpendicular to the interface (in saturation), its length is ~10-10 m and viscosity exceeds the solution viscosity by a factor of 102 – 104 [7], i.e. equals on average 1 kg/f·m; — a sublayer with increased concentration of surfactant molecules that are in dynamic equilibrium with the facial layer; the sublayer thickness is on average ~ 10–8 m; the sublayer existence is conditioned by the close distance to the interface and the long-range forces between molecules that are near the interface; viscosity of the sublayer lowers from 1 kg/f·m, which is the facial layer viscosity, to the sublayer bottom viscosity equal or less than viscosity of a 100 % surfactant, i.e. 0.1 kg/f·m, e.g. for the wetter DB; as per [7], 0.1 kg/f·m is the viscosity minimum for facial layer; — a diffusion layer with its bottom boundary of nearly 10-5 m and viscosity up to tenfold higher than viscosity of liquid according to different references [3, 8]; for instance, it is fourfold in [8]; it makes 0.01 kg/f·m for water and a dilute solution of wetter DB having concentration equal, less or a bit higher than the critical micelle concentration; according to [8], the diffusion layer viscosity is 4·10–3 kg/f·m; — the solution itself; its viscosity varies from 0.01 – 0.04 kg/f·m at x ≤ 10–5 m from the interface to 0.001 kg/f·m, which is viscosity of water and solution with concentration lower, equal or a bit higher that the critical micelle concentration, at x > 10–5 m from the interface. The above structuring in depth of a solution allows plotting its viscosity versus distance to interface as in Fig. 1 and expressing the solution viscosity as:

η = 10 −3 x −0.5 + η 0 ,

(2)

where η0 is the solution and water viscosity equaling 0.001 kg/f m. To fetch an equilibrium (or close to equilibrium) adsorption at an interface, molecules (micelles) must diffuse from the solution sufficient surfactant molecules to cover the interface surface. The adsorption layer formation is completed when the last molecule fills in the interface as a result of diffusion from a certain depth of the solution. This last molecule will make its way in the liquid with formed depth-wise growing viscosity. 454

Fig. 1. Viscosity η of a surfactant solution versus distance х to the interface of the solution with air or solid

For the surfactant molecules to cover 1 m2 of the interface surface, the limit depth is: x = 1 / sn , m. Here, n is calculated concentration of reagent molecules, m–3; s is seat of a molecule, m2. Replacing n by mass concentration с: c = Mn / N , where М is molecular weight of reagent; N is Avogadro constant, yields: x = M / sNc . (3) –20 2 For example, for wetter DB (М = 500, s = 30·10 m ): (4) x = 3 ⋅ 10 −9 / c , m. The concentration с is expressed in fractions. The Brownian diffusion rate of molecules and micelles can be presented as a result of differentiation of the Stokes – Einstein equation: x2 kT = , (5) t 3π rη where viscosity η is replaced by (2): dx kT 1 . (6) = ⋅ −5 dt 3π r 10 ⋅ 1.5 x 0.5 + 2 xη 0 Here, k is Boltzmann constant and equals 4·10–24 J/deg; T is absolute temperature, K; r is radius of a molecule or micelle, m; x is translatory Brownian displacement for the time t, s. Transforms and integration for х going from 10–10 cm to х and for t going from 0 to t bring about: 3π r −5 1,5 t= [10 ( x − 10 −15 ) + η0 ( x 2 − 10− 20 )] . (7) kT We assume that micelles of wetters (e.g., DB, DT-7, Aerosol OG, and others) equals 40·10–10 m (which is the length of the wetter molecules by [5]). Then, with the known r, k, Т = 300 K and η0 = 0.001 kg/f m, we have: 1.5 ⎤ ⎡⎛ M ⎞ 2 ⎤⎫ 3πr ⎧⎪ −5 ⎡⎛ M ⎞ −15 −20 ⎪ − + 10 10 η (8) ⎟ ⎟ − 10 ⎥ ⎬ . ⎢⎜ ⎥ 0 ⎢⎜ ⎨ kT ⎪ sNc sNc ⎝ ⎠ ⎝ ⎠ ⎪ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎭ ⎩ Relationship of the transit time of the “last” molecule to the solution – air interface and the concentration с results from replacing x in (8) by the expression (4): t = 46.8 ⋅10−6.5 ⋅ c −1.5 + 0.8 ⋅10−7 ⋅ c −2 − 9.01 ⋅10−8 . (9)

t=

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Fig. 2. Relation of the adsorption layer formation time t and concentration c of the molecules of wetters DB and Sintanol DT-7: 1 — calculation by (7); 2 — experimental data of the dynamic surface tension measurement RESULTS AND DISCUSSION

Figure 2 presents the calculation and experiment curves for the adsorption layer formation time of molecules of wetters DB and DT-7 at the solution and air interface versus the concentration of the wetters in the solution. The experimental curve was obtained after processing measurement data on dynamic surface tension [1]. It is worth noting, we took the averaged values of viscosity for the interface-close layers as we lacked reliable information on the link of viscosity of solutions and the distance to the interface. As a result, the calculated time of the adsorption layer formation and the experimental curve agree satisfactorily. It implies that the analyzed here mechanism of the adsorption layer formation at the surfactant solution and air interface is the predominant process. The Brownian diffusion of surfactant molecules toward the interface in a solution with depth-wise variable viscosity is of use to explaining the slower formation of the adsorption layer in case of ionizable wetter DS-Na as compared with the adsorption layer formation with molecules of nonionizable wetter DB [9]. The ionizable wetters have larger-size micelles due to solvate shells. The diffusion coefficient D = kT / 6πη r of micelles of the ionizable wetter is much lower than the nonionizable wetters having micelles nearly of the same weight. For example, ionizable sodium dodecyl sulfate has D = 6 ⋅ 10−12 m2/s, M = 40 kg/mole [3], non-ionizable methoxy polyoxyethylene caprinate has D ~ 8 ⋅ 10−11 m2/s, M = 40 kg/mole [5]. So, micelles of the first type have the effective size rion larger than the second type micelles, rnon , by a factor of 8 ⋅ 10−11 / 6 ⋅ 10−12 ≈ 13 , i.e., by an order of magnitude, which holds true for the adsorption layer formation times in solutions with equal concentrations of the surfactants, (7). The ionizable wetter concentration cion must exceed the nonionizable wetter concentration c non , given the equal t of the interface existence, as:

ci / cn = 1.5 r1 / r2 = 1.5 13 ≈ 5 times. This follows from (9), including micelle radius at c −1.5 ; the other members neglected, and we can express (9) as: t ~ rc −1.5 . To accelerate formation of an adsorption layer at a fresh surface of ionizable wetters, electrolyte should be added into the solution. Accelerated arrival of surfactant molecules to an interface with a lowering temperature is associated with the high viscosity of the solution, especially close to the air/solid and liquid interface due to the increased concentration of the surfactant in the near-surface layer. 456

Based on the considered mechanism of the surfactant molecule motion toward an interface, if the specific surface area of liquid is large, a wetter concentration may decrease owing to the molecule adsorption down to values at which the surface layer will not be filled with the wetter molecules, and the adsorption layer formation will get slower [10]. In order to prevent deceleration, the surfactant concentration in the solution must be constant or initially high. These outcomes are important to understand quick processes of interaction between surfactant solutions and gas or a solid, such as flotation of ores, foam formation, dispergating of solutions, capture of dust particles by drops, weakening of a well bottom in drilling, etc. The author is thankful to Academician V. A. Chanturia for the attention and concern. REFERENCES

1. L. D. Voronina, V. V. Kudryashov, and M. K. Shurinova, “Assessment of wetter solutions in dust control in dynamic conditions,” in: Gas Emission and Dust Control in Mines [in Russian], Nauka, Moscow (1972). 2. V. A. Chanturia and R. Sh. Shafeev, Chemistry of Surface Phenomena in Flotation [in Russian], Nedra, Moscow (1977). 3. A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces, Wiley-Interscience, New York (1997). 4. A. M. Gauden, “Saline flotation. Progress and problems present a challenge,” Engineering and Mining Journal, 157 (1956). 5. K. Shinoda, T. Nakagawa, B. Tamamusi, and T. Isemura, Colloidal Surfactants, Academic Press, New York (1963). 6. D. Netzel., G. Hoch, and T. Marx, “Adsorption studies of surfactants at the liquid-vapor interface: apparatus and method for rapidly determining the dynamic surface tension,” J. of Colloid Science, 5 (1964). 7. A. A. Trapeznikov, “Surface viscosity and its measurement methods,” in: Viscosity of Liquids and Colloidal Solutions [in Russian], AN SSSR, Moscow-Leningrad (1941). 8. B. V. Deryagin and M. M. Samygin, “Viscosity measurement in thin polymolecular layers of liquid,” in: Viscosity of Liquids and Colloidal Solutions [in Russian], AN SSSR, Moscow-Leningrad (1941). 9. V. V. Kudryashov, Scientific Basics for Hydraulic Dust Control in Mines in the North [in Russian], Nauka, Moscow (1984). 10. V. F. Skorokhodov, “Theory and practice advance in mineral separation in activated aqueous dispersions of air and the new flotation technique development,” Synopsis of Dr.Eng. Thesis [in Russian], Moscow (2003).

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