- Email: [email protected]
Mechanical model for two-dimensional coarsening foams
May 1994
Materials Letters 19 ( 1994) 3 17-323
Mechanical model for two-dimensional coarsening foams A. Jimtnez-Ceniceros,
G. VBzquez-Polo,
C. Renero, V.M. CastaiTo ’
Institute de Fisica, U.N.A.M., Apartado Postal 20-364, Mexico. D.F. 01000, Mexico
Received 11 November 1993; in final form 15 February 1994; accepted 17 February 1994
Abstract
A simple mechanical model for the structure of theoretical two-dimensional (2D) foams is presented. The proposed model is based on simple geometrical and energy considerations in foams. The structure and evolution of a theoretical system are also described as an example predicted by the model.
1. Introduction Cellular materials have represented a continuing area of interest for scientists for many years. There are several reasons for this to happen: firstly, the amazing fact of many quite different physical systems having basically the same cellular structure, ranging from the suds of a shampoo to the grain structure of metals and the arrangement of vegetable tissues [ l-8 ] ; and secondly, because of the potential technological applications of foams or cellular materials, such as stifling of explosions [ 1 ] or production of high-strength materials [ 9, lo]. In fact, much effort, both theoretical and experimental, has been dedicated throughout the years to the study of the structure first and then the dynamics of soap froths and other cellular systems in two and three dimensions. Thus, it is possible to find in the literature works on the behavior of the area of cells as a function of time [ 6,7,14], reporting the existence of two kinds of dynamics recognizable in a foam [ 7 1, or on the correlation between size and shape [4] through topological considerations, etc. Because of topological restrictions (basically Eu-
ler’s law), cellular systems are dimension-sensitive, that is, the dimension of the foam plays a definite role in some structural characteristics of the system. In particular, two-dimensional froths have attracted a great deal of attention because, among other reasons, of their relevance in metallurgy and biology [ 8 1. In the specific case of 2D foams, however, most articles deal either with geometry and topology alone or with the temporal evolution of the so-called topological valences [ 111 and scarce effort has been dedicated to describe the mechanical aspects present in 2D cellular systems, although those parameters are extremely relevant to the physico-chemical stability of foams, namely because the equilibrium of forces in foams is achieved through surfactants, foaming agents or stabilizers. In this article, we present a simple model of the mechanical structure of a theoretical 2D foam, fulfilling the topological restrictions and allowing one to evaluate the coarsening (i.e. time evolution indirectly) of this system. The aim is to establish a theoretical framework, based on simple mechanical considerations, for the understanding of properties and potential applications of 2D cellular systems.
’ Author to whom correspondence should be addressed. 0167-577x/94/$07.00
0 1994 Elsevier Science B.V. All rights reserved
SSDI0167-577x(94)00037-N
A. Jimhez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
318
2. The model
@,=v,(P,-Pa)
Let us consider a “cellular-like” 2D lattice formed with closely regular polygons with the same length side which obeys Plateau’s criteria [ 221. The general formula for the area A of an i regular polygon is given by: A, = $L,a, ,
(1)
where Lj is the perimeter
and a, the height. Obviously
Li=N,li) fi being the length of a side of the ith polygon and Ni the number of faces. From Fig. 1 it can be obtained that
and, since 8= x/N,, then A, =
NJ? 4 tan(x/N,)
(2)
’
For the surface of the cell, we notice from Fig. 1 that the interacting surfaces are the ones from one the side of each cell, but there is no interaction on the bottom and the top surfaces. So, the interaction surface will be Si=Lih or
where the exterior pression is PO and the interior pression is Pi and the volume of the ith cell is V,. Volume I’, is equal to Aih, A, being the area and h its thickness, which is constant. Remember that we assume that A, >> h. Also, the free surface energy Ei of the cell due to the walls that separate each cell from its neighbors is E,=cxS;,
(4)
where S, is the surface area and (Ythe surface tension. Eq. (4) explains why a foam shows a tendency to reduce the interfacial area, since any increase in this surface would represent an increase of the free surface energy of the system. Under mechanical equilibrium conditions, it must be true that the energy due to the pressure difference @ exerted by the air on the walls is approximately equal to the free energy E due to the surface tension over that same wall within each cell, that is Ei-@=O.
(5)
It is important to notice that Eq. (5) implies a quasistatic condition, that is, we are describing a metastable state at a given instant. This approximation is adequate for cellular systems with slow dynamics, which is the case for many practical examples. By using the above Eqs. (3 )- ( 5 ) and the fact that for polygonal cells we have the following relations:
St =Nilih . First, consider the energy situation in a cell. Since we have some gas pressure inside and outside each cell, we define the energy due to pressure difference as
(3)
1
K=Aih=
NJfh 4 tan(x/N,)
’
Si=Nllih, it is very simple to obtain that p _p I
0
= 4otan(xlNi)
(6)
4
and, by recalling that pressure is forcefper we get
T Oi 1
Fig. 1. Geometrical
parameters
of a 2D polygon.
1;=fo+4aN,
tan(x/N,)
.
unit area,
(7)
In this way, we have now an equation to calculate the force between cells in terms only of the surface tension, exterior pressure of the system and the number of sides of each cell. It must be noticed that in Eq. (7) we have eliminated I,, the length of each side. We
A. Jimknez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
must say that this is possible only if we consider this value as being approximately the same for all cells throughout the froth, although it can be increased globally through time, so implying that our model is valid for foams in which the scale of bubbles is of the same order of magnitude. This is consistent with Mullin’s statistical self-similarity hypothesis [ 12- 14 1. Moreover, we can get rid offo since one is basically interested in the difference of forces between two adjacent cells i and i. Thus, it is straightforward to obtain finally: .f;-J;=4c~[N,
tan(x/N,)-N,
tan(rr/N,)]
.
(8)
The model allows one to evaluate also the corresponding radii of curvature R,, of the surface between cells i and j, through the relation P,-Pj=2a/R,,
.
Thus R,,=f[tan(x/N,)-tan(n/N,)]-’
.
(9)
3. Results and discussion Eqs. (7), (8) and (9) of section 2 can now be utilized to obtain Tables 1, 2 and 3. In Table 1, for instance, it can be observed that the pressure in a cell decreases as the number of sides of that particular cell is increased; for example, a change from 3 to 10 sides represents a decrease of about 40% in the corresponding pressure. These first results provide a clue as to why the evolution of foams involves formation of cells with more and more sides, as has been observed experimentally. More interesting results are contained in Table 2, Table
I Number of sides N,
Pressure force in the cell f;-&=4olN,tan(n/N,)
3 4 5 6 7 8 9 10
20.74601 16.000~~ 14.53001 13.85601 13.484~~ 13.254~~ I3.109a 12.996a
319
which shows the pressure difference between cells of different number of sides. From Table 2 we can observe again that the smaller the number of sides of a given cell, the higher the pressure exerted by the neighboring cells on that particular cell. The sign of the pressure indicates the convexity or concavity of the wall between cells. Here, the values of pressure equal to zero indicate that the wall is in equilibrium, hence forming a flat interface. Thus, we observe that the more similar (in terms of number of sides) the cells, the smaller the pressure difference. This also explains some of the observed characteristics of foam coarsening. Table 3 contains the calculations of radii of curvature of the walls as a function of the number of sides N, and N, of the adjacent cells. The value co indicates, of course, flat interfaces. This is a particularly important result, since it allows one to begin to understand the dynamics of the foam. Indeed, the need of the system to relieve pressure from one cell to another, allows gas to diffuse from cells with less sides to those with more sides. From Tables l-3 one can observe that this process would take place around cells with six sides. That is, cells with less than six sides would eventually disappear whereas those with more than six sides would grow further. The process is rather interesting: when some cells disappear, the overall area is preserved, leading to some other cells to increase the length of their sides I,, producing less and larger cells. But, to minimize area in the new state, the system re-orders itself, repeating the process until the pressure difference is so small that the system reaches the steady state. In this way, our model can explain both the slow and fast dynamics reported previously. To show the feasibility of simulating coarsening of a given foam from the model, let us consider the series of simulated foams of Figs. 2a through 2f. The first corresponds to the initial state where we have all hexagonal cells except two (one is pentagonal and the other is heptagonal). By applying the above criteria and equations one can observe the coarsening until basically a single bubble is formed. The slow mechanism of evolution for bubbles with less than three sides is [ 15 ] : ( 1) Cells with three sides disappear, each leaving its three neighbors with one side less. (2) Cells with four sides disappear, each leaving
A. Jimknez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
320
Table 2 Relative pressure difference between adjacent cells Pressure difference
Ni
3 4 5 6 7 8 9 10
3-N,
4-N,
5-N,
6-N,
7-N,
8-N,
9-N,
10-N,
0.000 4.784 6.253 6.928 7.300 7.524 7.68 1 7.787
-4.785 0.000 1.469 2.143 2.515 2.745 2.897 3.003
-6.254 - 1.469 0.000 0.674 1.046 1.276 1.428 1.534
-6.928 -2.144 -0.675 0.000 0.372 0.601 0.753 0.859
-7.301 -2.516 - 1.047 -0.372 0.000 0.229 0.381 0.487
-7.525 -2.745 - 1.276 - 0.602 -0.229 0.000 0.151 0.258
- 7.682 - 2.897 - 1.428 -0.654 -0.381 -0.152 0.000 0.106
-7.788 - 3.003 - 1.534 -0.860 -0.487 -0.258 -0.106 0.000
Table 3 Radii of curvature
of the walls
R,
j
i
3
4
5
6
7
8
9
10
3 4 5 6 7 8 9 10
:683 0.497 0.433 0.399 0.379 0.365 0.355
-0.683 00 1.828 1.183 0.964 0.853 0.786 0.740
-0.497 - 1.828
-0.433 - 1.183 -3.351 00 5.220 3.064 2.342 1.980
- 0.400 - 0.965 -2.041 - 5.220 co 7.422 4.25 1 3.191
-0.379 -0.854 - 1.601 - 3.065 - 7.423 to 9.951 5.599
-0.366 -0.786 - 1.379 -2.343 -4.251 -9.952
-0.355 -0.741 - 1.245 - 1.981 -3.192 - 5.600 - 12.80 00
Y351 2.041 1.600 1.379 1.244
12;04
two of its four neighbors with one side less. (3) Cells with five sides disappear, each leaving two of its neighbors with one side less and one with one more side. It is possible to demonstrate that, if one has a 2D cellular array with N bubbles and a mean number of sides r equal to six exactly, this number will not change due to the slow dynamics, so we can presume that if this mean value changes, it will be due to the fast dynamics (rearrangement of bubbles or breakage of walls). Suppose we have cellular array with N bubbles, with one of them having three sides, N- 2 cells having six and the last one nine sides. The mean number of sides is
Immediately after the cell with three sides disappears, three of its neighbors will have lost a side, and the total number of bubbles will now be N- 1. It can happen that three cells with six sides lose one (and then the total number of cells with six sides becomes N-5) or two cells with six sides and the one with nine lose one side (in this case, the total number of cells with six sides becomes N-4). For the first case we have
[= 3+6(N-2)+9 N
I= 5~2+6(N-4)+8 N-l
=
12+6NN
12
6N =N=6.
[= 5x3+6(N-5)+9 N-l =
24+6N-30 N-l
6N-6 =- N-l
=6
-- 6N-6 - N-l
=6.
and for the second:
=
18+6N-24 N-l
A. ~~rn~nez-Ceni~~ros et al. /~a~er~a[s Letters I9 (1994) 317-323
321
Fig. 2. (a) through (f ). Coarsening of a two-dimensional theoretical foam, (a) being the initial configuration and (f) corresponding to the tinal state.
Now suppose we have a cellular array with N bubbles, where one cell has four sides, N- 2 cells six and one eight sides. The mean number of sides is
[_ 4+6(N-2)+8 N =
12+6NN
12 -- 6N -N=6,
comes N- 1. Again we have two cases: that two cells with six sides lose one side each, or the one with eight loses one side and one cell with six, too. So we have for the first case: Qx2+6(N-4)+8 N =
and after the cell with four sides disappears, N be-
18+6N-24 N-l
6N-6
=yyy=6,
and for the second: r
5x7+6(N-15)+13 N-7
=
[_ 5x1-6(N-3)+7 N-l = =
12+6N-
18
N-l
6N-6 --=6. - N-L
Finally, consider a cellular array with N bubbles, being one with five sides, N- 2 with six and one with seven. The mean number of sides is r
=
12+6N-12 N
-_ 6N =6. -N
=
18-24fSN N-l
6N-6 =N-_l
=6 *
Now let us follow a little more the evolution of this array, to show that N is a constant. We will assume that the ceil with larger number of sides is the one that grows, which is consistent with the theory. When the two five-sided bubbles disappear, there are now four with five sides (because four six-sided cells lose one side each), one with ten sides (because the one with eight sides gains one side from each of the five-sided ones), the total number of bubbles is now N- 3 and the total number of six-sided cells is N- 8, so 5~4+6(N-8)+10
/-_
=
19
6N+84168 6N-84 =N_14 N- 14
6N+ 120-282 N-27
=6 ’
+25
6N- 162 =6 = N-27 ’
[_ 5~25+6fN-72)+31 N-46 =
6N+432N-46
156
6N-276 N-46
=6
=
6N-426 N-71
=6
=
’
[_ 5x31+6fN-103)+37 N-71 =
5x2+6[N-4)+8 N-l
=
[_- 5x 13+6(N-28)+ N-14
=
When the bubble with five sides disappears, two bubbles will lose a side, and one will gain one side. For simplicity, we will consider here one case: we will assume that the one with seven gains a side, and two of six lose one, but it is easy to verify that N is constant. So we have, after the one with five sides disappears, that Nbecomes N- 1 and: r
6N-42 =-------=6, N-7
N-7
[_ 5x 19+6cN-47) N-27
5+6(N-2)+7 N
=
6N+48-90
6N+ 188-618 N-71
’
It is easy to notice when rchanges due to the diminishing of total number of bubbles N (if it is nut exactly six ) and when due to fast dynamics, and thus it is even possible to create an algorithm which, given the initial experimental values of N, each n (number of bubbles with i sides) and < provides us with the subsequent values of these variables, and try to see how and why it differs from the experimental data. Because variations in the experimental value of the mean number of sides [with the theoretical one depend only on the behavior of the fast dynamics, therefore one can obtain some rules for this behavior expe~mentally.
4. Final remarks
N-3 =
6N+30-48 N-3
6N-18 =N-3
and so on (see Figs. 2a-2f ):
=6
The structure and the basic dynamics of 2D foams can be explained by a simple mechanical model. The main features observed by previous researchers have been reproduced by the model. It is also interesting
A. Jimtkez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
to remark that the model can be experimentally tested since a quasi 2D foam can be produced by using two parallel plates separated by a distance smaller than the size of the smallest bubbles. In this case, currently under study and which will be reported separately, it is easy to demonstrate that the same equations as the ones presented here can be used. However, it will be most interesting to make a computer simulation of a 2D foam using the theoretical model and some of the results of experiment, in order to play with all variables involved, thus understanding better the phenomena through an “software experiment”. It is important to recall that these systems can be modelled mathematically as a cellular automata, because they behave according to their number of neighbors. (Cellular structures can be modelled as cellular automata! ) Some interesting experiments, however, remain to be performed. For instance, the equations for a statistical (random) distribution of cells can also be obtained from similar considerations, with more complicated mathematics (this also will be reported elsewhere). Also the effect of diffusion, temperature, exterior pressure and surface tension can also be modelled and tested experimentally, opening new and exciting possibilities for playing around with bubbles.
323
References [ 1 ] J.H. Aubert, A.N. Kryanik and P.B. Rand, Sci. Am. May (1986) 74. [ 21 H. Flyvbjerg and C. Jeppensen, Physica Scripta 38 ( 199 I ) 49. [ 31 D. Weaire and N. Rivier, Contemp. Phys. 25 (1984) 59. [4] N. Rivier and A. Lissowski, J. Phys. A IS (1982) L143. [ 51 J.E. Arron and D. Levine, Phys. Rev. Letters 69 ( 1992) 208. [ 61 J. Stavans and J.A. Glazier, Phys. Rev. Letters 62 (1989) 1318. [7] J. Glazier and J. Stavans, Phys. Rev. A 40 (1989) 7398. [S] J.C.M. Mombach, M.A.Z. Vasconcellons and R.M.C. de Almeida, J. Phys. D 23 (1990) 600. [9] L.J. Gibson, M.F. Ashby, G.S. Schajer and C.I. Robertson, Proc. Roy. Sot. A 382 (1982) 25. [ IO] L.J. Gibson and M.F. Ashby, Proc. Roy. Sot. A 382 (1982) 43. [ I 1 ] M.J. Mansfield, Introduction a la Topologia (Alhambra, Madrid, 1974). [ 121 F.J. Almgrem and J.E. Taylor, Sci. Am. July (1976) 82. [ 131 C.S. Smith, Sci. Am. January ( 1954) 58. [ 141 J. Glazier, S.P. Gross and J. Stavans, Physical Rev. A 36 (1987) 306. [ 151 J. Stavans, E. Domany and D. Mukamel, Europhys. Letters April ( 199 1) 479.
Materials Letters 19 ( 1994) 3 17-323
Mechanical model for two-dimensional coarsening foams A. Jimtnez-Ceniceros,
G. VBzquez-Polo,
C. Renero, V.M. CastaiTo ’
Institute de Fisica, U.N.A.M., Apartado Postal 20-364, Mexico. D.F. 01000, Mexico
Received 11 November 1993; in final form 15 February 1994; accepted 17 February 1994
Abstract
A simple mechanical model for the structure of theoretical two-dimensional (2D) foams is presented. The proposed model is based on simple geometrical and energy considerations in foams. The structure and evolution of a theoretical system are also described as an example predicted by the model.
1. Introduction Cellular materials have represented a continuing area of interest for scientists for many years. There are several reasons for this to happen: firstly, the amazing fact of many quite different physical systems having basically the same cellular structure, ranging from the suds of a shampoo to the grain structure of metals and the arrangement of vegetable tissues [ l-8 ] ; and secondly, because of the potential technological applications of foams or cellular materials, such as stifling of explosions [ 1 ] or production of high-strength materials [ 9, lo]. In fact, much effort, both theoretical and experimental, has been dedicated throughout the years to the study of the structure first and then the dynamics of soap froths and other cellular systems in two and three dimensions. Thus, it is possible to find in the literature works on the behavior of the area of cells as a function of time [ 6,7,14], reporting the existence of two kinds of dynamics recognizable in a foam [ 7 1, or on the correlation between size and shape [4] through topological considerations, etc. Because of topological restrictions (basically Eu-
ler’s law), cellular systems are dimension-sensitive, that is, the dimension of the foam plays a definite role in some structural characteristics of the system. In particular, two-dimensional froths have attracted a great deal of attention because, among other reasons, of their relevance in metallurgy and biology [ 8 1. In the specific case of 2D foams, however, most articles deal either with geometry and topology alone or with the temporal evolution of the so-called topological valences [ 111 and scarce effort has been dedicated to describe the mechanical aspects present in 2D cellular systems, although those parameters are extremely relevant to the physico-chemical stability of foams, namely because the equilibrium of forces in foams is achieved through surfactants, foaming agents or stabilizers. In this article, we present a simple model of the mechanical structure of a theoretical 2D foam, fulfilling the topological restrictions and allowing one to evaluate the coarsening (i.e. time evolution indirectly) of this system. The aim is to establish a theoretical framework, based on simple mechanical considerations, for the understanding of properties and potential applications of 2D cellular systems.
’ Author to whom correspondence should be addressed. 0167-577x/94/$07.00
0 1994 Elsevier Science B.V. All rights reserved
SSDI0167-577x(94)00037-N
A. Jimhez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
318
2. The model
@,=v,(P,-Pa)
Let us consider a “cellular-like” 2D lattice formed with closely regular polygons with the same length side which obeys Plateau’s criteria [ 221. The general formula for the area A of an i regular polygon is given by: A, = $L,a, ,
(1)
where Lj is the perimeter
and a, the height. Obviously
Li=N,li) fi being the length of a side of the ith polygon and Ni the number of faces. From Fig. 1 it can be obtained that
and, since 8= x/N,, then A, =
NJ? 4 tan(x/N,)
(2)
’
For the surface of the cell, we notice from Fig. 1 that the interacting surfaces are the ones from one the side of each cell, but there is no interaction on the bottom and the top surfaces. So, the interaction surface will be Si=Lih or
where the exterior pression is PO and the interior pression is Pi and the volume of the ith cell is V,. Volume I’, is equal to Aih, A, being the area and h its thickness, which is constant. Remember that we assume that A, >> h. Also, the free surface energy Ei of the cell due to the walls that separate each cell from its neighbors is E,=cxS;,
(4)
where S, is the surface area and (Ythe surface tension. Eq. (4) explains why a foam shows a tendency to reduce the interfacial area, since any increase in this surface would represent an increase of the free surface energy of the system. Under mechanical equilibrium conditions, it must be true that the energy due to the pressure difference @ exerted by the air on the walls is approximately equal to the free energy E due to the surface tension over that same wall within each cell, that is Ei-@=O.
(5)
It is important to notice that Eq. (5) implies a quasistatic condition, that is, we are describing a metastable state at a given instant. This approximation is adequate for cellular systems with slow dynamics, which is the case for many practical examples. By using the above Eqs. (3 )- ( 5 ) and the fact that for polygonal cells we have the following relations:
St =Nilih . First, consider the energy situation in a cell. Since we have some gas pressure inside and outside each cell, we define the energy due to pressure difference as
(3)
1
K=Aih=
NJfh 4 tan(x/N,)
’
Si=Nllih, it is very simple to obtain that p _p I
0
= 4otan(xlNi)
(6)
4
and, by recalling that pressure is forcefper we get
T Oi 1
Fig. 1. Geometrical
parameters
of a 2D polygon.
1;=fo+4aN,
tan(x/N,)
.
unit area,
(7)
In this way, we have now an equation to calculate the force between cells in terms only of the surface tension, exterior pressure of the system and the number of sides of each cell. It must be noticed that in Eq. (7) we have eliminated I,, the length of each side. We
A. Jimknez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
must say that this is possible only if we consider this value as being approximately the same for all cells throughout the froth, although it can be increased globally through time, so implying that our model is valid for foams in which the scale of bubbles is of the same order of magnitude. This is consistent with Mullin’s statistical self-similarity hypothesis [ 12- 14 1. Moreover, we can get rid offo since one is basically interested in the difference of forces between two adjacent cells i and i. Thus, it is straightforward to obtain finally: .f;-J;=4c~[N,
tan(x/N,)-N,
tan(rr/N,)]
.
(8)
The model allows one to evaluate also the corresponding radii of curvature R,, of the surface between cells i and j, through the relation P,-Pj=2a/R,,
.
Thus R,,=f[tan(x/N,)-tan(n/N,)]-’
.
(9)
3. Results and discussion Eqs. (7), (8) and (9) of section 2 can now be utilized to obtain Tables 1, 2 and 3. In Table 1, for instance, it can be observed that the pressure in a cell decreases as the number of sides of that particular cell is increased; for example, a change from 3 to 10 sides represents a decrease of about 40% in the corresponding pressure. These first results provide a clue as to why the evolution of foams involves formation of cells with more and more sides, as has been observed experimentally. More interesting results are contained in Table 2, Table
I Number of sides N,
Pressure force in the cell f;-&=4olN,tan(n/N,)
3 4 5 6 7 8 9 10
20.74601 16.000~~ 14.53001 13.85601 13.484~~ 13.254~~ I3.109a 12.996a
319
which shows the pressure difference between cells of different number of sides. From Table 2 we can observe again that the smaller the number of sides of a given cell, the higher the pressure exerted by the neighboring cells on that particular cell. The sign of the pressure indicates the convexity or concavity of the wall between cells. Here, the values of pressure equal to zero indicate that the wall is in equilibrium, hence forming a flat interface. Thus, we observe that the more similar (in terms of number of sides) the cells, the smaller the pressure difference. This also explains some of the observed characteristics of foam coarsening. Table 3 contains the calculations of radii of curvature of the walls as a function of the number of sides N, and N, of the adjacent cells. The value co indicates, of course, flat interfaces. This is a particularly important result, since it allows one to begin to understand the dynamics of the foam. Indeed, the need of the system to relieve pressure from one cell to another, allows gas to diffuse from cells with less sides to those with more sides. From Tables l-3 one can observe that this process would take place around cells with six sides. That is, cells with less than six sides would eventually disappear whereas those with more than six sides would grow further. The process is rather interesting: when some cells disappear, the overall area is preserved, leading to some other cells to increase the length of their sides I,, producing less and larger cells. But, to minimize area in the new state, the system re-orders itself, repeating the process until the pressure difference is so small that the system reaches the steady state. In this way, our model can explain both the slow and fast dynamics reported previously. To show the feasibility of simulating coarsening of a given foam from the model, let us consider the series of simulated foams of Figs. 2a through 2f. The first corresponds to the initial state where we have all hexagonal cells except two (one is pentagonal and the other is heptagonal). By applying the above criteria and equations one can observe the coarsening until basically a single bubble is formed. The slow mechanism of evolution for bubbles with less than three sides is [ 15 ] : ( 1) Cells with three sides disappear, each leaving its three neighbors with one side less. (2) Cells with four sides disappear, each leaving
A. Jimknez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
320
Table 2 Relative pressure difference between adjacent cells Pressure difference
Ni
3 4 5 6 7 8 9 10
3-N,
4-N,
5-N,
6-N,
7-N,
8-N,
9-N,
10-N,
0.000 4.784 6.253 6.928 7.300 7.524 7.68 1 7.787
-4.785 0.000 1.469 2.143 2.515 2.745 2.897 3.003
-6.254 - 1.469 0.000 0.674 1.046 1.276 1.428 1.534
-6.928 -2.144 -0.675 0.000 0.372 0.601 0.753 0.859
-7.301 -2.516 - 1.047 -0.372 0.000 0.229 0.381 0.487
-7.525 -2.745 - 1.276 - 0.602 -0.229 0.000 0.151 0.258
- 7.682 - 2.897 - 1.428 -0.654 -0.381 -0.152 0.000 0.106
-7.788 - 3.003 - 1.534 -0.860 -0.487 -0.258 -0.106 0.000
Table 3 Radii of curvature
of the walls
R,
j
i
3
4
5
6
7
8
9
10
3 4 5 6 7 8 9 10
:683 0.497 0.433 0.399 0.379 0.365 0.355
-0.683 00 1.828 1.183 0.964 0.853 0.786 0.740
-0.497 - 1.828
-0.433 - 1.183 -3.351 00 5.220 3.064 2.342 1.980
- 0.400 - 0.965 -2.041 - 5.220 co 7.422 4.25 1 3.191
-0.379 -0.854 - 1.601 - 3.065 - 7.423 to 9.951 5.599
-0.366 -0.786 - 1.379 -2.343 -4.251 -9.952
-0.355 -0.741 - 1.245 - 1.981 -3.192 - 5.600 - 12.80 00
Y351 2.041 1.600 1.379 1.244
12;04
two of its four neighbors with one side less. (3) Cells with five sides disappear, each leaving two of its neighbors with one side less and one with one more side. It is possible to demonstrate that, if one has a 2D cellular array with N bubbles and a mean number of sides r equal to six exactly, this number will not change due to the slow dynamics, so we can presume that if this mean value changes, it will be due to the fast dynamics (rearrangement of bubbles or breakage of walls). Suppose we have cellular array with N bubbles, with one of them having three sides, N- 2 cells having six and the last one nine sides. The mean number of sides is
Immediately after the cell with three sides disappears, three of its neighbors will have lost a side, and the total number of bubbles will now be N- 1. It can happen that three cells with six sides lose one (and then the total number of cells with six sides becomes N-5) or two cells with six sides and the one with nine lose one side (in this case, the total number of cells with six sides becomes N-4). For the first case we have
[= 3+6(N-2)+9 N
I= 5~2+6(N-4)+8 N-l
=
12+6NN
12
6N =N=6.
[= 5x3+6(N-5)+9 N-l =
24+6N-30 N-l
6N-6 =- N-l
=6
-- 6N-6 - N-l
=6.
and for the second:
=
18+6N-24 N-l
A. ~~rn~nez-Ceni~~ros et al. /~a~er~a[s Letters I9 (1994) 317-323
321
Fig. 2. (a) through (f ). Coarsening of a two-dimensional theoretical foam, (a) being the initial configuration and (f) corresponding to the tinal state.
Now suppose we have a cellular array with N bubbles, where one cell has four sides, N- 2 cells six and one eight sides. The mean number of sides is
[_ 4+6(N-2)+8 N =
12+6NN
12 -- 6N -N=6,
comes N- 1. Again we have two cases: that two cells with six sides lose one side each, or the one with eight loses one side and one cell with six, too. So we have for the first case: Qx2+6(N-4)+8 N =
and after the cell with four sides disappears, N be-
18+6N-24 N-l
6N-6
=yyy=6,
and for the second: r
5x7+6(N-15)+13 N-7
=
[_ 5x1-6(N-3)+7 N-l = =
12+6N-
18
N-l
6N-6 --=6. - N-L
Finally, consider a cellular array with N bubbles, being one with five sides, N- 2 with six and one with seven. The mean number of sides is r
=
12+6N-12 N
-_ 6N =6. -N
=
18-24fSN N-l
6N-6 =N-_l
=6 *
Now let us follow a little more the evolution of this array, to show that N is a constant. We will assume that the ceil with larger number of sides is the one that grows, which is consistent with the theory. When the two five-sided bubbles disappear, there are now four with five sides (because four six-sided cells lose one side each), one with ten sides (because the one with eight sides gains one side from each of the five-sided ones), the total number of bubbles is now N- 3 and the total number of six-sided cells is N- 8, so 5~4+6(N-8)+10
/-_
=
19
6N+84168 6N-84 =N_14 N- 14
6N+ 120-282 N-27
=6 ’
+25
6N- 162 =6 = N-27 ’
[_ 5~25+6fN-72)+31 N-46 =
6N+432N-46
156
6N-276 N-46
=6
=
6N-426 N-71
=6
=
’
[_ 5x31+6fN-103)+37 N-71 =
5x2+6[N-4)+8 N-l
=
[_- 5x 13+6(N-28)+ N-14
=
When the bubble with five sides disappears, two bubbles will lose a side, and one will gain one side. For simplicity, we will consider here one case: we will assume that the one with seven gains a side, and two of six lose one, but it is easy to verify that N is constant. So we have, after the one with five sides disappears, that Nbecomes N- 1 and: r
6N-42 =-------=6, N-7
N-7
[_ 5x 19+6cN-47) N-27
5+6(N-2)+7 N
=
6N+48-90
6N+ 188-618 N-71
’
It is easy to notice when rchanges due to the diminishing of total number of bubbles N (if it is nut exactly six ) and when due to fast dynamics, and thus it is even possible to create an algorithm which, given the initial experimental values of N, each n (number of bubbles with i sides) and < provides us with the subsequent values of these variables, and try to see how and why it differs from the experimental data. Because variations in the experimental value of the mean number of sides [with the theoretical one depend only on the behavior of the fast dynamics, therefore one can obtain some rules for this behavior expe~mentally.
4. Final remarks
N-3 =
6N+30-48 N-3
6N-18 =N-3
and so on (see Figs. 2a-2f ):
=6
The structure and the basic dynamics of 2D foams can be explained by a simple mechanical model. The main features observed by previous researchers have been reproduced by the model. It is also interesting
A. Jimtkez-Ceniceros et al. /Materials Letters 19 (1994) 317-323
to remark that the model can be experimentally tested since a quasi 2D foam can be produced by using two parallel plates separated by a distance smaller than the size of the smallest bubbles. In this case, currently under study and which will be reported separately, it is easy to demonstrate that the same equations as the ones presented here can be used. However, it will be most interesting to make a computer simulation of a 2D foam using the theoretical model and some of the results of experiment, in order to play with all variables involved, thus understanding better the phenomena through an “software experiment”. It is important to recall that these systems can be modelled mathematically as a cellular automata, because they behave according to their number of neighbors. (Cellular structures can be modelled as cellular automata! ) Some interesting experiments, however, remain to be performed. For instance, the equations for a statistical (random) distribution of cells can also be obtained from similar considerations, with more complicated mathematics (this also will be reported elsewhere). Also the effect of diffusion, temperature, exterior pressure and surface tension can also be modelled and tested experimentally, opening new and exciting possibilities for playing around with bubbles.
323
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