Fractally Aggregated Micro and Nanosystems Synthesized from Sols

ISSN 1087�6596, Glass Physics and Chemistry, 2014, Vol. 40, No. 2, pp. 190–202. © Pleiades Publishing, Ltd., 2014.Original Russian Text © I.E. Kononova, V.A. Moshnikov, M.B. Krishtab, I.A. Pronin, 2014, published in Fizika i Khimiya Stekla.

190

INTRODUCTION

Achievements of sol�gel technology [1–6] arecaused by the intensive development of technologieswhich provide the design of hybrid nanocomposites[7–9]. Depending on various experimental factors(temperature, catalysts, additives, and others), theformation and evolution of the structure in sol�gelprocesses proceed according to completely differentroutes. To gain a deeper insight into the origin of theprocesses, old models of growth and evolution of theaggregates are developed and new ones are suggested.At the stage of the formation and maturing of sol, thefollowing models of growth of fractal aggregates arethe most promising: diffusion�limited cluster�particleaggregation [10] and diffusion�limited cluster�clusteraggregates [11]. Moreover, the fractality of primaryelements depends substantially on the pH value andparticles acquire a globular view on reaching particularsizes.

The self�assembly of globules usually proceedsaccording to the mechanism which is described by themodel of cluster�cluster aggregation or other morecomplex models [12, 13]. In [14, 15], multilevel frac�tal nanostructures were detected.

The fractals of several types may simultaneouslyexist in multicomponent nanocomposites. Proceduresfor the assignment of small�angle X�ray scattering dataand determining fractal dimensionality of multilevelfractal objects were suggested and developed in [16,17].

In order to describe the physical properties ofnanocomposites, which are composed of phases withcontrasting properties, the models of percolation the�ory are widely used [18].

The formation of inorganic polymer networks incomplex systems can also be accompanied by spinodaldecomposition. This process is similar to the phenom�enon which is described by the Flory–Huggins theory,which is widely known from the physics of organicpolymers [19–21]. Finally, the consideration of realstructures requires the use of the models ofmicrophase separation [22].

Film formation during the sol�gel process. It shouldbe noted that the properties of composites in sol�gelprocesses are determined by the formation and evolu�tion of fractal aggregates at the step of sol preparation,as well as the conditions of spinodal decompositionduring the application of sol on the substrate surfaceand the heat treatment of film.

In this work, the description was performed withthe use of simplified approximations of the effect ofspinodal decomposition on the transformation ofnanocomposites both due to the concentration com�ponents and the growth of inorganic polymer. Silicondioxide was chosen as the object of study of globuleself�assembly.

Sols were prepared based on inorganic polymer of

the type. A representative of alkoxy�com�

pounds was chosen for film preparation, namely,orthosilicic acid ethyl ester tetraethoxysilane (TEOS).TEOS represents a hydrolytically unstable compoundfrom which polymer molecules are formed after thereaction with water and alcohol. Moreover, polymer�ization proceeds in an acidic medium. With theincrease in the time of the irreversible polycondensa�tion reaction (increased holding time of sol), thegrowth of fractal aggregates and the decrease in the

|

|Si O

n

⎡ ⎤⎢ ⎥⎣ ⎦

Fractally Aggregated Micro� and Nanosystems Synthesized from Sols

I. E. Kononova, V. A. Moshnikov, M. B. Krishtab, and I. A. ProninUl’yanov (Lenin) St. Petersburg State Electrotechnical University LETI, ul. Prof. Popova 5, St. Petersburg, 197376 Russia

e�mail: [email protected] November 7, 2011

Abstract—The stages of the evolution of fractally aggregated systems of tetraethoxysilane, which was hydro�lyzed in an acidic medium, have been considered. Spherical structures at nucleophilic growth and preperco�lation structures and percolation network materials on spinodal decomposition have been obtained. A pro�gram which enables one to simulate the growth of fractal aggregates both in two�dimensional and three�dimensional space has been designed.

Keywords: fractally aggregated systems, sol�gel method, silicon dioxide

DOI: 10.1134/S1087659614020102

GLASS PHYSICS AND CHEMISTRY Vol. 40 No. 2 2014

FRACTALLY AGGREGATED MICRO� AND NANOSYSTEMS 191

number of possible rearrangements between particlesduring the attachment of fractals on their frameworkoccurs (in addition, the average molecular weight ofpolymerized substances in the solution grows). Thisgives rise to a decrease in the entropy of mixing and,consequently, causes an increase in the energy of mix�ing (change of Gibbs energy) ΔGCM = ΔHCM – TΔSCM

(where ΔHCM is the enthalpy of mixing, ΔSCM is theentropy of mixing, and T is temperature), which cor�responds to the difference between the function ofsolution, , and the function of the mechanical mix�ture of components, GMEX: ΔGCM = Gp – GMEX. Thus,an increase in the free energy of system, ΔGCM, canproceed both upon cooling and polymerization, whichoccurs during the maturing of polymer sols and theapplication of substrates on their surface with the useof a centrifuge.

During the maturing of polymer sol, the change ofpolarizability of polymers presumably took place(configurations and conformations of macromole�cules); this would result in the increase in the enthalpyof mixing, ΔHCM. In any case, at positive values of thefunction of free energy of mixing, ΔGCM, derived fromgeneral principles of thermodynamics, the transitionof a homogeneous system to heterogeneous systemtook place.

At all stages of the sol�gel process, polymerizationoccurs, which affects the final composition and struc�ture of the xerogel. During the synthesis and maturingof the sol, fractal aggregates are formed (Fig. 1), whoseevolution depends on the composition of the initialsubstances, their concentration, order of mixing, thepH of the medium, temperature, etc. In Fig. 1, themicrographs of fractal aggregates that were formedduring acidic hydrolysis and polycondensation of theproducts of hydrolysis of tetraethoxysilane are givenfor illustration. The spherical particles which areformed during the sol�gel process and aggregated tosufficiently bulky formations, with the size rangingfrom units to dozens of micrometers, are clearlyobserved. The growth of these fractals in sol�gel pro�cesses according to the model of diffusion�limitedaggregation (DLA) naturally proceeds with the ran�dom character of cluster�cluster aggregation; randomcontacts of fractal aggregates and particles, thecrosslink of branched structures, and the formation ofa large percolation cluster occur.

Development of model considerations on the mech�anisms of growth of fractal aggregates in sol�gel pro�cesses and features of hierarchical self�assembly offractal clusters. In order to understand some physico�chemical processes, which proceed at the stage of solformation, a software program was created whichenables one to model the growth of fractal aggregatesboth in two� and three�dimensional space. Computerexperiments according to the DLA model and themodified DLA model, which differentiates the fea�

PG

tures of the formation of fractal aggregates in mul�tiphase systems, were conducted.

In Fig. 2, the results of the computer modeling aregiven. As follows from Figs. 1 and 2, the experimentaldata are closely correlated to the results of computermodeling.

The developed software program represents thepackage of the modules which are designed for theconstruction of fractal aggregates and analysis of theirstructure (the entire program code was written onPython 2,6 interpretable language). Two programshells were composed based on the written modules.The former represents a tool for the construction oftwo� and three�dimensional fractal aggregates andtheir visualization and the latter enables one to per�form their statistical analysis.

The DLFO (Deryagin–Landau–Ferway–Over�back) theory, which is used for the analysis of the sta�bility of lyophobic sols and assumes two types of inter�actions that arise at the approach of colloidal parti�cles–electrostatic, which is caused by the overlap ofdiffusion layers and prevents the thinning of the liquidlayer, as well as the molecular, which is determined bythe action of dispersion forces and causes the attrac�tion of particles–was used for the mathematicaldescription of the stability of colloidal systems and therationalization of coagulation (Fig. 3) [1, 23]. Thus,the final curve (Fig. 3) which characterizes the excessenergy in a thin film of liquid has two minima that areseparated by a potential barrier.

The possibility of the approach of particles duringthe elementary act of collision is determined by theheight of the energy barrier and depth of the potentialwells. Depending on the relationship between thementioned parameters of the curve and energy of heatmotion of a particle, the following situations are pos�sible: the system can be aggregately stable, and the

20 μm

Fig. 1. Fractal aggregates that are obtained on a silica sub�strate during hydrolytic polycondensation of tetraethox�ysilane.

192


KONONOVA et al.

coagulation of particles or the formation of a some�what ordered structure (gel) onthe transfer of particlesto the second minimum may proceed. In addition, aparticle is not mounted rigidly and separated from theneighboring particle by a thin film of liquid. This factwas used to design the model of evolution of the struc�ture that is formed at particle aggregation.

In order to describe the coagulation in sol�gel pro�cesses, the mechanism of evolution of the structure of

an aggregate, which is based on a free radial translationof the attached particle, was used.

The conventional DLA model was developed in theworks of Witten and Sunder and represented thegrowth of an aggregate due to the random walk of aparticle that was released from circumference, Rb

(Fig. 4), on a square lattice with the subsequentattachment to a cluster after contact. In order toincrease the rate of simulation, the second circumfer�ence was introduced, Rk (Fig. 4), after the intersectionof which, the particle begins to move from the randompoint on circumference Rb. In Fig. 4, two possible tra�jectories of the motion of a particle are given, t1 and t2.After this process is repeated many times, structuresare formed whose density decreases with the growth ofaggregate sizes, and the total number of particles, N, inthe aggregate is related to its inertia radius, Rg, by the

following relationship: where k0 is a

prefactor, à is the size of the primary particles, and Dis the weight fractal dimensionality of the formed clus�ter, which takes the values of 1.71 and 2.5, accordingto the studies in [24] for the two� and three�dimen�sional case, respectively.

The weight fractal dimensionality of a cluster is theprincipal parameter which enables one to compare theaggregates that are formed during computer simula�tion with the structures from the real experiment.Based on the results of comparison of the experimen�tal and model fractal dimensionality of the obtainedaggregates, one can state the features of aggregation.However, only simulation cannot unambiguously con�firm which mechanisms of aggregation form the basisof any process. This is related to the fact that the iden�tical value of fractal dimensionality can correspond tovarious features of aggregation; therefore, any resultsthat were obtained based on simulation require mani�fold experimental confirmation.

0 ,D

gRN k

a⎛ ⎞

= ⎜ ⎟⎝ ⎠

(a)

(b)

(c)

Fig. 2. Results of three computer experiments (a–c) on theformation of fractal aggregates.

ΔGfilm, h

h0

Fig. 3. Excess free energy, ΔGfilm, compared to thickness ofliquid layer, h.



One of the principal drawbacks of the Witten–Sun�der model is the anisotropy of the formed fractalaggregates. The aggregates which are obtained withthe use of this model inherit the symmetry of the crys�tal lattice where they were grown. For example, on asquare lattice, the fractal aggregate has four principaldirections of growth; this particularly manifests itselffor the aggregates which are composed of a large num�ber of particles (Fig. 5). This statement of the problem(Figs. 5–8) contradicts the real physical systemswhere aggregation can proceed due to the motion ofparticles in a random direction. The solution of thisproblem is the transition to the grateless model; thiswas achieved in the designed program.

The next modification of the conventional DLAmodel is the introduction of the possibility of restruc�turing a growing fractal aggregate. This provides amore adequate description of the real processes ofaggregation that takes place during sol�gel processes.It was mentioned above that, for typical materials thatare used in sol�gel processes, coagulation proceedswith a high degree of probability (depending on theparameters of the dispersion medium) without the for�mation of direct contact between particles, i.e., in thepresence of a liquid layer between them (in the first orsecond minimum of the curve which is given in Fig. 3).In order to describe coagulation, the following mech�anism of evolution of a growing cluster was used in thework: cluster growth proceeds on the attachment of awandering particle to it. In this case, a particle canrotate easily around an neighboring particle from acluster with which the bond was formed to look intothe more energetically stable position with the highest

number of neighbors. After this position is found, theparticle is fixed in it.

In addition, some modification was performed inthe model, which is required to accelerate the simula�tion process. The particles which start the motion at arandom point on the circumference of “birth”, Rb, farfrom a cluster, intersect the circumference that cir�cumscribes cluster, Rmax, also at a random point, withan identical homogeneous distribution angle; thisenables one to combine these circumferences with theaim of reducing the time on additional iterations,which describe the Brownian motion of a particle. Thesecond important modification, which is based on thehomogeneous distribution of a random walk, involvesthe possibility of an increase in the step size, when aparticle is far from the cluster. When the distance fromthe particle to the cluster is Δr, one can take the stepwhich is marginally smaller than Δr, instead of a singlestep. These procedures allow one to substantiallyaccelerate the work of the program without violatingthe principles of the Brownian motion of particlesand, consequently, the structure of the formed clus�ters.

With the use of the developed program software(statistical modulus), the fractal dimensionalities ofboth the two� and three�dimensional aggregates weredetermined. In addition, the number of particles inaggregate represented the only input parameter in themodel, while its structure was directly determined bythe wandering of a particle. The obtained values agreewell with the results of the study from [24] (D2D = 1.71,D3D = 2.5): = 1.724 ± 0.027, and = 2.52 ±0.034. The typical fractal aggregates which wereobtained with the use of a DLA�modelling programmodulus are given in Figs. 6 and 7.

2DD 3DD

t1

t2

Rb

Rk

Fig. 4. Conventional Witten–Sunder model (t1 and t2 arepossible trajectories of particles).

Fig. 5. Typical fractal cluster that is plotted within the con�ventional Witten–Sunder model.

194


KONONOVA et al.

Modeling the growth of an aggregate which is capa�ble of restructuring at its formation gives interestingresults. The view of the obtained structures dependsstrongly on the number of particles (N) that constitutethe aggregate. At N < 200, an object is not weight frac�tal and represents a sufficiently dense structure, whichis given in Fig. 8.

With the increase in the mass of an aggregate, thegradual return to a fractal structure takes place and thefractal dimensionality of the obtained structuresalmost corresponds to the dimensionality observed forthe case without restructuring. Moreover, the view ofthese aggregates is strongly reminiscent of hydraulicstructures (“viscous fingers”), which appear on forc�ing gas or liquid with low viscosity through a liquidwith high viscosity.

In order to rationalize this effect, one should con�sider the DLA model from the viewpoint of a mathe�matical tool, which lays its foundation from the view�point of frontier conditions of the Laplace equation,which determine the structure of the upcoming aggre�gate. In addition, it should be noted that the physicalsystem that is similar to DLA is characterized by someminimal scale of instability, which appears on the clus�ter surface (or at the interface between viscous andnonviscous liquids) and represents the minimum sizeof the branching of a structure. Although in the usualDLA aggregate, the minimum scale of instability is the

particle diameter; its value is 2–3 diameters for theevolving aggregate; i.e., its stability to branchingincreased. This results in the appearance of densenonfractal structures (until the achievement of a par�ticular aggregate mass).

Thus, in the work, the existence of some criticalvalue of mass of the aggregate that evolves during itsgrowth, which represents the threshold of the forma�tion of thin fractal structures, was shown.

Studies of the morphology of fractal structures.Studies of the morphology of fractal structure of filmsthat were synthesized from sols in a precursor mediumwere performed with the use of optical microscopy ona Polam series microscope with the system of output ofinformation to the monitor of a Videotest type com�puter, electron microscopy on a Quanta Ispect rasterelectron microscope, and atomic force microscopy(AFM) with the use of an Ntegra Terma nanolabora�tory (NT�MDT, Zelenograd) in semicontact mode. Inorder to diagnose the surface of the obtained samples,probe sensors with a cantilever that represents the rodof a rectangular section of NSG 01 series of NT�MDT(Zelenograd) with resonance frequencies of 150 kHzwere used in the work. On the whole, it was shownaccording to the results of the studies of the AFMmethod that, by varying the conditions of the sol�gelprocess (the composition of the initial components,the amount of solvent, the holding time of the sols, the

Fig. 6. Two�dimensional DLA aggregate from 50000 particles, which was obtained in the DLA modeling program.



temperature of annealing), one can control the evolu�tion of fractal aggregates of silicon dioxide.

In the work, various stages of evolution of fractallyaggregated systems of silicon dioxide were experimen�tally determined: the formation of spherical particlesat nucleophilic growth (Figs. 9a, 10a, 11a); develop�ment of labyrinth structures under the conditions ofspinodal decomposition, which is caused by polymer�ization (Figs. 9b, 10b); and the formation of percola�tion network structures (Fig. 11b).

We note that porous silicon dioxide (Fig. 11b),which was prepared via the sol�gel method can be usedas the matrix for synthesis. This synthesis is usuallyperformed at several steps: first, a porous matrix is pre�pared via the sol�gel method and, after that, nanopar�ticles are precipitated into the pores. Derived fromdoped siliceous, the advantage of the suggestedmethod of synthesis is the possibility to obtain nano�composites in a general process, whose structure isformed by nanoparticles that are homogeneously dis�tributed in a porous matrix [6, 25–27].

The size of the obtained particles depends on theconditions of synthesis. Systems from spherical parti�cles (Fig. 10a) were obtained from sols after holdingfor 1 h (molar ratio of TEOS : ethyl alcohol : deionized

water : nitric acid was 192 : 200 : 17 : 1, respectively),applied on silica substrates, and annealed at 600°C.Porous SiO2 (Fig. 11à), which is composed of aggre�gated fused particles and pores, which represent thegaps between these particles, was prepared from solsafter holding for 1 h (molar ratio of TEOS : ethyl alco�hol : water : nitric acid was 105 : 105 : 1 : 2.5, respec�tively), applied on silica substrates, and annealed at600°C. Percolation network structures (Fig. 10b)based on silicon dioxide were prepared from sols thathad been held for 1 h, where the molar ratio of TEOS: ethyl alcohol : water : nitric acid was 105 : 105 : 1 : 2.5,respectively, were formed on silica substrates andannealed at 600°C.

The special characteristics of the composed aggre�gates are related to their chemical structure; however,they also depend strongly on their size and shape.Thus, control of the size and shape of these aggregatesis of interest for the realization of the most recentfunctional devices. Microscopic images of spin�likesilicon dioxide particles are given for illustration pur�poses in Fig. 12 (Fig. 12a), which represent the aggre�gates of thread particles that are packed longwise asscissors (Fig. 12b).

Fig. 7. Three�dimensional DLA aggregate from 10000 particles, which was obtained in the DLA modeling program.

196


KONONOVA et al.

Studies of sol�gel processes on the physical coolingof polymer systems. In the general case, the kinetics ofcoagulation of particles in sol adhere to theSmolukhovskii equation, where temperature, T, is indirect view:

(1)

where is the rate of decay of the sol concentration,

k is Boltzmann’s constant, η is the dynamic viscosityof the sol, and c0 is the initial concentration.

Processes of coagulation were considered in thework within a mathematically determined Julian frac�tal (Fig. 13) [27].

It should be noted that the fundamental principlesof the structure of the substance in sol�gel nanotech�nology is the modular principle, which is based on therepresentation of structures as the compositions frommodules (fragments), which perform the functions ofbuilding blocks, and principle of structural hierarchy,which assumes that the structures of one type are mod�ules, building blocks, and the second�order structures,which have substantially larger sizes, are formed from

η

dd

20

4 ,3

c kT ct= −

dd

ct

them; and second�order structures, conversely,agglomerate to the third�order structures, etc.

Moreover, it should be noted that the properties ofcomplex hierarchical formations cannot be reduced tothe properties of modules.

Obviously, the simplest idealized illustration ofhierarchical self�assembly is the projection of thethree�dimensional determined Julian fractal aggregate(with the Hausdorf–Bezikovich fractal dimensionalityof 1.77), which is given in Fig. 13 and designed via theconsecutive connection of identical spherical parti�cles. Moreover, six other particles are connected to theinitial particles. Thus, during the first step (first itera�tion), the initial assembly from seven particles wasformed; at the second iteration, six identical assem�blies are connected to the obtained aggregate; at thethird iteration, six identical aggregates were connectedto six ends of the formed aggregate, etc. These fractalsmay form in physical systems at cluster�cluster aggre�gation under the conditions when the kinetics of theself�assembly of clusters drastically slows down on theincrease in the mass of the fractal aggregate. However,for physical systems, it is preferable to use model con�siderations on the building block, which is composedof 13 elements (in the three�dimensional case). Amodule from seven elements is more appropriate for

N = 200 N = 500

N = 1000 N = 3000

Fig. 8. Evolution of the structure of rearranged fractal during growth.



the two�dimensional case. The example is obvious forthe rationalization of the principal possibility of thedesign of materials with calibrated values of pore sizes.This is very important for the design of devices of thenew generation. The design of materials with the givenset of the pores of various sizes is of great practicalinterest due to the possibility of solving multifunc�tional problems. Pore sizes affect catalytic activity, theadsorption ability of a nanoreactor, the effectiveness ofthe supply of reacting substances and the withdrawalof side products of reactions, the dimensional proper�ties of obtained nanoparticles, etc.

For the given fractal, the relationship between itsdiameter, d, and number of particles, n, would look asfollows:

(2)

where α is the diameter of one elementary fractal unit.

αln( 13)3 ,nd −

=

Phase diagram (Fig. 14) of a mixture of inorganic

polymer of the type and a solvent has the

upper critical mixing temperature that was suggestedby the Flory–Huggins theory.

The binodal line emphasizes the range of stablesolutions from the metastable range; spinodal is theboundary of absolute instability of a system and its

|

|Si O

n

⎡ ⎤⎢ ⎥⎣ ⎦

(b)

(а) 50 µm

50 µm

Fig. 9. Optical images that illustrate (a) the formation ofspherical structures from sols based on TEOS that ishydrolyzed in an acidic medium at nucleophilic growthand (b) the formation of labyrinth structures at spinodaldecomposition.

(b)

(а) 3 µm

10 µm

Fig. 10. Electron microscope images (in secondary elec�trons) that illustrate (a) the formation of spherical struc�tures from sols based on TEOS that is hydrolyzed in anacidic medium at nucleophilic growth and (b) the forma�tion of labyrinth structures at spinodal decomposition.

198


KONONOVA et al.

only exit to the stable range is the critical solution tem�perature (Fig. 14).

The binodal line differentiates the ranges of stableand metastable states in the phase diagram of solutionsand is given by the equivalence of the first derivative ofGibbs’ energy of mixing to zero on a concentration atconstant temperature and pressure:

(3)

In the range of metastable states, the system is sta�ble to a small change of concentration (fluctuation),but it is not stable to a large change. There is also therange of labile states, where the system is absolutelyunstable and decays without the application of energy.This range is separated from the range of the metasta�ble state by the spinodal line, which is determinedfrom the following relationship:

(4)

In order to determine the thermodynamic parame�ters of Gibbs’ energy, the lattice model of Flory andHuggins was used in the work. Following their theory,the entire space of the solution is divided by elemen�tary cells, which are filled by either a polymer mole�cule or solvent molecule. Assuming this, the energy ofthe interaction for a two�component system, Δw, willbe determined by the following equation:

(5)

mix0.

,

Gn

T p

⎛ ⎞∂ =⎜ ⎟⎜ ⎟∂⎝ ⎠

mix20.

2,

G

n T p

⎛ ⎞∂ =⎜ ⎟⎜ ⎟∂⎝ ⎠

12 11 221( ),2

w w w wΔ = − +

20

2

020180 161412108642

18

16

14

12

10

8

6

4

800

700

600

500

400

300

200

100

0

80

10

0

8070605040302010

70

60

50

40

30

20

2.0

1.5

1.0

0.5

0

667

77

2.30

00.

312

nm

µm

Height

µm

Height

µm

µm

(a) (b)

Fig. 11. Atomic force images that illustrate (a) the surface geometry of spherical structures from sols based on TEOS that is hydro�lyzed in acidic medium at nucleophilic growth and (b) formation of percolation network structures at spinodal decomposition.

(b)

(а) 50 µm

50 µm

Fig. 12. Optical images of spin�like particles (a) based onsilicon dioxide and (b) scissors�like structure.



where is the energy of interaction of the unit of amonomer chain with solvent molecules, is theenergy of interaction of solvent molecules to eachother, and is the energy of interaction of a mono�mer unit of a chain with other elements of this chain.

For this reason, the enthalpy of dissolution of poly�mer is

(6)

where k is the Boltzmann constant; is th enumberof solvent molecules; is the number of polymermolecules with the degree of polymerization, ; isthe total number of interactions; and is the Flory andHuggins parameter.

Let us denote the entropy value in an analogousmanner using the lattice model from Boltzmann’sequation:

(7)

where is the number of microstates which canimplement the given macrostate of the system. Then,following this relationship, we obtain the followingequation:

(8)

12w

11w

22w

( ) χmix 21 ,

xNH kTN

NΔ =

1N

2Nx N

χ

mix ln ,S k W=

W

[ ]mix ln ln(1 ) (1 ) ,R x x x xS = − + − −

where is the concentration that is represented inmolar fractions.

x

Fig. 13. Projection of three�dimensional determined Julian fractal aggregate.

Tem

per

atu

re

Composition of mixture

1

2

Fig. 14. Phase state diagram of the mixture of two compo�nents at physical cooling with the dome of decomposition.(1) binodal and (2) spinodal.

200


KONONOVA et al.

Thus,

(9)

It follows from the given dependences that, in theentire range between the minima of the dependence ofGibbs’ energy vs. composition, the system is prone tothe decomposition to two phases. This is mathemati�cally represented in Eq. (3); i.e., the system is in ametastable state between the critical points of energy.Let us consider Eq. (3) for the solution of polyortho�silicic acid in 2�butanol (Fig. 15).

However, if one considers the system between zerovalues of function (4), the energy of the mechanicalmixture is always less than the energy of the single�phase solution. In addition, system delayers almostinstantaneously without any external consumption ofenergy. This process is referred to as spinodal decom�position and is characterized by the appearance of lab�yrinth structures.

( ) [ ]χmix ln ln2

1 (1 ) (1 ) .xN

kTN TR x x x xN

G = + + − −

Consequently, the range of the spinodal decompo�sition is localized between the inflection points on theenergy curve of the system; this corresponds to Eq. (4).Let us consider this relationship for the SiO2–2�butanol system quantitatively (Fig. 16).

This dependence shows that spinodal decomposi�tion can be realized by the following routes: cooling ofthe system, polymerization with a decrease in the con�figuration entropy or rapid vaporization of solvent.However, this phenomenon is observed at moderatecooling of sol that does not amount to phase transi�tion. It should be noted that silica sol were obtainedafter the acidic hydrolysis of TEOS, which was per�formed in the medium where 2�butanol was used asthe organic solvent.

In the work, on spinodal decomposition, silicondioxide films were prepared from the sols that werecooled to the temperature of liquid nitrogen. InFig. 17, a typical AFM image of the film that wasobtained from the sol which was prepared in themedium of 2�butanol and TEOS, which illustrates theranges with a high and low concentration of nanopar�ticles of the order of 10 nm, is given. The denselypacked formations with a high concentration of spin�like nanoparticles represent self�organizing sphericalranges.

CONCLUSIONS

It was shown according to AFM that, by varyingthe thermodynamic and kinetic conditions of sol�gel

5

–5

0.50

molar fractions

x, kJ/mol

100 K200 K300 K400 K

∂Gmix

∂n T, p

Fig. 15. First derivative of Gibbs energy of mixing on con�centration compared to composition (t, p = const).

5

–5

0.50

molar fractions

x, kJ/mol

100 K200 K300 K400 K

∂2Gmix

∂n2T, p

Fig. 16. Second derivative of Gibbs energy of mixing onconcentration compared to composition (t, p = const).

100

0

200

9008007006005004003002001000

300

400

500

600

700

800

900

нм

нм

Fig. 17. Morphology of surface of the structure that wasprepared from sol, which was preliminarily cooled to thetemperature of liquid nitrogen.The size of the scanning range is 900 × 900 nm.



synthesis, one can control the evolution of fractalaggregates, which are formed at acidic hydrolysis andpolycondensation of tetraethoxysilane. In the work,various stages of evolution of fractally aggregated sys�tems were experimentally determined: the formationof spherical particles at nucleophilic growth; thedevelopment of labyrinth structures at spinodaldecomposition, which is caused by the processes ofpolymerization (pre�percolation structures); and theformation of percolation network structures.

Under the conditions of spinodal decompositionfrom sols that are cooled to the temperature of liquidnitrogen, densely packed formations with a high con�centration of spin�like nanoparticles, with a size of10 nm, which represent self�organizing sphericalranges, were obtained based on TEOS, which washydrolyzed in an acidic medium.

As the result of the cooling of sol, the followingphenomena are observed: a slowdown of the growth ofnucleating seeds in the range of nucleophilic growth;with the cooling of the system below the temperatureof spinodal, the spinodal decomposition of the solu�tion occurs, a new disordered state. However, coolingsol below the phase transition temperature, leads totheopposite phenomenon being observed, namely, theformation of highly ordered aggregates. This phenom�enon is related to the transitive processes of the rigidcooling–heating of sol.

ACKNOWLEDGMENTS

The work was supported by the Federal TargetedProgramme “Scientific and Scientific PedagogicalStaff of Innovative Russia” for 2009–2013 throughstate contract P1249 issued on June 7, 2010.

REFERENCES

1. Brinker, C.J. and Scherer, G.W., Sol–Gel Science: ThePhysics and Chemistry of Sol�Gel Processing, San Diego,California, United States: Academic, 1990.

2. Nakanishi, K., Takahashi, R., Nagakane, T.,Kitayama, K., Koheiya, N., Shikata, H., and Soga, N.,Formation of hierarchical pore structure in silica gel,J. Sol–Gel Sci. Technol., 2000, vol. 17, pp. 191–210.

3. Handbook of SolGel Science and Technology: Processing,Characterization, and Applications, Sumio Sakka, Ed.,New York, Springer�Verlag, 2004, vols. 1–3.

4. Hüsing, N., Design of inorganic and inorganic–organic hybrid materials by sol–gel processing–fromnanostructures to hierarchical networks, in Sol–GelMethods for Materials Processing, Innocenzi, P.,Zub, Yu.L., and Kessler, V.G., Eds., Dordrecht, TheNetherlands: Springer�Verlag, 2008, pp. 91–104.

5. Nakanishi, K., Sol–gel process of oxides accompaniedby phase separation, Bull. Chem. Soc. Jpn., 2006,vol. 79, no. 5, pp. 673–691.

6. Maksimov, A.I., Moshnikov, V.A., Tairov, Yu.M., andShilova, O.A., Osnovy zol’–gel’ tekhnologii nanoko�mpozitov (Fundamentals of the Sol–Gel Technology of

Nanocomposites), 2nd ed., St. Petersburg: Tekhnome�dia, 2008.

7. Gracheva, I.E., Moshnikov, V.A., Karpova, S.S., andMaraeva, E.V., Net�like structured materials for gassensors, J. Phys.: Conf. Ser., 2011, vol. 291, p. 012017.http://iopscience.iop.org/1742�6596/291/1/012017.

8. Moshnikov, V.A., Gracheva, I.E., Kuznezov, V.V.,Maximov, A.I., Karpova, S.S., and Ponomareva, A.A.,Hierarchical nanostructured semiconductor porousmaterials for gas sensors, J. Non�Cryst. Solids, 2010,vol. 356, nos. 37–40, pp. 2020–2025.

9. Gracheva, I.E., Maksimov, A.I., and Moshnikov, V.A.,Analysis of the structural features of tin dioxide�basedfractal nanocomposites using atomic�force microscopyand X�ray diffraction, J. Surf. Invest., 2009, vol. 3,no. 5, pp. 761–768.

10. Jullien, R. Fractal aggregates, Comments Condens. Mat�ter Phys., 1987, vol. 13, no. 4, pp. 177–205.

11. Zhabrev, V.A., Moshnikov, V.A., Tairov, Yu.M., andShilova, O.A., Zol’–gel’ tekhnologiya. Uchebnoe posobie(Sol–Gel Technology: A Guidebook), St. Petersburg:St. Petersburg Electrotechnical University “LETI,”2004.

12. Roldugin, V.I., The characteristics of fractal dispersesystems, Usp. Khim., 2003, vol. 72, no. 11, pp. 1027–1054.

13. Roldugin, V.I., Self�organization of nanoparticles atinterfacial surfaces, Usp. Khim., 2004, vol. 73, no. 2,pp. 123–156.

14. Shilova, O.A. and Shilov, V.V., Nanocomposite oxideand hybrid organicinorganic materials prepared by thesol–gel method: Synthesis, properties, and applica�tions, in Nanosistemy, Nanomaterialy, Nanotekhnologii(Nanosystems, Nanomaterials, and Nanotechnolo�gies), Shpak, A.P., Ed., Kiev: Akademperiodika, 2003,vol. 1, no. 1, pp. 983.

15. Shpak, A.P., Shilov, V.V., Shilova, O.A., andKunitskii, Yu.A., Diagnostika nanosistem. Mno�gourovnevye fraktal’nye nanostruktury (Diagnosis ofNanosystems: Multilevel Fractal Nanostructures),Kiev: Akademperiodika, 2004, part 2.

16. Beaucage, G., Approximation leading to a unifiedexponential/power�law approach to small�angle scat�tering, J. Appl. Crystallogr., 1995, vol. 28, pp. 717–728.

17. Beaucage, G., Small�angle scattering from polymericmass fractals of arbitrary mass�fractal dimension,J. Appl. Crystallogr., 1996, vol. 29, pp. 134–146.

18. Efros, A.L., Fizika i geometriya besporyadka (Physicsand Geometry of Disorder), Moscow: Nauka, 1982,issue 19.

19. Huggins, M., Some properties of solutions of long�chain compounds, J. Phys. Chem., 1942, vol. 46,pp. 151–158.

20. Flory, P.J., Principles of Polymer Chemistry, Ithaca, NewYork: Cornell University Press, 1953.

21. Grosberg, A.Yu. and Khokhlov, A.R., Statisticheskayafizika makromolekul: Uchebnoe rukovodstvo (StatisticalPhysics of Macromolecules: A Guidebook), Moscow:Nauka, 1989.

22. Rozenberg, B.A., Microphase separation in solidifiedmulticomponent polymeroligomer systems, Ross. Khim.Zh., 2001, vol. 45, nos. 5–6, pp. 23–31.

202


KONONOVA et al.

23. Khamova, T.V., Shilova, O.A., and Golikova, E.V.,Investigation of the structuring in the sol–gel systemsbased on tetraethoxysilane, Glass Phys. Chem., 2006,vol. 32, no. 4, pp. 448–459.

24. Meakin, P., A historical introduction to computermodels for fractal aggregates, J. Sol–Gel Sci. Technol.,1999, vol. 15, no. 2, p. 9799.

25. Moshnikov, V.A., Gracheva, I.E., and An’chkov, M.G.,Investigation of sol–gel derived nanomaterials with ahierarchical structure, Glass Phys. Chem., 2011, vol. 37,no. 5, pp. 485–495.

26. Gracheva, I.E., Moshnikov, V.A., Maraeva, E.V., Kar�pova, S.S., Alexsandrova, O.A., Alekseyev, N.I., Kuz�netsov, V.V., Olchowik, G., Semenov, K.N.,

Startseva, A.V., Sitnikov, A.V., and Olchowik, J.M.,Nanostructured materials obtained under conditions ofhierarchical self�assembly and modified by derivativeforms of fullerenes, J. Non�Cryst. Solids, 2012, vol. 358,no. 2, pp. 433–439.

27. Moshnikov, V.A. and Gracheva, I.E., Gas�sensitivenetwork nanocomposites based on dioxides of tin andsilicon, in Prilozhenie k zhurnalu “Vestnik Ryazanskogogosudarstvennogo radiotekhnicheskogo universiteta”(Supplement to the Journal “Bulletin of the RyazanState Radio Engineering University”), Ryazan, 2009,no. S30, pp. 92–98.

Translated by A. Murav’ev

SPELL: 1. Bystrov

Fractally Aggregated Micro and Nanosystems Synthesized from Sols

Documents

Transcript of Fractally Aggregated Micro and Nanosystems Synthesized from Sols