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Structure of foams produced by agitation
Colloids and Surfaces A: Physicochemical and Engineering Aspects 149 (1999) 19–22
Structure of foams produced by agitation Pedro Licinio *, Jose M.A. Figueiredo Departamento de Fı´sica, ICEx, UFMG, C.P. 702, 30.123–970 Belo Horizonte/MG, Brazil Received 28 August 1997; accepted 25 March 1998
Abstract A statistical description of foaming processes is proposed. It should apply whenever bubble creation involves strong gas–liquid interface fluctuations as in the case of agitations or wave break ups. The present calculations lead to distributions which are completely different from the maximum entropy distributions found for coarsened foams. Three-dimensional foams were also produced by mixing air and water+Extran solutions in a shaker. The cumulative bubble distribution from these experiments show a good agreement with the scaling form predicted by the present theory. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Foams; Fragmentation; Statistical distributions
1. Introduction Foams are complex metastable two bulk-phase (plus interface) systems. They present continuous maturation processes which show wet–dry concentration transitions with polyhedra formation at a first stage. Then slow diffusive ripening or spontaneous bursts may also occur. Up today most studies on foams concern two-dimensional asymptotic equilibrium properties of cellular structures which presents great interest for physics, topology and metallurgy [1]. Foams have broad technological importance (among the large uses of foams are effective fire extinction, bakery and transport of granular media in pipes) [2]. In this paper a statistical description is given for bubble distributions obtained under conditions where strong instabilities in the liquid–gas interfaces are the main mechanism for bubble production. Such instabilities are believed to occur during ocean * Corresponding author. Fax: 0055 31 499 5600; e-mail: [email protected]
wave break-ups mixtures.
or
agitation
of
liquid–gas
2. Experimental Experiments were carried out with water–Extran foams [3] being produced in a shaker. The foam structure was then investigated using an image analysis technique, as described in Ref. [3]. The methodology employed basically consists of: (1) controlling foam production in a shaker; (2) flattening the foam between glass plates; (3) capture of the foam image; and (4) image analysis. The image analysis step gives volumes of individual bubbles. In further numerical analysis, bubble occurrence histograms were multiplied by bubble volumes and divided by bin interval and observed foam volume before integration. By such a procedure, cumulative histograms of partial volume
0927-7757/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0 9 2 7- 7 7 5 7 ( 9 8 ) 0 03 9 9 - 9
20
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
fraction w(v), of foam occupied by bubble species of volume smaller than v were obtained.
3. Results In the present experiments two different scaling forms of the foam statistics for cumulative bubble distributions were identified. In a first moment, foam is produced by shaking. In this case bubble radius distribution is probably determined by the competition between input energy and dissipation [3]. Later, after the shaking is stopped, the maturation process starts. In this coarsening regime, gas diffusion between neighboring bubbles driven by surface tension and geometric constraints become important in defining the bubble distribution. Fig. 1 shows a cumulative distribution for a fresh foam sample produced after agitation for 90 s at a frequency of 2.0 Hz observed within 2 min of preparation. It also shows a distribution obtained for another foam sample coarsened for 200 min between glass plates. An initial and naive approach to foam statistics would consider a random partitioning of the available volume V into i species of n cells or bubbles i with volume v . Entropy maximization would then i lead to the exponential distribution [1]: n 3exp(−kv ) (1) i i The cumulative distribution in this case reads w(x)=1−(1+x) exp(−x)
(2)
where x=v/v is a normalized bubble volume. 0 This kind of distribution has been found for the asymptotic regime during the slow maturation of quasi two-dimensional foams (soap froth between glass plates). This is surprising since the partitioning is not a priori random, but results from complex interactions while gas diffusion between neighboring cells takes place. Nevertheless, as can be seen in Fig. 1, the integrated bubble distribution obtained for slow maturation shows a good fitting to Eq. (2). A simple formalism is presented next to describe the first process, that is, production. It is a strongly non-equilibrium process and presently no theory exits to describe the three-dimensional foam formation. Wave break-up fed by shaking
Fig. 1. Cumulative bubble volume distribution obtained for: #, foams immediate after creation by shaking at a frequency of 2.0 Hz; and +, after coarsening (another preparation) for 200 min between glass plates. The fits to Eqs. (10) and (2) are shown as continuous curves in the log–log plot. It is also shown, for comparison, plots of the: – – –, Mott; and – · –, Gaussian distributions.
energy is probably the main mechanism responsible to foam creation. The fresh foam created then acts as a dissipator for the wave kinetic energy, decreasing break-ups. A steady state is expected to be built after some shaking time. This steady state is analyzed by considering a partitioning of the available surface energy of a breaking wave. The problem of three-dimensional foam creation, leads to the consideration of the total foam area or energy U, that is, the sum of the surface energies u =sv2/3 (3) i i from each bubble, where s is the interface surface tension. So U3∑ n u i i i
(4)
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
It is assumed that bubbles are created with equal a priori probability, so each bubble species should possess a degeneracy g given by the ratio of the i breaking wave excess energy (potential surface) to the surface of a single closing bubble, that is, a bubble can be created with a degeneracy inversely proportional to its area: g 3v−2/3 (5) i i The number of available configurations for a given distribution is then gn (6) V=a i i i ni ! The standard statistical mechanics technique of entropy maximization under the constraint of constant total foam energy (area) then leads to the distribution
CAB D
v 2/3 n 3g exp(−ku )3v−2/3 exp − i (7) i i i i v 0 where k is a Lagrange multiplier. Passing to the continuous limit and under proper normalization, the volume distribution per unit volume then reads:
AB
AAB B
2 v −2/3 v 2/3 n(v)= v−2 (8) exp − 3 0 v v 0 0 The partial volume occupied by bubble species smaller than v can be obtained by integration as
P
v
dv∞v∞n(v∞)=1−(1+x)exp(−x) (9) 0 which has the same form of the distribution obtained for random volume equipartition, Eq. (2), but now with x=(v/v )2/3. This function 0 is fitted to the experimental curves as shown in Fig. 1. The fit to the data taken after shaking at a frequency of 2.0 Hz is remarkably good at overall extent. The coarsened regime fits well to the random partition maximum entropy model with an initial slope of 2. The initial slope for the foaming model introduced is 4/3. Dynamic fragmentation has been studied in many contexts, as rock fracture, fuel spray and emulsification [4,5]. The simplest fragmentation
w(x)=
21
approach considers Poisson variates, which is analogous to the random partitioning described at the beginning of this article and leads to exponential distributions. The well known Gaussian (and log– normal ) distributions have also been often invoked. An integrated zero-centered Gaussian distribution is plotted in Fig. 1 for comparison. Note that the initial slope is also 2, but it saturates even faster than the exponential distribution. Of course, a change in width gives only a horizontal displacement, conserving form, in this logarithmic representation. Mott and Linfoot [4] give a distribution obtained after partitioning a fixed volume with random planes as
AAB B
v 1/3 n 3v−2/3 exp − i i i v 0
(10)
This distribution resembles Eq. (7), except for the exponent in the exponential. In particular the cumulative distribution also has an initial log–log slope of 4/3. As can be seen in Fig. 1, the distribution is much too broad and will not fit the experimental data.
4. Conclusion In conclusion, it has been shown that foams created by agitation have a stationary distribution which is completely different from the coarsened distributions described to date. Before coarsening, foams can be modeled by a wave breaking mechanism where surface energy is distributed with maximum configurational entropy. Volume effects have been neglected here, since the authors worked with a total foam volume much greater than mean bubble volume. It should be interesting to investigate finite volume effects in order to test the range of applicability of this model.
Acknowledgment This work was supported by the Brazilian agency FAPEMIG.
22
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
References [1] J. Stavans, The evolution of cellular structures, Rep. Prog. Phys. 56 (1993) 733–789. [2] A.J. Wilson ( Ed.), Foams: Physics, Chemistry and Structure, Springer Verlag, Berlin, 1989.
[3] P. Licinio, J.M.A. Figueiredo, Steady foam states, Europhys. Lett. 36 (3) (1996) 173–178. [4] D.E. Grady, M.E. Kipp, Geometric statistics and dynamic fragmentation, J. Appl. Phys. 58 (3) (1985) 1210–1222. [5] T. Ishii, M. Matsushita, J. Phys. Soc. Jpn. 61 (1992) 3474.
Structure of foams produced by agitation Pedro Licinio *, Jose M.A. Figueiredo Departamento de Fı´sica, ICEx, UFMG, C.P. 702, 30.123–970 Belo Horizonte/MG, Brazil Received 28 August 1997; accepted 25 March 1998
Abstract A statistical description of foaming processes is proposed. It should apply whenever bubble creation involves strong gas–liquid interface fluctuations as in the case of agitations or wave break ups. The present calculations lead to distributions which are completely different from the maximum entropy distributions found for coarsened foams. Three-dimensional foams were also produced by mixing air and water+Extran solutions in a shaker. The cumulative bubble distribution from these experiments show a good agreement with the scaling form predicted by the present theory. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Foams; Fragmentation; Statistical distributions
1. Introduction Foams are complex metastable two bulk-phase (plus interface) systems. They present continuous maturation processes which show wet–dry concentration transitions with polyhedra formation at a first stage. Then slow diffusive ripening or spontaneous bursts may also occur. Up today most studies on foams concern two-dimensional asymptotic equilibrium properties of cellular structures which presents great interest for physics, topology and metallurgy [1]. Foams have broad technological importance (among the large uses of foams are effective fire extinction, bakery and transport of granular media in pipes) [2]. In this paper a statistical description is given for bubble distributions obtained under conditions where strong instabilities in the liquid–gas interfaces are the main mechanism for bubble production. Such instabilities are believed to occur during ocean * Corresponding author. Fax: 0055 31 499 5600; e-mail: [email protected]
wave break-ups mixtures.
or
agitation
of
liquid–gas
2. Experimental Experiments were carried out with water–Extran foams [3] being produced in a shaker. The foam structure was then investigated using an image analysis technique, as described in Ref. [3]. The methodology employed basically consists of: (1) controlling foam production in a shaker; (2) flattening the foam between glass plates; (3) capture of the foam image; and (4) image analysis. The image analysis step gives volumes of individual bubbles. In further numerical analysis, bubble occurrence histograms were multiplied by bubble volumes and divided by bin interval and observed foam volume before integration. By such a procedure, cumulative histograms of partial volume
0927-7757/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0 9 2 7- 7 7 5 7 ( 9 8 ) 0 03 9 9 - 9
20
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
fraction w(v), of foam occupied by bubble species of volume smaller than v were obtained.
3. Results In the present experiments two different scaling forms of the foam statistics for cumulative bubble distributions were identified. In a first moment, foam is produced by shaking. In this case bubble radius distribution is probably determined by the competition between input energy and dissipation [3]. Later, after the shaking is stopped, the maturation process starts. In this coarsening regime, gas diffusion between neighboring bubbles driven by surface tension and geometric constraints become important in defining the bubble distribution. Fig. 1 shows a cumulative distribution for a fresh foam sample produced after agitation for 90 s at a frequency of 2.0 Hz observed within 2 min of preparation. It also shows a distribution obtained for another foam sample coarsened for 200 min between glass plates. An initial and naive approach to foam statistics would consider a random partitioning of the available volume V into i species of n cells or bubbles i with volume v . Entropy maximization would then i lead to the exponential distribution [1]: n 3exp(−kv ) (1) i i The cumulative distribution in this case reads w(x)=1−(1+x) exp(−x)
(2)
where x=v/v is a normalized bubble volume. 0 This kind of distribution has been found for the asymptotic regime during the slow maturation of quasi two-dimensional foams (soap froth between glass plates). This is surprising since the partitioning is not a priori random, but results from complex interactions while gas diffusion between neighboring cells takes place. Nevertheless, as can be seen in Fig. 1, the integrated bubble distribution obtained for slow maturation shows a good fitting to Eq. (2). A simple formalism is presented next to describe the first process, that is, production. It is a strongly non-equilibrium process and presently no theory exits to describe the three-dimensional foam formation. Wave break-up fed by shaking
Fig. 1. Cumulative bubble volume distribution obtained for: #, foams immediate after creation by shaking at a frequency of 2.0 Hz; and +, after coarsening (another preparation) for 200 min between glass plates. The fits to Eqs. (10) and (2) are shown as continuous curves in the log–log plot. It is also shown, for comparison, plots of the: – – –, Mott; and – · –, Gaussian distributions.
energy is probably the main mechanism responsible to foam creation. The fresh foam created then acts as a dissipator for the wave kinetic energy, decreasing break-ups. A steady state is expected to be built after some shaking time. This steady state is analyzed by considering a partitioning of the available surface energy of a breaking wave. The problem of three-dimensional foam creation, leads to the consideration of the total foam area or energy U, that is, the sum of the surface energies u =sv2/3 (3) i i from each bubble, where s is the interface surface tension. So U3∑ n u i i i
(4)
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
It is assumed that bubbles are created with equal a priori probability, so each bubble species should possess a degeneracy g given by the ratio of the i breaking wave excess energy (potential surface) to the surface of a single closing bubble, that is, a bubble can be created with a degeneracy inversely proportional to its area: g 3v−2/3 (5) i i The number of available configurations for a given distribution is then gn (6) V=a i i i ni ! The standard statistical mechanics technique of entropy maximization under the constraint of constant total foam energy (area) then leads to the distribution
CAB D
v 2/3 n 3g exp(−ku )3v−2/3 exp − i (7) i i i i v 0 where k is a Lagrange multiplier. Passing to the continuous limit and under proper normalization, the volume distribution per unit volume then reads:
AB
AAB B
2 v −2/3 v 2/3 n(v)= v−2 (8) exp − 3 0 v v 0 0 The partial volume occupied by bubble species smaller than v can be obtained by integration as
P
v
dv∞v∞n(v∞)=1−(1+x)exp(−x) (9) 0 which has the same form of the distribution obtained for random volume equipartition, Eq. (2), but now with x=(v/v )2/3. This function 0 is fitted to the experimental curves as shown in Fig. 1. The fit to the data taken after shaking at a frequency of 2.0 Hz is remarkably good at overall extent. The coarsened regime fits well to the random partition maximum entropy model with an initial slope of 2. The initial slope for the foaming model introduced is 4/3. Dynamic fragmentation has been studied in many contexts, as rock fracture, fuel spray and emulsification [4,5]. The simplest fragmentation
w(x)=
21
approach considers Poisson variates, which is analogous to the random partitioning described at the beginning of this article and leads to exponential distributions. The well known Gaussian (and log– normal ) distributions have also been often invoked. An integrated zero-centered Gaussian distribution is plotted in Fig. 1 for comparison. Note that the initial slope is also 2, but it saturates even faster than the exponential distribution. Of course, a change in width gives only a horizontal displacement, conserving form, in this logarithmic representation. Mott and Linfoot [4] give a distribution obtained after partitioning a fixed volume with random planes as
AAB B
v 1/3 n 3v−2/3 exp − i i i v 0
(10)
This distribution resembles Eq. (7), except for the exponent in the exponential. In particular the cumulative distribution also has an initial log–log slope of 4/3. As can be seen in Fig. 1, the distribution is much too broad and will not fit the experimental data.
4. Conclusion In conclusion, it has been shown that foams created by agitation have a stationary distribution which is completely different from the coarsened distributions described to date. Before coarsening, foams can be modeled by a wave breaking mechanism where surface energy is distributed with maximum configurational entropy. Volume effects have been neglected here, since the authors worked with a total foam volume much greater than mean bubble volume. It should be interesting to investigate finite volume effects in order to test the range of applicability of this model.
Acknowledgment This work was supported by the Brazilian agency FAPEMIG.
22
P. Licinio, J.M.A. Figueiredo / Colloids Surfaces A: Physicochem. Eng. Aspects 149 (1999) 19–22
References [1] J. Stavans, The evolution of cellular structures, Rep. Prog. Phys. 56 (1993) 733–789. [2] A.J. Wilson ( Ed.), Foams: Physics, Chemistry and Structure, Springer Verlag, Berlin, 1989.
[3] P. Licinio, J.M.A. Figueiredo, Steady foam states, Europhys. Lett. 36 (3) (1996) 173–178. [4] D.E. Grady, M.E. Kipp, Geometric statistics and dynamic fragmentation, J. Appl. Phys. 58 (3) (1985) 1210–1222. [5] T. Ishii, M. Matsushita, J. Phys. Soc. Jpn. 61 (1992) 3474.