- Email: [email protected]
Relationship between the microstructure and the water permeability of transparent gas barrier coatings
Surface
and Coatings
Technology
100-101
(1998)
459-462
Relationship between the microstructure and the water permeability of transparent gas barrier coatings G. Garcia-Ayuso
b, R. Salvarezza ‘, J.M. Martinez-Duart
b, 0. Sknchez a, L. Vizquez a3*
a Institute de Ciencia de Materiales de Madrid (CSIC). Cantoblanco, 28049 Madrid. Spain b Departamento Fisica Aplicada C-XII, Unitlersidad Autdnoma de Madrid, Cantoblanco. 28049 Madrid, ’ Visiting researcher from INIFTA, La Plata, Argentina
Spain
Abstract
The relationship betweenthe microstructure and the water vapor permeability of transparent barrier coatings on polymeric films hasbeenstudiedby atomic force microscopy(AFM). The averagegrain size,d, and the surfacepore fractal dimension,D, , of the different coatingsweredeterminedfrom the AFM images.It wasobservedthat thosefilms with low d and high D, values presenteda low water vapor permeability whereaswhen either d washigh or D, was low the water vapor permeability increased. Thesedifferencesare explained on the basisof the different dependences of the two main water moleculetransport mechanisms (Knudsen and the surfacediffusion) on the coating microstructure.0 1998Elsevier ScienceS.A. Keywords:
Atomic force microscopy; Barrier coatings;Water vapor permeability
expressed, in a first approximation, as:
1. Introduction Microporous barrier coatings are widely usedin many technological and scientific applications. In particular, transparent barrier coatings (silica, alumina, etc.) on polymeric flexible substrates are mainly used for the packaging industry. Such coatings must have a low permeability of gas molecules such as water and oxygen in order to preserve the product quality. It is well known that there are different permeation mechanismsdepending on both the coating pore size and the nature of the diffusion molecule [ 1,2]. For pores with size from 2 to 100 nm, oxygen and water molecules are transported by the Knudsen mechanism (i.e. gas molecule flow through the pore volume) but water molecules permeation may have also a strong contribution from the surface diffusion transport mechanism. It has been shown that the Knudsen permeability is proportional to r/tg whereas the surface assisted permeability changes as l/(rz,), where r is the pore size, ~~is the pore network tortuosity and 7S is the pore surface tortuosity [3,4]. It can be assumedthat the pore size is proportional to the grain size d; thus, the coating permeability, P, can be
* Corresponding author. e-mail: [email protected]
Tel:
34-l-334-9084;
0257-8972/98/$19.00 0 1998 Elsevier PII SO257-8972(97)00671-3
Fax:
34-I-372-0623;
Science B.V. All rights reserved.
(1) where A and B are constants. Eq. (l), which contains both the surface diffusion transport and the Knudsen transport structural dependences,shows the strong relationship between the coating microstructure and P. It is well established [5] that surface irregularity is quantitatively described in terms of fractal geometry through the concept of fractal surface dimension D,. Thus, when OS+2 the surface is smooth and no traps for surface diffusion of molecules exist, whereas when OS+3 the surface tends to be rather irregular with many traps, so that the net displacement of diffusing molecules through the barrier over the surface by random walk is largely hindered [6]. As the fractal dimensional increment, AD,, defined as AD,= D,-2 [7], increases from 0 to 1 the surface irregularities become increasingly predominant. Thus, the determination of D, could be a way of calculating Z, since both parameters reflect the degree of surface pore complexity (z,cc D,). In this work we have calculated the values of d and D, from the atomic force microscopy (AFM) images of the barrier coating. However, the determination of zg from AFM data is not possible. Thus, we have studied the relation-
ship between the water vapor coating permeability and the coating microstructure, the latter analyzed by AFM.
2. Experimental 2.1. Smiple prepurution Two types of coatings were investigated. First. silica/alumina films deposited by co-evaporation of SiO, (electron beam gun evaporated ) and Al (thermally evaporated ) in a roll-to-roll coater. Secondly, silica films produced by plasma-enhanced chemical vapor deposition (PECVD) in a batch coater from a gas mixture of silane and nitrous oxide as precursors gases and nitrogen as carrier gas. In both cases, the substrate was a commercial polyester [polyethylene terephthalate (PET)] film 12 urn thick. The permeability of the samples to moisture was measured using a Permatran W-200 water vapor permeation system under tropical conditions (38 ‘C and 90% relative humidity). 2.2. AFM chacterirution AFM measurements were made with a Nanoscope III (Digital Instruments, Santa Barbara, CA) microscope operating in the tapping mode with silicon cantilevers. All images were taken in air and consisted of 5 12 x 512 data points. The grain size distribution and the average grain size, d are directly obtained from the AFM images. The D, determination is based on the “slit island analysis” method proposed by Mandelbrot et al. [7]. The application of this analysis procedure to AFM images has been explained elsewhere [8,9]. Briefly, it consists of “sectioning” the surface images obtained by AFM with a plane at a given height (equivalent to “fill with water” the structure up to a given level). This procedure defines “lake looking” areas (water-filled regions) surrounded by the surface morphology. Then, in these “lakes” it is analyzed both the “lake” area, A, as well as the corresponding “lake” perimeter, L. This procedure is repeated for all the “lakes” defined for the different height values and the data pairs (A,L) are plotted in a double logarithmic graph. These points are arranged along a straight line with a slope M. Finally, the values of the fractal dimension of the pore surface D, (i.e. 5,) can be obtained from the relation D, =2M+ 1 [6,7].
3. Results Fig. 1 shows typical AFM images of a silica/alumina coating with low Pwater values (Fig. la). Fig. 1b displays the same image of Fig. la but after “water filling” the sample surface up to a given level in order to show the
perimeter and area of the “lakes”. Similar images are shown in Fig. lc and d for the PECVD samples. From the AFM images the grain size distribution of all the samples was determined (Fig. 2). It was found that the co-evaporated samples presented a narrower grain size distribution and a smaller, even 10 times smaller, average grain size than the PECVD samples. Fig. 3 depicts the Log L vs Log A plots for a silica/alumina sample with low P,,,,, and for a PECVD sample. Both samples present D, values close to 2.55. A similar analysis was carried out on all the samples. The P water vs D, data are plotted in Fig. 4 (the permeabilities are normalized to the water vapor permeability of the PET substrate) where it is observed that for the silica/alumina samples Pwaterx l/(AD,)‘.‘. However, for the PECVD samples P,,,,, does not depend on D, (the P water values are high despite of the high D, value).
4. Discussion The data obtained for both types of samples look contradictory at first glance, as samples with similar D, values show quite different values of the permeability. It should be noted that no dependence of Pw,,,r on either coating thickness or silica/alumina content was previously found [9]. However, this apparent contradiction can be explained by means of Eq. ( 1) where P,,,,, has two contributions, the surface diffusion term (PK l/(z,d)) and the Knudsen term (Peed/z,). Thus, if d is too large the Knudsen term dominates, despite the value of r5, (i.e. D,). In the PECVD samples it can be observed from the AFM images that the pores defined between grains are quite large (Z 100 nm). In this case, the Knudsen transport mechanism should be the predominant one [ 1,2], leading to high P,,,,, values. For the co-evaporated coatings the situation is rather different as all the samples, despite their P,,,,, value, have a similar, small, average grain size. It is evident that Pwater decreases as D,, and therefore AD,, increases. This effect can be explained in terms of the different surface pore tortuosity: for those samples with D, values close to 2 (ADS-O, implying low t,) the internal surface will be less irregular and less tortuous than for those samples with higher D, values which are more irregular (AD,+1 and, therefore, high ts). Then, the water molecules would diffuse faster on a smooth surface (as that of the samples with low D, values) than on a fractal one [6]. Furthermore, it was confirmed that in the behavior observed for the co-evaporated samples in Fig. 4 there was not a strong contribution from the Knudsen transport mechanism. which operates in parallel to the surface dilfusion mechanism [9]. The previous explanation of our experimental results can be better understood from Fig. 5 where the d vs
G. Garcia-A?uso
et al. J Swfbce
utd Coatings
(4
Technology
(1998)
459-462
461
(b)
Fig. I. (a) 0.48 x 0.48 pm2 AFM image of a silica/alumina co-evaporated sample given height level. The “lakes” so formed are clearly shown. (c) I x I pm* AFM (c> but “5Se6 wjfn waler” up 10 a @Yen hti&t 1evd.
1/(d(A0,)‘.4) values for all coatings, PECVD (marked as n ) and co-evaporated (marked 0) are plotted. Those films with PwaterjPPET CO.36 are indicated with the corresponding symbol “open”. It can be observed in Fig. 5 that two regions can be clearly distinguished, delimited by the arch of a circle. Inside the circle we have low R GrtT Gha wi& ha& se2.2~~s%ixx size {i.e. s.8~ pax\\ ad3 ‘%i&s3 23s 3’l.e. g3. T-w s&s %ms ‘he %k&Een ~ wmW&&ion is rl&igibl~ arid Chc s&ace d&s& W&n%port mechanism dominates leading to the observed low
100-101
with low fWater. (b) Same as (a) but “filled with water” up to a image of a silica PECVD sample with high P,.,,,. (d) Same as
P,,,,, values because of the high surface pore irregularity. Outside the limiting arch of the circle the points correspond to film coatings with high P,,,, (poor barrier properties). Among these coatings, two types of samples can be distinguished: the co-evaporated coatings, with small grain (i.e. pore) size and low D, values, ad the PEC\ID ones wi& page ST&in~gj~yl -*& r_+. xns\‘T co awii ,$xeE& ‘&3 ?-d,&qr -+ahzs bxaz of the hi4 3urCxe pore Gx
0.6
: z
0.8
5
0.6
1,
a 0.4 zi‘m n g 0.2
"s 0 0.4 .g $
.
.
.
7
0.2 0.0
.
I
0
100
200 Grain
Fig. 2. Typical grain PECVD (.) samples.
300 size
size distributions
400
500
(- ) and
4’s,
2.5 2.0
i
r
1 3.0
I
3.5
4.0
Lo~,~
4.5
/ 5.0
I
150
I
200
(nm)
Fig. 5. l/(rl(AD,)l-‘) vs ti plot for the co-evaporated (0) and PECVD ( n ) samples. “Open” symbols indicate those co-evaporated coatings The arch of the circle delimits the low P,,,,, with pwater /P,,,
’
5.5
Acknowledgement
(A (11111~))
Fig. 3. Log L vs Log A plots for a co-evaporated sample with low P,,,,, (0) and a PECVD sample with high Pwate, (0). The corresponding D, values, obtained from the slopes of the curves, are indicated.
.
1 .o
.
. 0.8 L P
I
100
one. In this case, P,,,,, reaches high values despite the high value of the surface tortuosity. In conclusion, this work shows that AFM imaging can be a useful technique to study the relationship between the barrier coating surface microstructure and the water vapor permeability. This technique allows one to evaluate the relative contribution of the Knudsen and surface diffusion transport mechanisms to the total permeability.
4.0
2.5
50
d
(nm)
of the co-evaporated
l. . .
.
This work was financially supported through BRITE EURAM II project BRE-2-CT92-0237 and the CSIC/CONICET cooperation program. The authors are indebted to CE.TE.V (Italy) for providing the samples and performing the water vapor diffusivity measurements. They are also indebted to Otto Nielsen Emballage (Denmark) for performing the oxygen permeability measurements.
0.6-
p. -La
Q 3
0.4 0.2 -
-I
O.O1 Fig. 4. Normalized water permeability Pwater/PPET vs D, plot for the co-evaporated (2) and PECVD samples ( l ), The continuous line indicates the best fit (P Waters ( I/(AD,)‘+‘)) for the co-evaporated samples.
diffusion one but, in this case, the water molecules transport is enhanced due to the smooth and Euclidean pore morphology. For the PECVD coatings the situation is quite different, as the pores are large enough for the Knudsen transport mechanism to become the prevalent
[I] K.-H. Lee, S.-T. Hwang, J. Colloid Interface Sci. I10 ( 1986) 544. [2] R. Pfetler. M. Ohring. J. Appl. Phys. 52 (1981) 777. [3] R. Datta, S. Dechapanichkul. J.S. Kim, L.Y. Fang, H. Uehara, J. Membrane Sci. 75 ( 1992) 245. [4] Y. Horiguchi, R.R. Hudgins, P.L. Silveston, Can. J. Chem. Eng. 49 (1971) 76. [5] B.B. Mandelbrot,
The Fractal Geometry of Nature. W.H. Freeman. New York, 1982. [6] P. Pfeifer and M. Obert, in: D. Avnir (Ed.), The fractal approach to hetereogeneous chemistry, Wiley. New York, 1989. p. I I. [7] B.B. Mandelbrot. D.E. Passoja. A.J. Paullay. Nature 308 (1984) 721.
[S] J.M. Gomez-Rodriguez. L. Vazquez, A.M. Barb. R.C. Salvarezza, J.M. Vara. A.J. Arvia. J. Phys. Chem. 96 ( 1992) 347. [9] G. Garcia-Ayuso, R. Salvarezza. J.M. Martinez-Duart. 0. Sanchez. L. Vazquez, Adv. Mater. 9 (1997) 6544658.
and Coatings
Technology
100-101
(1998)
459-462
Relationship between the microstructure and the water permeability of transparent gas barrier coatings G. Garcia-Ayuso
b, R. Salvarezza ‘, J.M. Martinez-Duart
b, 0. Sknchez a, L. Vizquez a3*
a Institute de Ciencia de Materiales de Madrid (CSIC). Cantoblanco, 28049 Madrid. Spain b Departamento Fisica Aplicada C-XII, Unitlersidad Autdnoma de Madrid, Cantoblanco. 28049 Madrid, ’ Visiting researcher from INIFTA, La Plata, Argentina
Spain
Abstract
The relationship betweenthe microstructure and the water vapor permeability of transparent barrier coatings on polymeric films hasbeenstudiedby atomic force microscopy(AFM). The averagegrain size,d, and the surfacepore fractal dimension,D, , of the different coatingsweredeterminedfrom the AFM images.It wasobservedthat thosefilms with low d and high D, values presenteda low water vapor permeability whereaswhen either d washigh or D, was low the water vapor permeability increased. Thesedifferencesare explained on the basisof the different dependences of the two main water moleculetransport mechanisms (Knudsen and the surfacediffusion) on the coating microstructure.0 1998Elsevier ScienceS.A. Keywords:
Atomic force microscopy; Barrier coatings;Water vapor permeability
expressed, in a first approximation, as:
1. Introduction Microporous barrier coatings are widely usedin many technological and scientific applications. In particular, transparent barrier coatings (silica, alumina, etc.) on polymeric flexible substrates are mainly used for the packaging industry. Such coatings must have a low permeability of gas molecules such as water and oxygen in order to preserve the product quality. It is well known that there are different permeation mechanismsdepending on both the coating pore size and the nature of the diffusion molecule [ 1,2]. For pores with size from 2 to 100 nm, oxygen and water molecules are transported by the Knudsen mechanism (i.e. gas molecule flow through the pore volume) but water molecules permeation may have also a strong contribution from the surface diffusion transport mechanism. It has been shown that the Knudsen permeability is proportional to r/tg whereas the surface assisted permeability changes as l/(rz,), where r is the pore size, ~~is the pore network tortuosity and 7S is the pore surface tortuosity [3,4]. It can be assumedthat the pore size is proportional to the grain size d; thus, the coating permeability, P, can be
* Corresponding author. e-mail: [email protected]
Tel:
34-l-334-9084;
0257-8972/98/$19.00 0 1998 Elsevier PII SO257-8972(97)00671-3
Fax:
34-I-372-0623;
Science B.V. All rights reserved.
(1) where A and B are constants. Eq. (l), which contains both the surface diffusion transport and the Knudsen transport structural dependences,shows the strong relationship between the coating microstructure and P. It is well established [5] that surface irregularity is quantitatively described in terms of fractal geometry through the concept of fractal surface dimension D,. Thus, when OS+2 the surface is smooth and no traps for surface diffusion of molecules exist, whereas when OS+3 the surface tends to be rather irregular with many traps, so that the net displacement of diffusing molecules through the barrier over the surface by random walk is largely hindered [6]. As the fractal dimensional increment, AD,, defined as AD,= D,-2 [7], increases from 0 to 1 the surface irregularities become increasingly predominant. Thus, the determination of D, could be a way of calculating Z, since both parameters reflect the degree of surface pore complexity (z,cc D,). In this work we have calculated the values of d and D, from the atomic force microscopy (AFM) images of the barrier coating. However, the determination of zg from AFM data is not possible. Thus, we have studied the relation-
ship between the water vapor coating permeability and the coating microstructure, the latter analyzed by AFM.
2. Experimental 2.1. Smiple prepurution Two types of coatings were investigated. First. silica/alumina films deposited by co-evaporation of SiO, (electron beam gun evaporated ) and Al (thermally evaporated ) in a roll-to-roll coater. Secondly, silica films produced by plasma-enhanced chemical vapor deposition (PECVD) in a batch coater from a gas mixture of silane and nitrous oxide as precursors gases and nitrogen as carrier gas. In both cases, the substrate was a commercial polyester [polyethylene terephthalate (PET)] film 12 urn thick. The permeability of the samples to moisture was measured using a Permatran W-200 water vapor permeation system under tropical conditions (38 ‘C and 90% relative humidity). 2.2. AFM chacterirution AFM measurements were made with a Nanoscope III (Digital Instruments, Santa Barbara, CA) microscope operating in the tapping mode with silicon cantilevers. All images were taken in air and consisted of 5 12 x 512 data points. The grain size distribution and the average grain size, d are directly obtained from the AFM images. The D, determination is based on the “slit island analysis” method proposed by Mandelbrot et al. [7]. The application of this analysis procedure to AFM images has been explained elsewhere [8,9]. Briefly, it consists of “sectioning” the surface images obtained by AFM with a plane at a given height (equivalent to “fill with water” the structure up to a given level). This procedure defines “lake looking” areas (water-filled regions) surrounded by the surface morphology. Then, in these “lakes” it is analyzed both the “lake” area, A, as well as the corresponding “lake” perimeter, L. This procedure is repeated for all the “lakes” defined for the different height values and the data pairs (A,L) are plotted in a double logarithmic graph. These points are arranged along a straight line with a slope M. Finally, the values of the fractal dimension of the pore surface D, (i.e. 5,) can be obtained from the relation D, =2M+ 1 [6,7].
3. Results Fig. 1 shows typical AFM images of a silica/alumina coating with low Pwater values (Fig. la). Fig. 1b displays the same image of Fig. la but after “water filling” the sample surface up to a given level in order to show the
perimeter and area of the “lakes”. Similar images are shown in Fig. lc and d for the PECVD samples. From the AFM images the grain size distribution of all the samples was determined (Fig. 2). It was found that the co-evaporated samples presented a narrower grain size distribution and a smaller, even 10 times smaller, average grain size than the PECVD samples. Fig. 3 depicts the Log L vs Log A plots for a silica/alumina sample with low P,,,,, and for a PECVD sample. Both samples present D, values close to 2.55. A similar analysis was carried out on all the samples. The P water vs D, data are plotted in Fig. 4 (the permeabilities are normalized to the water vapor permeability of the PET substrate) where it is observed that for the silica/alumina samples Pwaterx l/(AD,)‘.‘. However, for the PECVD samples P,,,,, does not depend on D, (the P water values are high despite of the high D, value).
4. Discussion The data obtained for both types of samples look contradictory at first glance, as samples with similar D, values show quite different values of the permeability. It should be noted that no dependence of Pw,,,r on either coating thickness or silica/alumina content was previously found [9]. However, this apparent contradiction can be explained by means of Eq. ( 1) where P,,,,, has two contributions, the surface diffusion term (PK l/(z,d)) and the Knudsen term (Peed/z,). Thus, if d is too large the Knudsen term dominates, despite the value of r5, (i.e. D,). In the PECVD samples it can be observed from the AFM images that the pores defined between grains are quite large (Z 100 nm). In this case, the Knudsen transport mechanism should be the predominant one [ 1,2], leading to high P,,,,, values. For the co-evaporated coatings the situation is rather different as all the samples, despite their P,,,,, value, have a similar, small, average grain size. It is evident that Pwater decreases as D,, and therefore AD,, increases. This effect can be explained in terms of the different surface pore tortuosity: for those samples with D, values close to 2 (ADS-O, implying low t,) the internal surface will be less irregular and less tortuous than for those samples with higher D, values which are more irregular (AD,+1 and, therefore, high ts). Then, the water molecules would diffuse faster on a smooth surface (as that of the samples with low D, values) than on a fractal one [6]. Furthermore, it was confirmed that in the behavior observed for the co-evaporated samples in Fig. 4 there was not a strong contribution from the Knudsen transport mechanism. which operates in parallel to the surface dilfusion mechanism [9]. The previous explanation of our experimental results can be better understood from Fig. 5 where the d vs
G. Garcia-A?uso
et al. J Swfbce
utd Coatings
(4
Technology
(1998)
459-462
461
(b)
Fig. I. (a) 0.48 x 0.48 pm2 AFM image of a silica/alumina co-evaporated sample given height level. The “lakes” so formed are clearly shown. (c) I x I pm* AFM (c> but “5Se6 wjfn waler” up 10 a @Yen hti&t 1evd.
1/(d(A0,)‘.4) values for all coatings, PECVD (marked as n ) and co-evaporated (marked 0) are plotted. Those films with PwaterjPPET CO.36 are indicated with the corresponding symbol “open”. It can be observed in Fig. 5 that two regions can be clearly distinguished, delimited by the arch of a circle. Inside the circle we have low R GrtT Gha wi& ha& se2.2~~s%ixx size {i.e. s.8~ pax\\ ad3 ‘%i&s3 23s 3’l.e. g3. T-w s&s %ms ‘he %k&Een ~ wmW&&ion is rl&igibl~ arid Chc s&ace d&s& W&n%port mechanism dominates leading to the observed low
100-101
with low fWater. (b) Same as (a) but “filled with water” up to a image of a silica PECVD sample with high P,.,,,. (d) Same as
P,,,,, values because of the high surface pore irregularity. Outside the limiting arch of the circle the points correspond to film coatings with high P,,,, (poor barrier properties). Among these coatings, two types of samples can be distinguished: the co-evaporated coatings, with small grain (i.e. pore) size and low D, values, ad the PEC\ID ones wi& page ST&in
0.6
: z
0.8
5
0.6
1,
a 0.4 zi‘m n g 0.2
"s 0 0.4 .g $
.
.
.
7
0.2 0.0
.
I
0
100
200 Grain
Fig. 2. Typical grain PECVD (.) samples.
300 size
size distributions
400
500
(- ) and
4’s,
2.5 2.0
i
r
1 3.0
I
3.5
4.0
Lo~,~
4.5
/ 5.0
I
150
I
200
(nm)
Fig. 5. l/(rl(AD,)l-‘) vs ti plot for the co-evaporated (0) and PECVD ( n ) samples. “Open” symbols indicate those co-evaporated coatings The arch of the circle delimits the low P,,,,, with pwater /P,,,
’
5.5
Acknowledgement
(A (11111~))
Fig. 3. Log L vs Log A plots for a co-evaporated sample with low P,,,,, (0) and a PECVD sample with high Pwate, (0). The corresponding D, values, obtained from the slopes of the curves, are indicated.
.
1 .o
.
. 0.8 L P
I
100
one. In this case, P,,,,, reaches high values despite the high value of the surface tortuosity. In conclusion, this work shows that AFM imaging can be a useful technique to study the relationship between the barrier coating surface microstructure and the water vapor permeability. This technique allows one to evaluate the relative contribution of the Knudsen and surface diffusion transport mechanisms to the total permeability.
4.0
2.5
50
d
(nm)
of the co-evaporated
l. . .
.
This work was financially supported through BRITE EURAM II project BRE-2-CT92-0237 and the CSIC/CONICET cooperation program. The authors are indebted to CE.TE.V (Italy) for providing the samples and performing the water vapor diffusivity measurements. They are also indebted to Otto Nielsen Emballage (Denmark) for performing the oxygen permeability measurements.
0.6-
p. -La
Q 3
0.4 0.2 -
-I
O.O1 Fig. 4. Normalized water permeability Pwater/PPET vs D, plot for the co-evaporated (2) and PECVD samples ( l ), The continuous line indicates the best fit (P Waters ( I/(AD,)‘+‘)) for the co-evaporated samples.
diffusion one but, in this case, the water molecules transport is enhanced due to the smooth and Euclidean pore morphology. For the PECVD coatings the situation is quite different, as the pores are large enough for the Knudsen transport mechanism to become the prevalent
[I] K.-H. Lee, S.-T. Hwang, J. Colloid Interface Sci. I10 ( 1986) 544. [2] R. Pfetler. M. Ohring. J. Appl. Phys. 52 (1981) 777. [3] R. Datta, S. Dechapanichkul. J.S. Kim, L.Y. Fang, H. Uehara, J. Membrane Sci. 75 ( 1992) 245. [4] Y. Horiguchi, R.R. Hudgins, P.L. Silveston, Can. J. Chem. Eng. 49 (1971) 76. [5] B.B. Mandelbrot,
The Fractal Geometry of Nature. W.H. Freeman. New York, 1982. [6] P. Pfeifer and M. Obert, in: D. Avnir (Ed.), The fractal approach to hetereogeneous chemistry, Wiley. New York, 1989. p. I I. [7] B.B. Mandelbrot. D.E. Passoja. A.J. Paullay. Nature 308 (1984) 721.
[S] J.M. Gomez-Rodriguez. L. Vazquez, A.M. Barb. R.C. Salvarezza, J.M. Vara. A.J. Arvia. J. Phys. Chem. 96 ( 1992) 347. [9] G. Garcia-Ayuso, R. Salvarezza. J.M. Martinez-Duart. 0. Sanchez. L. Vazquez, Adv. Mater. 9 (1997) 6544658.