UNIVERZA V LJUBLJANI FAKULTETA ZA MATEMATIKO IN FIZIKO ODDELEK ZA FIZIKO Seminar 2 DIFUSSION LIMITED AGGREGATION Elizabeth Dafne Di Rocco Mentor: Prof. dr. Rudolf Podgornik November 25th, 2009 Abstract This article’s main topics are fractals and Diffusion Limited Aggregation (DLA) model. DLA model describes how a fractal is built from particles in low concentrations. The DLA cluster formed through DLA is formed by particles moving due to Brownian motion (diffusion) which meet and stick together randomly (aggregation) to form the cluster. Fractals can be built using this model from nanomaterials (i.e. TiO 2 ). There are several experimental methods for growing fractals. This article also includes a short description of TiO 2 photocatalytic properties and the advantage of using nano-TiO 2 fractals grown using DLA method.
Transcript
UNIVERZA V LJUBLJANI
FAKULTETA ZA MATEMATIKO IN FIZIKO
ODDELEK ZA FIZIKO
Seminar 2
DIFUSSION LIMITED AGGREGATION
Elizabeth Dafne Di Rocco
Mentor: Prof. dr. Rudolf Podgornik
November 25th, 2009
Abstract
This article’s main topics are fractals and Diffusion Limited Aggregation (DLA) model. DLA modeldescribes how a fractal is built from particles in low concentrations. The DLA cluster formed throughDLA is formed by particles moving due to Brownian motion (diffusion) which meet and stick togetherrandomly (aggregation) to form the cluster. Fractals can be built using this model from nanomaterials(i.e. TiO2). There are several experimental methods for growing fractals. This article also includes ashort description of TiO2 photocatalytic properties and the advantage of using nano-TiO2 fractals grownusing DLA method.
In nature we find curious structures that are called fractals. Fractals can not be described by classicalgeometry since they are irregular. Fractals exhibit interesting properties that can be used for differentapplications, the most important of which is the fact that a fractal has a very great surface-to-volumeratio.
The process by which a fractal can be grown from a solution is called Diffusion Limited Aggregation(DLA) and it is the main topic of this article. DLA uses two processes to generate a fractal: diffusionand aggregation. The solution must have a very low concentration of particles in order for a fractal togrow. The particles in the solution move around (Brownian motion) and they can stick together slowlyforming a cluster.
DLA can be used to generate fractals from different materials including nanoparticles. Fractals grownusing DLA can afterwards be used instead of the more common forms: powdered materials, crystals,solutions, etc. This article will present how a fractal can be grown from nanomaterials. As mentionedabove fractals have a high surface-to-volume ratio, a fact that is nowadays used to increase photocatalyticactivity of some materials. The photocatalytic activity of a sample is proportional to the surface of thephotocatalityc material. Using fractals we can increase the exposed surface of the photocatalytic material.The last part of this article explains the photocatalytic properties of nano-TiO2 which are improved bygrowing fractals using DLA.
2 FRACTALS 3
2 Fractals
Fractals are irregular shapes that can not be represented by classical geometry. The geometric patternof the fractal is repeated at ever smaller scales. Fractals are scale invariant and self-similar. Thereare several examples of fractal-like structures in living as well in nonliving nature: mineral deposition,snowflake growth, fjords made by glaciers, corals, lichen, etc. (see Figures 1, 2, 3, 4).
Figure 3: Snowflake under a microscope [1]Figure 4: Lightning path [1]
A fractal often has the following features:
• It has a fine structure at arbitrarily small scales.
• It is too irregular to be easily described in traditional Euclidean geometric language.
• It is self-similar (at least approximately or stochastically).
• It has a simple and recursive definition
Because they appear similar at all levels of magnification, fractals are often considered to be infinitelycomplex (in informal terms). However, not all self-similar objects are fractals. For example, the real
2 FRACTALS 4
line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; forinstance, it is regular enough to be described in Euclidean terms. Images of fractals can be created usingfractal-generating software. Images produced by such software are normally referred to as being fractalseven if they do not have the above characteristics, as it is possible to zoom into a region of the imagethat does not exhibit any fractal properties.
Figure 5: Fractal pentagram drawn with a vectoriteration program. [1]
Figure 6: A fractal flame created with the programApophysis. [1]
Figure 7: Even 2000 times magnification ofthe Mandelbrot set uncovers fine detail resem-bling the full set. [1]
3 DIFFUSION LIMITED AGGREGATION 5
2.1 Fractal dimension
Fractals have a finite area but an infinite perimeter. In contrast to regular shapes (triangles, squares. . . )fractals have non-integer dimension D. Fractional dimension D is a statistical quantity that gives anindication of how completely the fractal fills space. How is D defined?
If we take an object with linear size equal to 1 residing in Euclidean dimension D, and reduce its linearsize by the factor 1/r in each spatial direction, it takes N = rD number of self similar objects to coverthe original object (Figure 8). r is called the characteristic linear dimension.
We can use ln on both sides of this equation and weget the definition for D.
lnN = D ln r
D =lnN
ln r(1)
By applying the above equation to a fractal struc-ture, we can get the dimension of the fractal struc-ture:
D = limε→0
lnN(ε)
ln 1ε
(2)
where N(ε) is the number of self-similar structuresof linear size ε needed to cover the whole structure.
Figure 8: Defining dimension from a unit object.[1]
2.2 Classification
Fractals can be classified according to their self-similarity:
• Exact self-similarity is the strongest type of self-similarity. The fractal appears identical at allscales.
• Quasi self-similarity is a loose form of self-similarity. The fractal appears approximately identicalat different scales.
• Statistical self-similarity is the weakest type of self-similarity. The fractal has numerical orstatistical measures which are preserved across scales. All DLA fractals belong to this type.
3 Diffusion Limited Aggregation
DLA theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system wherediffusion is the primary mean of transport in the system. DLA can be observed in the laboratory inmany systems such as electrodeposition and dielectric breakdown. Let us first take a look at the twomechanisms in DLA.
3 DIFFUSION LIMITED AGGREGATION 6
3.1 Diffusion
Diffusion is a net transport of molecules froma region of higher concentration to one of lowerconcentration by random molecular motion. Theresult of diffusion is a gradual mixing of materialin fluid systems. In a phase with uniform tem-perature and absent external net forces acting onthe particles, the diffusion process will eventuallyresult in complete mixing or a state of equilib-rium.
Figure 9: Diffusion of (a) one type of particles and(b) two types of particles in a solution. [2]
Diffusion is the macroscopic result of random thermal motion on a microscopic scale. If the distributionof all types of particles in the solution are not uniform, there will be a net flux even though the motion ofeach individual molecule is completely random. The flux is proportional to the gradient in concentration(molar or molecular).
~j = −D~∇n, (3)
where D is the diffusion constant and n is the concentration. We also know the following relation:
~∇ ·~j = −∂n
∂t+ q, (4)
where q stands for sources. From equations 3 and 4 and assuming that D is constant it is possible towrite the diffusion equation as such:
∂n
∂t= D∇2n− q. (5)
3.2 Aggregation
If particles have the possibility to attract each other and stick together, they form aggregates. Theforces between the particles may be weak or strong. For particles which carry electrical charge the forcesare strong. Aggregates represent a preferred state compared to spread particles that can stick together.Aggregates are usually well ordered.
Without an electrical charge the forces are much weaker i.e. the sticking force is Van der Waals force.Each aggregate is unique because there is no ordering force of the electrical field. Such a building is alsonamed a cluster.
3.3 DLA cluster
A DLA-cluster (also known as Brownian tree) is a fractal aggregate made by DLA, where the shape ofthe cluster is controlled by the possibility of particles to reach the cluster via Brownian motion. Startingwith a uniform distribution, some particles might meet. The aggregates may grow as long as there are
3 DIFFUSION LIMITED AGGREGATION 7
particles moving around. “Arms” of the cluster “catch” particles so that they can’t reach inner parts ofthe cluster. During the diffusion of a particle through the solution it is more likely, that the particleattaches to the outer regions than to the inner ones of the cluster - a ffluffy shape occurs, with manyarms (see Figure 10).
Figure 10: A DLA cluster grown from a copper sulphate solution in an electrodeposition cell. [1]
A single-particle bump on a straight edge of thecluster is more likely to catch a wandering par-ticle also due to the fact that it has three unoc-cupied neighbours while each particle along theedge has only one unoccupied neighbour (see Fig-ure 11).
Figure 11: Higher probability to glue to bumps inthe cluster. [3]
DLA is “Diffusion-limited” because the particles are considered to be in low concentrations so they don’tcome in contact with each other and the structure grows one particle at a time rather than by chunks ofparticles.
For every solution we should also consider the sticking coefficient i.e. when a particle reaches the clusterit will not always stick. Thus, when it doesn’t stick immediately, it moves along in the vicinity of thecluster’s arms until it either finally sticks somewhere or gets lost. If the wandering particle strikes partof the existing structure and always sticks, then stickiness is 1. Otherwise is less than 1. Low stickinessprobability gives rise to denser clusters.
Fractal growth has been observed after certain growth time under a field emission scanning electronmicroscope (SEM) which gave direct proof of the DLA process (see Figure 12).
Figure 12: The growth-time dependent morphology of the silver structures, demonstrating DLA process.The growth time is 1, 5, 10 and 60 min from left to right. [4]
3 DIFFUSION LIMITED AGGREGATION 8
3.4 Differential equation for DLA
The dynamics of deformable bodies with a well defined surface can be represented by a gauge field Ψ [5]:inside the body is Ψ ≤ 0, outside Ψ > 0, on the surface, which grows by deposition of diffusing particles,is
Ψ(x, y, z; t) = 0 . (6)
The equation of motion for Ψ (changing of surface’s topology) is
∂Ψ
∂t+ (~V · ~∇)Ψ = 0 , (7)
where ~V (~r, t) is the velocity field.
In the region where Ψ > 0 particles with concentration n around the surface diffuse and are absorbed atthe surface boundary. The surface absorbs all the particles that hit it. This is described with equation5. The boundary conditions on the surface are n = 0. The particles that disappear from the surface’senvironment become part of the solid and change the surface which has “concentration” n0. The equation5 changes when considering a given rate of absorption per unit time Φ(~r, t) at a given boundary point ~rand time t into
∂n
∂t= D∇2n− q − lim
ǫ→0+Φ(~r, t)δ(Ψ − ǫ)|~∇Ψ| , (8)
The amount of material absorbed from the environment at a point ~r of the boundary per unit time andunit area is
Φ(~r, t) = D(~∇n)+ · ~m , (9)
where + denotes approaching the boundary from the region of positive Ψ and ~m is a unit vector normalto the surface and pointing into the region of positive Ψ. The normal velocity ~Vm(~r) of the growingsurface is then
Vm(~r) =1
n0D(~∇n)+ · ~m . (10)
The velocity field ~V (~r) throughout the space is
~V (~r) =1
n0D(~∇n)+ . (11)
With equations 8 through 11 it is possible to write the solution for n
n = ns + limǫ→0+
∫dt′d~r′G[~r − ~r′, t− t′]
∂Ψ
∂t′(~r′, t′)δ[Ψ(~r′, t′) − ǫ)] , (12)
where ns is the concentration that would have existed in the presence of just the sources and G is theGreen function. Equation 7 can be rewritten into
∂Ψ
∂t− lim
ǫ→0+D
∫dt′d~r′~∇G(~r − ~r′, t− t′) · ~∇Ψ(~r, t)
∂Ψ
∂t′(~r′, t′)δ[Ψ(~r′, t′) − ǫ] = −
1
n0D~∇ns · ~∇Ψ . (13)
The solution of equation 13 gives the description of the surface growth.
3 DIFFUSION LIMITED AGGREGATION 9
3.5 Computer simulations of DLA
Computer simulation of DLA is one of the primary means of studying DLA model. Simulations canbe done on a lattice or along the lines of a standard molecular dynamics simulation where a particle isallowed to freely random walk until it gets within a certain critical range at which time it is pulled ontothe cluster. Of critical importance is that the number of particles undergoing Brownian motion in thesystem is kept very low so that only the diffusion controls the aggregation.
The simplest computer growth (using a lattice) starts with an initial seed particle at some origin andanother particle somewhere on the lattice (see Figures 13 and 14). Then the second particle moves aroundin random motion (in 2D: up, down, left, right), step by step from lattice site to lattice site. It can meetthe first particle or move out from the lattice and another particle is introduced (the first particle eitherbounces off the edge or the image is toroidally bound; however, new points can be seeded anywhere in theimage area). If the particle touches the initial particle, it is immobilized instantly and becomes part ofthe aggregate. Then another particle is thrown onto the lattice, it walks around and after a while meetsthe first two or moves out from the lattice. The action is repeated as long as particles are available.
Figure 13: Initial seed particle is in the middleof the lattice. Second particle is introduced ran-domly. [3]
Figure 14: Fourth particle is introduced to a three-particle aggregate. [3]
Within DLA simulation models there are some variations. For example:
• You can introduce a sticking coefficient and when a particle reaches the cluster it will not alwaysstick. As mentioned before the sticking coefficient can have a value between 1 (particles alwaysstick) and 0 (particles never stick). Low stickiness probability gives rise to more dense clusters.The fractal dimension does not change much until the sticking coefficient becomes less than 0.1.As the sticking coefficient vanishes the fractal dimension becomes close to the spatial dimension -close to 2 in 2D (see Figure 15).
Figure 15: Fractal grown from a point with different stickiness probability (S). The higher thestickiness, the lower the fractal dimension, the lower the density of the fractal. [6]
A further modification is using different attaching probabilities depending on the current geometricalenvironment i.e. the more neighbours that are already present, the more likely it is for a particleto attach.
4 FRACTALS AND NANOPARTICLES 10
• The lattice geometry can be varied: a square lattice with four neighbour sites, a triangular latticewith six neighbours. The overall shape of the cluster is related to the shape of the lattice.
• Movement over any distance (off-lattice DLA) is allowed (there is no lattice). This approach allowsthe creation of very large clusters.
Figure 16: Initial seed is apoint. [7]
Figure 17: Initial seed is a line.[7]
Figure 18: Initial seed is the in-ner part of a square. [7]
Figure 19: Initial seed is theouter part of a circle. [7] Figure 20: Initial seed is the in-
ner part of a circle. [7]
• If there are several simultaneously growing clusters within a “solution” cluster-cluster-aggregation(CCA) is introduced.
• A starting line can be used instead of a starting point as a seed line thus resulting in forest likeclusters (diffusion-limited deposition, DLD) (see Figures 16 - 20).
4 Fractals and nanoparticles
4.1 DLA and nanotubes
Van der Waals force is not present only when we are dealing with spherical particles, but also when theshape of particles is non-spherical. Nowadays are “in” all materials which are nanotube-like. They alsoaggregate due to mentioned force and DLA. But when investigating nanotube’s properties it is better tohave a single nanotube then a whole aggregate of nanotubes. This is the reason why physicist startedto investigate the Van der Waals interaction between two nanotubes at an arbitrary angle Θ (see Figure21).
4 FRACTALS AND NANOPARTICLES 11
Figure 21: Two cylinders at an arbitrary angle. [8]
The interaction free energy G for two cylinders with anisotropic dielectric properties yields [8]
G(l,Θ) = −(πa2)2(A(0) + A(2) cos2 Θ)
2πl4 sinΘ(14)
(15)
where a is the radius of the cylinders, l is the separation, A is the Hamaker coefficient which quantifiesthe magnitude of the Van der Waals interaction. This coefficients represent the material properties ofthe interacting bodies.
The molecular structure of the nanotubes was ignored and the interactions between them were derivedin terms of dielectric constant and indices of refraction. See below:
A0 =3
2kBT
∞∑n=0
1
2π
∫ 2π
0
∆Lm(Φ)∆Rm(Φ −π
2)dΦ (16)
A0 + A2 =3
2kBT
∞∑n=0
1
2π
∫ 2π
0
∆Lm(Φ)∆Rm(Φ)dΦ . (17)
The summation in the expressions above is not continuous but rather over a discrete set of Matsubarafrequencies ωn = 2πnkBT
~. The spectra functions are
∆Lm(Φ) = −(∆⊥(L) +1
4(∆‖(L) − 2∆⊥(L)) cos2 Φ) (18)
∆Rm(ψ) = −(∆⊥(R) +1
4(∆‖(R) − 2∆⊥(R)) cos2 ψ) (19)
(20)
where anisotropy parts are
∆⊥ ≡ǫc⊥ − ǫmǫc⊥ + ǫm
(21)
∆‖ ≡ǫc‖ − ǫm
ǫm. (22)
All the dielectric functions (ǫ⊥, ǫm, ǫ‖) have to be taken as their Kramers-Kronig transforms at iωn, i.e.ǫ(iωn), where ωn are the Matsubara frequencies mentioned above. The Kramers-Kronig transform is:
4 FRACTALS AND NANOPARTICLES 12
ǫ(iω) = 1 +2
π
∫ ∞
0
ǫ”(ξ)dξ
ξ2 + ω2(23)
where ǫ”(ξ) is the imaginary part of the dielectric response function, i.e. ǫ(ξ) = ǫ′
(ξ) + iǫ”(ξ). ǫ(iω) isreferred to as the van der Waals-London dispersion spectrum (or vdW-Ld spectrum). The magnitude ofǫ(iω) essentially describes how well the material responds and is polarized by fluctuations up to the givenfrequency [9].
The dielectric and vdW-Ld spectra are different for each type of nanotubes. In the figures below (22and 23) you can see both spectra for two types of carbon nanotubes (CNT): one is a semiconductingnanotube [6, 5, s] and the other is a metallic nanotube [9, 3, m]. In both graphs both the radial (normalto the nanotube cylinder) and the axial (paralell to the nanotube cylinder) spectra are shown.
Figure 22: The imaginary part of the dielectric spectrum versus frequency for the (a) [6, 5, s] and (b) [9,3, m] single walled CNTs in their radial and axial directions. [9]
Figure 23: The vdW-Ld spectrum of both single walled CNTs in their radial and axial directions for the(a) [6, 5, s] and (b) [9, 3, m]. [9]
This is the physics behind aggregation of dielectrical nanotubes.
4.2 DLA and thin films
There have been several reports on DLA growth of thin films [10, 11, 12]. One of the methods wherefractals successfully grew on the substrate is using bulk selenium (Se) powder and silver (Ag) foil in asolvothermal process in autoclave at 160 ◦C for 10 hours with alcohol as a solvent [10]. Dendrites ofAg2Se nanocrystals were formed, with each dendritic branch often in single-crystal form and nearly allbranches having their (001) crystal direction pointing along the surface normal of the Ag foil substrate.The Ag2Se reaction product aggregated first into nanoparticles that were solvated.
4 FRACTALS AND NANOPARTICLES 13
The diffusion and DLA of these nanoparticles gave rise to the formation of dendrites from an Ag nanowire(see Figure 24). Similar to this case is when replacing Ag nanowire with Ag foil (see Figure 25): arrivalof Se to the silver surface, the formation of Ag2Se and the nucleation of Ag2Se nanocrystals. Differentfrom the formation of Ag2Se nanocrystals on a Ag nanowire, the formation of Ag2Se nanocrystals on aAg foil is supported by surface energy minimization toward the formation of Ag2Se nanocrystals withtheir (001) crystal orientation preferentially aligned with the normal direction of the flat surface. Evenmore, the diffusion of the Ag2Se nanocrystals is also mediated by the flat surface, which leads to thetendency of forming 2D dendrites parallel to the flat surface. The diffusion of Ag2Se nanocrystals on topof first layer of 2D dendrites leads to the growth of the dendritic film in a 3D island growth mode (seeFigure 26).
Figure 24: Schematic diagram of the Ag2Se dendrite growth mechanism from an Ag nanowire with nosurface-mediated support. [10]
Figure 25: Schematic diagram of the Ag2Se dendrite growth mechanism from an Ag foil with surface-mediated nucleation to enable (001)-oriented dendrite growth. [10]
4 FRACTALS AND NANOPARTICLES 14
Figure 26: SEM morphologies: a) nanocrystals of (001)-oriented Ag2Se after 1 h of solvothermal growthwith methanol as the solvent; b) oriented attachment toward the formation of the trunk of a dendriteafter 3 h of solvothermal growth with methanol as the solvent; c) close-up of a full dendrite formed after12 h of solvothermal growth with methanol as the solvent; d) close-up of a full dendrite formed after 12h of solvothermal growth with dodecanol as the solvent; e) large-field view of dendrites formed under theconditions of part -c- for the case of a relatively high nucleation density; f) large-field view of dendritesformed under the condition of part -d- for the case of a relatively low nucleation density. [10]
Figure 27: Schematic illustration of the growth process of silver dendrites [11]. The layer of a silvernanoparticles/nanoclusters and the layer of synchronized silver dendrites were named as the Volmer-Weber (VW) layer and the DLA layer respectively. DLA layer derived from the continuous aggregationgrowth of small particles on the VW layer.
Another method for growing fractals on film is on a thin porous silicon layer which was immersed in a2.5 M NH4F solution containing 0.01 M silver nitrate (AgNO3) at 50 ◦C (see Figure 27) [11]. After theetching process, the silicon wafers were cleaned. In NH4F solution, the etched silicon substrates werealways wrapped with a layer of thick silver film, which was rather loose and could be easily detachedfrom the surface of silicon substrates. A morphological evolution of silver dendrites was followed by atime-dependent process (see Figure 28).
5 PHOTOCATALYTIC ACTIVITY OF TIO2 15
Figure 28: SEM images of the silver films prepared in a 2.5 M NH4F solution containing 0.01 M silvernitrate at 50 ◦C for a) 5 min B9 15 min and c) 60 min. [11]
5 Photocatalytic activity of TiO2
Titanium dioxide, also known as titanium (IV) oxide or titania, is the naturally occurring oxide oftitanium with chemical formula TiO2.This oxide is widely used and studied due to its properties. I willfocus on its use as a photocatalyst.
Photocatalysis is a reaction which uses light to activate a substance which modifies the rate of a chemicalreaction without being involved itself. Today, semiconductors are usually selected as photocatalysts,because semiconductors have a narrow gap between the valence and conduction bands. In order forphotocatalysis to proceed, the semiconductors need to absorb energy equal to or more than its energygap. Photocatalytic activity (PCA) is the ability of a material to create an electron hole pair as a resultof exposure to EM radiation. The resulting free-radicals are very efficient oxidizers of organic matter.
The superhydrophilicity phenomenon of glass coated with TiO2 and exposed to sun light was discovered in1995. TiO2 incorporated into outdoor building materials and paints reduces concentrations of airbornepollutants. Even more, using photocatalysis, there is no need to use conventional cleaning chemicals.Result of this knowledge was development of self-cleaning glass and anti-fogging coatings.
5.1 TiO2 nanomaterials
Titanium dioxide occurs in nature as three different minerals: rutile, anatase and brookite. The mostcommon form is rutile, which is also the most stable form. Anatase and brookite both convert to rutileupon heating. Brookite is in nature the least common between all of them.
Rutile represents the stable phase of TiO2 at high temperatures. On the other hand anatase andbrookite are common in nanoscale grained samples. When heating different transformations can occur:anatase→brookite→rutile, brookite→anatase→rutile, anatase→rutile, brookite→rutile. Which transfor-mation will occur depends on initial particle size. For equally sized nanoparticles, anatase is thermody-namically stable for sizes below 11 nm, brookite is stable for sizes between 11 and 35 nm, rutile is stablefor sizes above 35 nm.
5.2 Photon-induced electron and hole properties
TiO2 is a n-type semiconductor which is widely used as a photocatalyst. Energy gap of rutile is 3.0 eV,of anatase 3.2 eV. This is the lowest energy of impact photons needed to excite electrons from valance tounoccupied conducting band leaving behind positive holes. These photons come from UV spectra (387.5nm for anatase [13]).
Negative electron in conductive band and positive hole in valance band represent charge carriers. They canrecombine nonradiatively, radiatively (heat) or get trapped and react with electron donors or acceptorsabsorbed on the surface of the photocatalyst. If charge separation is maintained, the electrons and holescan migrate to the catalyst surface where they participate in redox reactions with adsorbed species. In
6 CONCLUSION 16
particular, hole h+V B
may combine with H2O or OH− to produce the hydroxyl radical
H2O + h+V B
→ HO.
ads +H+ads
. (24)
e−CB
can be picked up by oxygen to generate superoxide radicals which can in turn generate hydroperoxideand hydrogen peroxide, decomposed at the semiconductor surface into hydroxyl radicals.
O2(ads) + e−CB
→ O.−2(ads) (25)
O.−2(ads) +H+
ads→ HO.
2(ads) (26)
HO.
2(ads) + e−CB
+H+ads
→ H2O2(ads) → 2HO.
ads (27)
These very reactive radicals can oxidize the adsorbed organic pollutants to achieve complete decomposi-tion [14].
Photocatalytic activity γ is defined as
γ =C
3tmI(28)
C is the concentration of the pollutant in units mol/l, t is the irradiation time of UV light, m is theconcentration of TiO2 particles in units g/l, I is the intensity of adsorbed UV light [15].
The photocatalytical properties of TiO2 are affected by several factors: crystal structure, morphologyand surface area. The highest photoactivity between all 3 crystal structures is shown by anatase. Propermixture of anatase and rutile TiO2 gives higher photocatalytic activity than pure anatase TiO2[16].
5.3 TiO2 fractals
Fractals are used to increase the photocatalytic activity of TiO2. Rougher surface means larger surfacearea and more photocatalytic reactions take place. Fractals have a large surface-to-volumen ratio whichas mentioned improves photocatalytic activity. Additionally fractals are usually easier to filter since theyare usually much larger than nanoparticles. It has also been reported that the fractal surfaces absorblight more efficiently and produce more photocurrent than smooth surfaces. For these reasons DLA isused to obtain fractal clusters of TiO2.
Nano-TiO2 can be coated on many building materials and for this coatings fractals give better results.The advantages of using nano-TiO2 fractals are currently being researched.
6 Conclusion
Fractals are often found in nature. The advantages of such structures are nowadays being used for differentpurposes in science, medice, construction, etc. With DLA it is possible to construct fractals either asmodels by using computers or by actually growing them in very low concentration solutions. DLA wasproven by observing growth of the silver structures under SEM. It is important that concentration ofparticles is low enough because diffusion in fractal growth should represent the main transport. UsingDLA we can grow fractals from nanomaterials.
Nowadays, when population is environmentally aware, photocatalytic activity is one of the most appre-ciated properties of TiO2. Due to this property there are several applications concerning cleaning ofsurfaces and cleaning of waste-water. Fractals of anastase TiO2 can be used to increase photocalyticactvity and at the same time simplify filtering of TiO2.
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