TEM MICROGRAPHS FRACTAL ANALYSIS
TEM MICROGRAPHS FRACTAL ANALYSIS Gianina Dobrescu, Maria Crisan, Maria Zaharescu, N.I.Ionescu, M.Rusu* Institute of Physical Chemistry "I.G.Murgulescu", Romanian Academy, Spl.Independentei 202, 77208, Bucharest, Romania *University of Bucharest, Physics Department, Magurele, Bucharest, Romania
Abstract: TEM micrographs can be used to compute fractal dimensions of 3-dimensional objects. In order to covert the fractal dimension of TEM micrograph to the 3-dimensional fractal dimension, a modified sphere method was developed. Using this method, fractal dimensions of some sol-gel powders were computed Key words: fractal dimension, sphere method, TEM micrograph, sol-gel powders.
1. INTRODUCTION Fractal theory was developed in the last two decades in order to achieve a better characterization of different phenomena in physics, chemistry, biology, medicine and so on [1, 2]. A direct method to determine fractal dimension is to analyze images obtained from scanning tunneling microscopy [3-6], scanning probe microscopy [7, 8], transmission electron microscopy (TEM) [9]. TEM micrographs can provide high-resolution pictures of 3-dimensional objects, such as sol-gel powders. The aim of this paper is to analyze TEM images in order to obtain fractal dimensions of 3-dimensional objects and to apply the method to some sol-gel powders. The “mass-radius” relation is used to compute fractal dimension of TEM micrograph. Some assumptions are to be made to obtain fractal dimension of 3-dimensional object from TEM micrograph fractal dimension.
2. THEORY Self-similarity, that is the main property of fractal objects, has a mathematical description [1,2]:
N (r / R) ~ (r / R) − D
(1)
where D is fractal dimension and N(r,R) is the number of boxes of size r which cover the object of linear size R. Equation (1) leads to the following methods for fractal dimension determination [11]: - the “box counting” method [12-14], when R is defined as the largest distance between any two points belonging to the system:
N box (r ) ~ r − D -
(2)
the mass-radius method:
N sites ( R) ~ R D
(3)
where Nsites(R) is the number of sites located within distance R from a given site, R>>r, (the mass in a sphere of radius R). We used equation (3) to determine TEM micrographs fractal dimensions, for R varying in a broad domain, from 1 pixel to the maximum radius of the cluster. The image fractal dimension characterizes the plane object projection. Therefore, fractal dimension of 3-dimensional object will be greater then TEM micrograph fractal dimension [15]:
D = D TEM + D o
(4)
where D is powder fractal dimension, DTEM is image fractal dimension and Do is a correction factor with fractal dimension behavior. As we already saw, to compute fractal dimension of a 3-dimensional object, we measure total number of r size boxes that covered the object. The number of r x r x r boxes needed to covered the r x r x h parallelepiped in fig.1 will be Nz(xr, yr, r), where xr and yr are center coordinates of parallelepiped xy-projection. In this moment, two different cases are to be considered: 2.1 Fractal distribution along z axes If, along z axes, 3-dimensional object exhibit a fractal behavior, it is easy to understand we can write the following relation:
N z ( xr , y r , r ) = Az r − Dz
(5)
where Dz is fractal dimension along z-axes which means is the exponent of scaling along z-axes. Equation (5) is obeyed even for r=1, and, therefore:
N z ( xr , y r ,1) = Az ( xr , y r ) = ρ ( xr , y r )
(6)
where A(xr,yr) is a constant depending on chosen center xr, yr, and it is equal with the total sum of particles along zaxes described by coordinates xr,yr. This total sum of particles can be experimentally computed from gray-level of xr,yr pixel of TEM micrograph, ρ(xr,yr) . From equation (5) and (6):
N z ( x r , y r , r ) = ρ ( x r , y r ) r − Dz
(7)
Total number of r size boxes needed to covered the whole object, N(r) can be computed as:
N (r ) =
∑N
z
( x r , y r , r ) = r − Dz
xr , y r
∑ ρ(x , y ) r
r
(8)
xr , y y
Assuming that the gray level sum ρ(xr,yr) scaling with r like:
θ (r ) =
∑ ρ ( x , y ) = Br r
− Dθ
r
(9)
xr , yt
equation (8) becomes:
N (r ) = Br − ( Dz + Dθ )
(10)
D = Dz + Dθ
(11)
From equation (2) and equation (10):
For isotrope fractals equation (5) can be written even for ox and oy axes, and Dx=Dy=Dz=D0. So:
N ~ NxNyNz ~ r
− ( Dx + D y + Dz )
D = Dx + D y + Dz = 3D0
= r −D ,
(12)
Combining equations (11) and (12) the result is straightforward:
D=
3Dθ 2
(13)
Equation (13) will be used in the following for computing fractal dimension of some sol-gel monocomponent powders, where Dθ will be TEM micrograph fractal dimension computed from gray-level distribution, using equation (9).
2.2 Uniform distribution along z axes If uniform distribution along z axes occurs, the 3-dimensional object has a fractal surface, is a self-affine object rather than a self-similar one, and equation (5) becomes:
N z ( xr , y r , r ) =
Az h r
(14)
As we discussed above:
N z ( x r , y r ,1) = A z h = ρ ( x r , y r ),
∑
N (r ) = x
r
, y
N z (x r , y r , r) = r
1 r
∑ x
r
, y
ρ(x r , y r ) = r
1 r
Br − D θ
= Kr − D
(15)
meaning that:
D = Dθ + 1 .
(16)
Fig. 1. Grey parallelepiped r x r x h
3. FRACTAL DIMENSION COMPUTATION OF SOL-GEL POWDERS Sol-gel monocomponent SiO2, AlO(OH), TiO2 oxide powders were prepared using the alkoxide route [16-18]. The size and the shape of the powders were determined by TEM using a performant JEOL-FX-2000 equipment.
Fig.2 TEM micrograph of a SiO2 powder sample The SiO2 powder shows a bimodal distribution of particles size. The particles with low dimension are of about 20 nm and the high dimension particles exceed 200 nm (Fig.2). AlO(OH) powder consists of nano-sized particles with highest tendency to aggregation (Fig.3). TiO2 powder presents aggregates of about 50 nm consisting from smaller particles (bellow 10 nm) with high tendency to crystallisation (Fig.4). In this case the low temperature of preannealing thermal treatment (300°C) required for water and organic residues evolution has induced the nanocrystallisation of the powders.
Fig. 3. and 4. TEM micrograph of a TiO2 powder sample (left) and of a AlO(OH) powder sample (right) .
For every sample 3 different TEM micrographs were analyzed [19]. Results are presented in table I. Table I Monocomponent powders fractal dimensions No.
Sample
1. 2. 3. 4. 5. 6. 7. 8. 9.
TiO2 300oC AlO(OH) 450oC SiO2 300oC
Fractal dimension Fractal dimension -short scale -large scale 2. 8 2.9 2.7 2. 8 2.7 3.0 2.7 3.0 2.8 2.9 2.8 3.0 2.8 2.9 2.8 3.0
Phase composition anatase weak crystallized boehmite weak crystallized and η(γ)-Al2O3 amorphous
Figures 5 show log-log plot of number of occupied sites within R radius sphere for micrographs in figure 2. Sol-gel powders obtained have fractal structures: TiO2 and SiO2 samples have fractal dimension of 2.8 and AlO(OH) has fractal dimension of 2.7.
12.00
logN(R)
10.00 8.00 6.00 4.00 2.00 0.00 0.00
2.00
4.00
6.00
log(R)
Fig. 5. Log-log plot of number of occupied sites within R radius sphere for micrograph in figure 2. Generally, we obtained two different fractal dimensions: one for short scales and another for long scales. For SiO2 samples (no.7-9) two types of particles characterize the system: the small ones of 10nm and the big ones of 200nm, each having different kind of aggregation. For AlO(OH) sample (no.4-6) the high aggregation lead to fractal dimensions around Euclidean dimension of 3.0. For TiO2 sample TEM micrograph no.2 there are a lot of island-like aggregations scaling different from component particles. This behavior leads to different long-range and short-range correlation and, finally, at two different fractal dimensions 2.9 and 2.7. The fractal dimension is related to the aggregation processes. These high fractal dimensions 2.7 –2.8, for all analyzed samples indicate high aggregation when clusters grow independent and high aggregation of clusters together. The fractal dimension depends also on processes that occur during thermal treatment, when adsorbed and structural water, as well as organic residues are eliminated.
4. CONCLUSIONS A method for computing fractal dimensions from TEM micrographs was developed. This method was applied to some monocomponent sol-gel powders. The investigated monocomponent SiO2, TiO2 and AlO(OH) obtained by solgel method, present good fractal structures with high fractal dimensions, between 2.7 and 3.0. This fractality
indicates high aggregation during hydrolysis-polycondensation process and during the thermal treatment at low temperature.
REFERENCES [1] B.B.Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977. [2] B.B.Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [3] M.W.Mitchell and D.A.Bonnell, Journal of Material Research, 5, 1990, 2244 . [4] J.P.Carrejo, T.Thundat, L.A.Nagahara, S.M.Lindsay and A.Majumdar, Journal of.Vacuum Science and Technology B, 9, 1991, 955. [5] D.R.Denley, Journal of Vacuum Science and Technology A, 8, 1990, 603. [6] J.M.Gomez-Rodriguez, A.M.Baro, L.Vasquez, R.C.Salvarezza, J.M.Vara and A.J.Arvia, Journal of Physical Chemistry 96, 1992, 347. [7] J.M.Williams and T.B.Beebe Jr., Journal of Physical Chemistry 97, 1993, 6249; Journal of Physical Chemistry 97, 1993, 6255. [8] J.J.Friel and C.S.Pande, Journal of Matererial Research. 8, 1993, 100. [9] G.Dobrescu, M. Zaharescu, M.Zaharescu, N.I.Ionescu, submitted to J.Sol-Gel Sci.Technol. [10] D.I.dos Santos, M.A.Aegerter, A.F.Craievich, T.Lours and J.Zarzycki, Journal of Non-Crystalline Solids 95 & 96 , 1987, 1143. [11] P.Pfeifer and M.Obert in The Fractal Approach to Heterogeneous Chemistry edited by D.Avnir, J.Wiley & Sons Ltd, 1989. [12] P.Grassberger, Physics Letters A, 97, 1983, 224. [13] P.Grassberger and I.Procaccia, Physica 9D, 1983,189. [14] J.Samios, P.Pfeifer, U.Mittag, M.Obert and T.Dorfmuler, Chemical Physics Letters, 128, 1986, 545. [15] G.Dobrescu and M.Rusu, in press. [16] Maria Zaharescu, Maria Preda, Maria Crişan, D.Crişan and N.Drăgan, Key Engineering Materials, 132-136, 1997, 852. [17] Maria Zaharescu, Maria Crişan, D.Crişan, N.Drăgan and A.Jitianu, Journal of European Ceramic Society, 18, 1998, 1257. [18] Maria Crişan, A.Jitianu, D.Crişan, Marina Bălăşoiu, N.Drăgan and Maria Zaharescu, Journal of Optoelectronics and Advances Materials, 2, 2000, 339. [19] Maria Crişan, A. Jitianu, Maria Zaharescu, F.Mizukami and Shu-ichi Niwa, Journal of Disperse Science and Technology, accepted for publication.
Abstract: TEM micrographs can be used to compute fractal dimensions of 3-dimensional objects. In order to covert the fractal dimension of TEM micrograph to the 3-dimensional fractal dimension, a modified sphere method was developed. Using this method, fractal dimensions of some sol-gel powders were computed Key words: fractal dimension, sphere method, TEM micrograph, sol-gel powders.
1. INTRODUCTION Fractal theory was developed in the last two decades in order to achieve a better characterization of different phenomena in physics, chemistry, biology, medicine and so on [1, 2]. A direct method to determine fractal dimension is to analyze images obtained from scanning tunneling microscopy [3-6], scanning probe microscopy [7, 8], transmission electron microscopy (TEM) [9]. TEM micrographs can provide high-resolution pictures of 3-dimensional objects, such as sol-gel powders. The aim of this paper is to analyze TEM images in order to obtain fractal dimensions of 3-dimensional objects and to apply the method to some sol-gel powders. The “mass-radius” relation is used to compute fractal dimension of TEM micrograph. Some assumptions are to be made to obtain fractal dimension of 3-dimensional object from TEM micrograph fractal dimension.
2. THEORY Self-similarity, that is the main property of fractal objects, has a mathematical description [1,2]:
N (r / R) ~ (r / R) − D
(1)
where D is fractal dimension and N(r,R) is the number of boxes of size r which cover the object of linear size R. Equation (1) leads to the following methods for fractal dimension determination [11]: - the “box counting” method [12-14], when R is defined as the largest distance between any two points belonging to the system:
N box (r ) ~ r − D -
(2)
the mass-radius method:
N sites ( R) ~ R D
(3)
where Nsites(R) is the number of sites located within distance R from a given site, R>>r, (the mass in a sphere of radius R). We used equation (3) to determine TEM micrographs fractal dimensions, for R varying in a broad domain, from 1 pixel to the maximum radius of the cluster. The image fractal dimension characterizes the plane object projection. Therefore, fractal dimension of 3-dimensional object will be greater then TEM micrograph fractal dimension [15]:
D = D TEM + D o
(4)
where D is powder fractal dimension, DTEM is image fractal dimension and Do is a correction factor with fractal dimension behavior. As we already saw, to compute fractal dimension of a 3-dimensional object, we measure total number of r size boxes that covered the object. The number of r x r x r boxes needed to covered the r x r x h parallelepiped in fig.1 will be Nz(xr, yr, r), where xr and yr are center coordinates of parallelepiped xy-projection. In this moment, two different cases are to be considered: 2.1 Fractal distribution along z axes If, along z axes, 3-dimensional object exhibit a fractal behavior, it is easy to understand we can write the following relation:
N z ( xr , y r , r ) = Az r − Dz
(5)
where Dz is fractal dimension along z-axes which means is the exponent of scaling along z-axes. Equation (5) is obeyed even for r=1, and, therefore:
N z ( xr , y r ,1) = Az ( xr , y r ) = ρ ( xr , y r )
(6)
where A(xr,yr) is a constant depending on chosen center xr, yr, and it is equal with the total sum of particles along zaxes described by coordinates xr,yr. This total sum of particles can be experimentally computed from gray-level of xr,yr pixel of TEM micrograph, ρ(xr,yr) . From equation (5) and (6):
N z ( x r , y r , r ) = ρ ( x r , y r ) r − Dz
(7)
Total number of r size boxes needed to covered the whole object, N(r) can be computed as:
N (r ) =
∑N
z
( x r , y r , r ) = r − Dz
xr , y r
∑ ρ(x , y ) r
r
(8)
xr , y y
Assuming that the gray level sum ρ(xr,yr) scaling with r like:
θ (r ) =
∑ ρ ( x , y ) = Br r
− Dθ
r
(9)
xr , yt
equation (8) becomes:
N (r ) = Br − ( Dz + Dθ )
(10)
D = Dz + Dθ
(11)
From equation (2) and equation (10):
For isotrope fractals equation (5) can be written even for ox and oy axes, and Dx=Dy=Dz=D0. So:
N ~ NxNyNz ~ r
− ( Dx + D y + Dz )
D = Dx + D y + Dz = 3D0
= r −D ,
(12)
Combining equations (11) and (12) the result is straightforward:
D=
3Dθ 2
(13)
Equation (13) will be used in the following for computing fractal dimension of some sol-gel monocomponent powders, where Dθ will be TEM micrograph fractal dimension computed from gray-level distribution, using equation (9).
2.2 Uniform distribution along z axes If uniform distribution along z axes occurs, the 3-dimensional object has a fractal surface, is a self-affine object rather than a self-similar one, and equation (5) becomes:
N z ( xr , y r , r ) =
Az h r
(14)
As we discussed above:
N z ( x r , y r ,1) = A z h = ρ ( x r , y r ),
∑
N (r ) = x
r
, y
N z (x r , y r , r) = r
1 r
∑ x
r
, y
ρ(x r , y r ) = r
1 r
Br − D θ
= Kr − D
(15)
meaning that:
D = Dθ + 1 .
(16)
Fig. 1. Grey parallelepiped r x r x h
3. FRACTAL DIMENSION COMPUTATION OF SOL-GEL POWDERS Sol-gel monocomponent SiO2, AlO(OH), TiO2 oxide powders were prepared using the alkoxide route [16-18]. The size and the shape of the powders were determined by TEM using a performant JEOL-FX-2000 equipment.
Fig.2 TEM micrograph of a SiO2 powder sample The SiO2 powder shows a bimodal distribution of particles size. The particles with low dimension are of about 20 nm and the high dimension particles exceed 200 nm (Fig.2). AlO(OH) powder consists of nano-sized particles with highest tendency to aggregation (Fig.3). TiO2 powder presents aggregates of about 50 nm consisting from smaller particles (bellow 10 nm) with high tendency to crystallisation (Fig.4). In this case the low temperature of preannealing thermal treatment (300°C) required for water and organic residues evolution has induced the nanocrystallisation of the powders.
Fig. 3. and 4. TEM micrograph of a TiO2 powder sample (left) and of a AlO(OH) powder sample (right) .
For every sample 3 different TEM micrographs were analyzed [19]. Results are presented in table I. Table I Monocomponent powders fractal dimensions No.
Sample
1. 2. 3. 4. 5. 6. 7. 8. 9.
TiO2 300oC AlO(OH) 450oC SiO2 300oC
Fractal dimension Fractal dimension -short scale -large scale 2. 8 2.9 2.7 2. 8 2.7 3.0 2.7 3.0 2.8 2.9 2.8 3.0 2.8 2.9 2.8 3.0
Phase composition anatase weak crystallized boehmite weak crystallized and η(γ)-Al2O3 amorphous
Figures 5 show log-log plot of number of occupied sites within R radius sphere for micrographs in figure 2. Sol-gel powders obtained have fractal structures: TiO2 and SiO2 samples have fractal dimension of 2.8 and AlO(OH) has fractal dimension of 2.7.
12.00
logN(R)
10.00 8.00 6.00 4.00 2.00 0.00 0.00
2.00
4.00
6.00
log(R)
Fig. 5. Log-log plot of number of occupied sites within R radius sphere for micrograph in figure 2. Generally, we obtained two different fractal dimensions: one for short scales and another for long scales. For SiO2 samples (no.7-9) two types of particles characterize the system: the small ones of 10nm and the big ones of 200nm, each having different kind of aggregation. For AlO(OH) sample (no.4-6) the high aggregation lead to fractal dimensions around Euclidean dimension of 3.0. For TiO2 sample TEM micrograph no.2 there are a lot of island-like aggregations scaling different from component particles. This behavior leads to different long-range and short-range correlation and, finally, at two different fractal dimensions 2.9 and 2.7. The fractal dimension is related to the aggregation processes. These high fractal dimensions 2.7 –2.8, for all analyzed samples indicate high aggregation when clusters grow independent and high aggregation of clusters together. The fractal dimension depends also on processes that occur during thermal treatment, when adsorbed and structural water, as well as organic residues are eliminated.
4. CONCLUSIONS A method for computing fractal dimensions from TEM micrographs was developed. This method was applied to some monocomponent sol-gel powders. The investigated monocomponent SiO2, TiO2 and AlO(OH) obtained by solgel method, present good fractal structures with high fractal dimensions, between 2.7 and 3.0. This fractality
indicates high aggregation during hydrolysis-polycondensation process and during the thermal treatment at low temperature.
REFERENCES [1] B.B.Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977. [2] B.B.Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [3] M.W.Mitchell and D.A.Bonnell, Journal of Material Research, 5, 1990, 2244 . [4] J.P.Carrejo, T.Thundat, L.A.Nagahara, S.M.Lindsay and A.Majumdar, Journal of.Vacuum Science and Technology B, 9, 1991, 955. [5] D.R.Denley, Journal of Vacuum Science and Technology A, 8, 1990, 603. [6] J.M.Gomez-Rodriguez, A.M.Baro, L.Vasquez, R.C.Salvarezza, J.M.Vara and A.J.Arvia, Journal of Physical Chemistry 96, 1992, 347. [7] J.M.Williams and T.B.Beebe Jr., Journal of Physical Chemistry 97, 1993, 6249; Journal of Physical Chemistry 97, 1993, 6255. [8] J.J.Friel and C.S.Pande, Journal of Matererial Research. 8, 1993, 100. [9] G.Dobrescu, M. Zaharescu, M.Zaharescu, N.I.Ionescu, submitted to J.Sol-Gel Sci.Technol. [10] D.I.dos Santos, M.A.Aegerter, A.F.Craievich, T.Lours and J.Zarzycki, Journal of Non-Crystalline Solids 95 & 96 , 1987, 1143. [11] P.Pfeifer and M.Obert in The Fractal Approach to Heterogeneous Chemistry edited by D.Avnir, J.Wiley & Sons Ltd, 1989. [12] P.Grassberger, Physics Letters A, 97, 1983, 224. [13] P.Grassberger and I.Procaccia, Physica 9D, 1983,189. [14] J.Samios, P.Pfeifer, U.Mittag, M.Obert and T.Dorfmuler, Chemical Physics Letters, 128, 1986, 545. [15] G.Dobrescu and M.Rusu, in press. [16] Maria Zaharescu, Maria Preda, Maria Crişan, D.Crişan and N.Drăgan, Key Engineering Materials, 132-136, 1997, 852. [17] Maria Zaharescu, Maria Crişan, D.Crişan, N.Drăgan and A.Jitianu, Journal of European Ceramic Society, 18, 1998, 1257. [18] Maria Crişan, A.Jitianu, D.Crişan, Marina Bălăşoiu, N.Drăgan and Maria Zaharescu, Journal of Optoelectronics and Advances Materials, 2, 2000, 339. [19] Maria Crişan, A. Jitianu, Maria Zaharescu, F.Mizukami and Shu-ichi Niwa, Journal of Disperse Science and Technology, accepted for publication.
