- Email: [email protected]
Twofold description of disordered structures
J O U R N A L OF
ELSEVIER
Journal of Non-Crystalline Solids 192 & 193 (1995) 111-115
Twofold description of disordered structures Vicente Mayagoitia *, Fernando Rojas, Isaac Kornhauser Departamento de QMmica, Universidad Aut6noma Metropolitana-lztapalapa, Apartado Postal 55-534, M~xico 13, D.F. 09340, Mexico
Abstract A new statistical theory, which has been applied initially to the description of porous media and heterogeneous adsorbent surfaces, is proposed to be extended to the morphology of nanoscale disordered materials. 1. Introduction The structure and topological aspects of many types of amorphous media, which influence their physical and chemical properties, stabilities and phase transitions, can be studied by means of what we have termed a twofold description. This method has been firstly applied to assess the morphology of (i) porous media [1] (for the study of capillary processes in such media as, for instance, capillary condensation [2] and evaporation [3] or textural determinations [4]); and (ii) heterogeneous surfaces of adsorbents [5] (for the treatment of adsorption equilibrium and surface diffusion [6], chemisorption [7], reaction rate in the adsorbed phase [8] and surface characterization [9]). More recently, this theory has been also applied to the description of other types of disordered system such as arboreous [10] and dense [11] aggregates resulting from the sol-gel transition, as well as to the assessment of the morphology of polymers [12]. It seems to us that this approach could be extended to treat many other different kinds of disordered media and that other nanostructures could be usefully described by means of this theory.
* Corresponding author. Tel: + 52-5 724 4672. Telefax: + 52-5 724 4666. E-mail: [email protected].
For clarity, the foundations of the twofold description are first presented for the special case of porous media - the field of its initial application and then the general principles of this theory are formulated. Finally, a preliminary discussion of the application of this theory to nanostructures is outlined, and visual representations of some disordered structures are presented.
2. Twofold description of the topology of porous media Every porous network can be visualized as being comprised of two types of element: sites (antrae, cavities) and bonds (capillaries, passages), alternating unavoidably to form the connected network. The connectivity, C, is the number of bonds meeting at a site, while every bond is delimited by two sites. For simplicity, the size of each entity is expressed using a unique quantity, R, defined in the following way: for sites considered as hollow spheres, R is the radius of the sphere, while for bonds idealized as hollow cylinders open at both ends, R is the radius of the cylinder. Instead of considering only the size distribution of voids, without regard to the type of element (site or bond) to which they belong, it would seem more
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 4 3 3 - 5
V. Mayagoitia et al. / Journal of Non-Crystalline Solids 192&193 (1995) 111-115
112
appropriate to establish a double distribution of sizes. In this scheme, Fs(R) and FB(R) will be the size distribution functions, for sites and bonds, on a number of elements basis and normalized so that the probabilities of finding a site or a bond having a size, R, or smaller are, respectively,
S(R)= forCFs(R) dR; B(R)= forCFB(R) dR. (1) From the very definitions of 'site' and 'bond' emerges, in a natural way, the following construction principle: the size of any bond is always smaller than or at most equal to the size of the site to which it leads. Two self-consistency laws guarantee the fulfillment of the above principle. The first law establishes that bonds must be sufficiently small to be associated with the sites belonging to a given size distribution, expressed by
B(R) >~S(R)
forevery R.
(2)
A second law is still required since, when there is a considerable overlap between the size distributions, there appear topological size correlations between neighbouring elements. Thus, the probabilities of finding a size R s for a site and a size R B for a given one of its bonds are not independent. The probability density for this joint event is
F(R s NRB) =Fs(Rs)FB(RB)Ck(Rs, RB).
(3)
The second law can be expressed as 6 ( R s, R B ) = 0
f o r R B > R s.
(4)
If the randomness in the topological assignation of sizes is maximized, while applying the restriction imposed by the construction principle, the most verisimilar form of ~b in the correct case, R B > Rs, is obtained [1] as
(
B- S
exp --~S(RO 6(Rs, RB)=
B(Rs )
S(Rs)
B- S S(RB)
exp --~B(R~) =
B(RB) -
(5)
The topological size correlations promote a size segregation effect, in that sites and bonds of the larger size join together to form regions of large elements, while elements of the smaller size associate to comprise alternating regions of small entities. This effect is the more important the greater the overlap. The consequences of this effect in the development of capillary processes are of the utmost importance. We have proposed a classification of porous structures based upon the relative positions of the size distributions of sites and bonds [2]; five types have been recognized, each exhibiting a characteristic behaviour during capillary condensation and evaporation [2,3]. More generally, percolation processes have been analyzed for correlated porous media [13].
3. General foundations of the twofold description
In order to obtain a convenient statistical description of other complex systems, it is necessary to (1) decompose the system into a collection of two types of interrelated element, 'sites' and 'bonds', each having a different function in the network that must be clearly established; (2) recognize a 'metric', or a property such as size, number of repetitive units, energy, etc., that characterizes each element. The nature of this metric must be the same for sites and bonds; (3) point out, from the very definitions of 'site' and 'bond', a 'construction principle': an obvious and constant inequality in the metric of every site compared with those of its corresponding bonds (networks observing this principle are termed as ' self-consistent'); (4) propose a distribution of this metric for each type of element. The twofold distribution must be set in a number of elements basis and normalized. If the distributions of sites and bonds are inappropriate or their overlap is considerable, the construction principle risks violation. In order to avoid this, two 'self-consistency laws' must be imposed: the first deals with restrictions imposed to distributions as a whole, in order to have a permitted collection of sites and bonds, while the second is of a local character and prevents the union of elements that could violate the construction principle.
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192& 193 (1995) 111-115
A balance of sites and bonds, when performing a self-consistent assignation of metrics to directly related elements, allows a ready determination of the function expressing the correlation of metrics throughout the network. This function is similar for different types of network. When the randomness in this assignation of metric to the elements is maximized, with self-consistency remaining as the only restriction, the structure is termed 'verisimilar', or that having the most expected morphology in the absence of other particular information. Conversely, a comparison of the verisimilar model with experiment permits an assessment of the physicochemical constraints that, acting during the formation of the network, lower the randomness. Application of this function, either by analytical (probabilistic) or by digital (Monte Carlo) methods [8-11] provides information about the presence of a 'segregation effect' of the metric throughout the network, i.e., usually networks are non-fully random media. This procedure yields a more detailed dual-description about the distribution of the metric, but most of all gives precious information about the 'morphology', or the precise sequence of values of the metric throughout the network, statistically expressed. The term 'twofold description' is related to the consideration of two types of element, but the treatment could also be multivariate, as for arboreous aggregates [10,11]. Treatments, such as percolation theory and fractal approaches, are general enough as to be applicable to many phenomena, apparently unrelated at first sight. Our approach similarly offers the possibility of describing intricate topological properties of many systems in terms of a twofold diagram and a very simple correlation function. Verisimilarity, the basic property sought in the twofold analysis, appears to be much more fundamental than autosimilarity, the key property in fractal studies. We term as verisimilar models of disordered media those that are conceived by introducing the minimum sort of restrictions, e.g., wherein the randomness is shared alike by all types of constitutive element. The correlation function expressed by Eq. (5) corresponds to this case, but this function has a remarkable property of adaptability to much more restrictive situations. Overlap between element distributions seems to play a central role in this descrip-
113
tion. An increasing overlap leads to short-range and then to medium-range order in structures. When overlap tends to completeness, long-range order arises.
4. Three applications The twofold description, when used in conjunction with Monte Carlo methods, allows the visualization of diverse disordered structures, such as porous media, heterogeneous adsorption surfaces or aggregation products resulting from the sol-gel transition (Figs. l(a)-(c)). In the case of porous media mentioned above, the construction principle refers to the impossibility of a site (a circle representing a sphere in Fig. l(a)) to be smaller (in radius) than any of its C surrounding bonds (the cylinders around each sphere). Expressions such as Eq. (3) are necessary for obtaining the required conditional probability densities, F(Rs, R~) (to find a site of size R s connected to a bond of size R B) and F(RB, R s) (to find a bond of size R B connected to a site of a given size Rs), and then to implement the desired Monte Carlo approach. Size correlations between neighbouring elements arise by means of the function ~b, the segregation effect being one consequence. For heterogeneous adsorption surfaces, the construction principle indicates that sites (potential energy wells at the solid surface at which vapour molecules can adsorb) are always deeper (i.e., always of a larger absolute energy) than any of the C energy summits (bonds) that surround it. Expressions giving the probabilities of finding a site linked to a bond of a given energy or vice-versa can be setup via the appropriate correlation energy function similar to that in Eq. (5), now expressed in terms of the absolute potential energies of sites and bonds. Fig. l(b) is a view of a heterogeneous surface with valleys (sites) and summits (bonds) of potential energy. For aggregates appearing by gelation of a sol precursor, it is necessary to establish a simplified model. The gel originates from a network of varying energy, which comprises potential energy minima or sites (where particles have a certain probability to appear, depending on the energy of this potential well), each one delimited by potential energy barriers
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192&193 (1995) 111-115
114
(bonds) that govern the difficulty of adding the particle to a growing cluster. Again, the energies of the sites are deeper than the energies of their bonds. Special considerations concerning the possibilities of a particle being close to an existing cluster and surmounting an energy barrier in order to aggregate with it lead to the implementation of a Monte Carlo method that allows the representation of a frozen picture at a particular stage of the gelation process (Fig. l(c)). Depending on the particular conditions imposed on the aggregation mechanism (e.g., the parameters of the twofold energy distribution, the number of seed particles, the orientation restraints for a particle to aggregate into a cluster, etc.), the structures can range between loose (diffusion-limited aggregation) and compact (reaction-limited aggregation). Future work will extend farther the twofold description theory to other kinds of nanoscale manifestations, such as precursors to sol-gel materials and polymers, clusters or microcrystallites in media such as amorphous silica, and disorder accompanying structural relaxation.
(a)
This work was supported by CONACyT (M6xico) and CONICET (Argentina) under the joint project titled 'Cat~lisis, Fisicoqulmica de Superficies e Interfases Gas-S61ido.'
References
|
I
Fig. 1. Morphologies of some disordered structures: (a) a porous medium, (b) a heterogeneous adsorption surface and (c) a gel produced by the aggregation of particles.
[1] V. Mayagoitia, M.J. Cruz and F. Rojas, J. Chem. Soc., Faraday Trans. 1 85 (1989) 2071. [2] V. Mayagoitia, F. Rojas and I. Kornhauser, J. Chem. Soc., Faraday Trans. 1 84 (1988) 785. [3] V. Mayagoitia, B. Gilot, F. Rojas and I. Kornhauser, J. Chem. Soc., Faraday Trans. 1 84 (1988) 801. [4] V. MayagoJtia and F. Rojas, in: Fundamentals of Adsorpton III, ed. A.B. Mersmann and S.E. Scholl (The Engineering Foundation, New York, 1991) p. 563. [5] V. Mayagoitia, F. Rojas, V. Pereyra and G. Zgrablich, Surf. Sci. 221 (1989) 394. [6] V. Mayagoitia, F. Rojas, J.L. Riccardo, V. Pereyra and G. Zgrablich, Phys. Rev. B41 (1990) 7150. [7] V. Mayagoitia and I. Kornhauser, in: Fundamentals of Adsorption IV, ed. M. Suzuki (Kodansha, Tokyo, 1993) p. 421. [8] V. Mayagoitia, F. Rojas and I. Kornhauser, Langmuir 9 (1993) 2748.
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192&193 (1995) 111-115 [9] J.L. Riccardo, V. Pereyra, G. Zgrablich, V. Mayagoitia, F. Rojas and I. Komhauser, Langmuir 9 (1993) 2730. [10] V. Mayagoitia, A. Domlnguez and F. Rojas, J. Non-Cryst. Solids 147&148 (1992) 183. [11] V. Mayagoitia, F. Rojas and I. Kornhauser, J. Sol-Gel Sci. Technol. 2 (1994) 259.
115
[12] G.B. Kuznetsova, V. Mayagoitia and I. Kornhauser, J. Polym. Mater. 19 (1993) 19. [13] R.J. Faccio, G. Zgrablich and V. Mayagoitia, J. Phys.: Condens. Matter 5 (1993) 1823.
ELSEVIER
Journal of Non-Crystalline Solids 192 & 193 (1995) 111-115
Twofold description of disordered structures Vicente Mayagoitia *, Fernando Rojas, Isaac Kornhauser Departamento de QMmica, Universidad Aut6noma Metropolitana-lztapalapa, Apartado Postal 55-534, M~xico 13, D.F. 09340, Mexico
Abstract A new statistical theory, which has been applied initially to the description of porous media and heterogeneous adsorbent surfaces, is proposed to be extended to the morphology of nanoscale disordered materials. 1. Introduction The structure and topological aspects of many types of amorphous media, which influence their physical and chemical properties, stabilities and phase transitions, can be studied by means of what we have termed a twofold description. This method has been firstly applied to assess the morphology of (i) porous media [1] (for the study of capillary processes in such media as, for instance, capillary condensation [2] and evaporation [3] or textural determinations [4]); and (ii) heterogeneous surfaces of adsorbents [5] (for the treatment of adsorption equilibrium and surface diffusion [6], chemisorption [7], reaction rate in the adsorbed phase [8] and surface characterization [9]). More recently, this theory has been also applied to the description of other types of disordered system such as arboreous [10] and dense [11] aggregates resulting from the sol-gel transition, as well as to the assessment of the morphology of polymers [12]. It seems to us that this approach could be extended to treat many other different kinds of disordered media and that other nanostructures could be usefully described by means of this theory.
* Corresponding author. Tel: + 52-5 724 4672. Telefax: + 52-5 724 4666. E-mail: [email protected].
For clarity, the foundations of the twofold description are first presented for the special case of porous media - the field of its initial application and then the general principles of this theory are formulated. Finally, a preliminary discussion of the application of this theory to nanostructures is outlined, and visual representations of some disordered structures are presented.
2. Twofold description of the topology of porous media Every porous network can be visualized as being comprised of two types of element: sites (antrae, cavities) and bonds (capillaries, passages), alternating unavoidably to form the connected network. The connectivity, C, is the number of bonds meeting at a site, while every bond is delimited by two sites. For simplicity, the size of each entity is expressed using a unique quantity, R, defined in the following way: for sites considered as hollow spheres, R is the radius of the sphere, while for bonds idealized as hollow cylinders open at both ends, R is the radius of the cylinder. Instead of considering only the size distribution of voids, without regard to the type of element (site or bond) to which they belong, it would seem more
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 4 3 3 - 5
V. Mayagoitia et al. / Journal of Non-Crystalline Solids 192&193 (1995) 111-115
112
appropriate to establish a double distribution of sizes. In this scheme, Fs(R) and FB(R) will be the size distribution functions, for sites and bonds, on a number of elements basis and normalized so that the probabilities of finding a site or a bond having a size, R, or smaller are, respectively,
S(R)= forCFs(R) dR; B(R)= forCFB(R) dR. (1) From the very definitions of 'site' and 'bond' emerges, in a natural way, the following construction principle: the size of any bond is always smaller than or at most equal to the size of the site to which it leads. Two self-consistency laws guarantee the fulfillment of the above principle. The first law establishes that bonds must be sufficiently small to be associated with the sites belonging to a given size distribution, expressed by
B(R) >~S(R)
forevery R.
(2)
A second law is still required since, when there is a considerable overlap between the size distributions, there appear topological size correlations between neighbouring elements. Thus, the probabilities of finding a size R s for a site and a size R B for a given one of its bonds are not independent. The probability density for this joint event is
F(R s NRB) =Fs(Rs)FB(RB)Ck(Rs, RB).
(3)
The second law can be expressed as 6 ( R s, R B ) = 0
f o r R B > R s.
(4)
If the randomness in the topological assignation of sizes is maximized, while applying the restriction imposed by the construction principle, the most verisimilar form of ~b in the correct case, R B > Rs, is obtained [1] as
(
B- S
exp --~S(RO 6(Rs, RB)=
B(Rs )
S(Rs)
B- S S(RB)
exp --~B(R~) =
B(RB) -
(5)
The topological size correlations promote a size segregation effect, in that sites and bonds of the larger size join together to form regions of large elements, while elements of the smaller size associate to comprise alternating regions of small entities. This effect is the more important the greater the overlap. The consequences of this effect in the development of capillary processes are of the utmost importance. We have proposed a classification of porous structures based upon the relative positions of the size distributions of sites and bonds [2]; five types have been recognized, each exhibiting a characteristic behaviour during capillary condensation and evaporation [2,3]. More generally, percolation processes have been analyzed for correlated porous media [13].
3. General foundations of the twofold description
In order to obtain a convenient statistical description of other complex systems, it is necessary to (1) decompose the system into a collection of two types of interrelated element, 'sites' and 'bonds', each having a different function in the network that must be clearly established; (2) recognize a 'metric', or a property such as size, number of repetitive units, energy, etc., that characterizes each element. The nature of this metric must be the same for sites and bonds; (3) point out, from the very definitions of 'site' and 'bond', a 'construction principle': an obvious and constant inequality in the metric of every site compared with those of its corresponding bonds (networks observing this principle are termed as ' self-consistent'); (4) propose a distribution of this metric for each type of element. The twofold distribution must be set in a number of elements basis and normalized. If the distributions of sites and bonds are inappropriate or their overlap is considerable, the construction principle risks violation. In order to avoid this, two 'self-consistency laws' must be imposed: the first deals with restrictions imposed to distributions as a whole, in order to have a permitted collection of sites and bonds, while the second is of a local character and prevents the union of elements that could violate the construction principle.
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192& 193 (1995) 111-115
A balance of sites and bonds, when performing a self-consistent assignation of metrics to directly related elements, allows a ready determination of the function expressing the correlation of metrics throughout the network. This function is similar for different types of network. When the randomness in this assignation of metric to the elements is maximized, with self-consistency remaining as the only restriction, the structure is termed 'verisimilar', or that having the most expected morphology in the absence of other particular information. Conversely, a comparison of the verisimilar model with experiment permits an assessment of the physicochemical constraints that, acting during the formation of the network, lower the randomness. Application of this function, either by analytical (probabilistic) or by digital (Monte Carlo) methods [8-11] provides information about the presence of a 'segregation effect' of the metric throughout the network, i.e., usually networks are non-fully random media. This procedure yields a more detailed dual-description about the distribution of the metric, but most of all gives precious information about the 'morphology', or the precise sequence of values of the metric throughout the network, statistically expressed. The term 'twofold description' is related to the consideration of two types of element, but the treatment could also be multivariate, as for arboreous aggregates [10,11]. Treatments, such as percolation theory and fractal approaches, are general enough as to be applicable to many phenomena, apparently unrelated at first sight. Our approach similarly offers the possibility of describing intricate topological properties of many systems in terms of a twofold diagram and a very simple correlation function. Verisimilarity, the basic property sought in the twofold analysis, appears to be much more fundamental than autosimilarity, the key property in fractal studies. We term as verisimilar models of disordered media those that are conceived by introducing the minimum sort of restrictions, e.g., wherein the randomness is shared alike by all types of constitutive element. The correlation function expressed by Eq. (5) corresponds to this case, but this function has a remarkable property of adaptability to much more restrictive situations. Overlap between element distributions seems to play a central role in this descrip-
113
tion. An increasing overlap leads to short-range and then to medium-range order in structures. When overlap tends to completeness, long-range order arises.
4. Three applications The twofold description, when used in conjunction with Monte Carlo methods, allows the visualization of diverse disordered structures, such as porous media, heterogeneous adsorption surfaces or aggregation products resulting from the sol-gel transition (Figs. l(a)-(c)). In the case of porous media mentioned above, the construction principle refers to the impossibility of a site (a circle representing a sphere in Fig. l(a)) to be smaller (in radius) than any of its C surrounding bonds (the cylinders around each sphere). Expressions such as Eq. (3) are necessary for obtaining the required conditional probability densities, F(Rs, R~) (to find a site of size R s connected to a bond of size R B) and F(RB, R s) (to find a bond of size R B connected to a site of a given size Rs), and then to implement the desired Monte Carlo approach. Size correlations between neighbouring elements arise by means of the function ~b, the segregation effect being one consequence. For heterogeneous adsorption surfaces, the construction principle indicates that sites (potential energy wells at the solid surface at which vapour molecules can adsorb) are always deeper (i.e., always of a larger absolute energy) than any of the C energy summits (bonds) that surround it. Expressions giving the probabilities of finding a site linked to a bond of a given energy or vice-versa can be setup via the appropriate correlation energy function similar to that in Eq. (5), now expressed in terms of the absolute potential energies of sites and bonds. Fig. l(b) is a view of a heterogeneous surface with valleys (sites) and summits (bonds) of potential energy. For aggregates appearing by gelation of a sol precursor, it is necessary to establish a simplified model. The gel originates from a network of varying energy, which comprises potential energy minima or sites (where particles have a certain probability to appear, depending on the energy of this potential well), each one delimited by potential energy barriers
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192&193 (1995) 111-115
114
(bonds) that govern the difficulty of adding the particle to a growing cluster. Again, the energies of the sites are deeper than the energies of their bonds. Special considerations concerning the possibilities of a particle being close to an existing cluster and surmounting an energy barrier in order to aggregate with it lead to the implementation of a Monte Carlo method that allows the representation of a frozen picture at a particular stage of the gelation process (Fig. l(c)). Depending on the particular conditions imposed on the aggregation mechanism (e.g., the parameters of the twofold energy distribution, the number of seed particles, the orientation restraints for a particle to aggregate into a cluster, etc.), the structures can range between loose (diffusion-limited aggregation) and compact (reaction-limited aggregation). Future work will extend farther the twofold description theory to other kinds of nanoscale manifestations, such as precursors to sol-gel materials and polymers, clusters or microcrystallites in media such as amorphous silica, and disorder accompanying structural relaxation.
(a)
This work was supported by CONACyT (M6xico) and CONICET (Argentina) under the joint project titled 'Cat~lisis, Fisicoqulmica de Superficies e Interfases Gas-S61ido.'
References
|
I
Fig. 1. Morphologies of some disordered structures: (a) a porous medium, (b) a heterogeneous adsorption surface and (c) a gel produced by the aggregation of particles.
[1] V. Mayagoitia, M.J. Cruz and F. Rojas, J. Chem. Soc., Faraday Trans. 1 85 (1989) 2071. [2] V. Mayagoitia, F. Rojas and I. Kornhauser, J. Chem. Soc., Faraday Trans. 1 84 (1988) 785. [3] V. Mayagoitia, B. Gilot, F. Rojas and I. Kornhauser, J. Chem. Soc., Faraday Trans. 1 84 (1988) 801. [4] V. MayagoJtia and F. Rojas, in: Fundamentals of Adsorpton III, ed. A.B. Mersmann and S.E. Scholl (The Engineering Foundation, New York, 1991) p. 563. [5] V. Mayagoitia, F. Rojas, V. Pereyra and G. Zgrablich, Surf. Sci. 221 (1989) 394. [6] V. Mayagoitia, F. Rojas, J.L. Riccardo, V. Pereyra and G. Zgrablich, Phys. Rev. B41 (1990) 7150. [7] V. Mayagoitia and I. Kornhauser, in: Fundamentals of Adsorption IV, ed. M. Suzuki (Kodansha, Tokyo, 1993) p. 421. [8] V. Mayagoitia, F. Rojas and I. Kornhauser, Langmuir 9 (1993) 2748.
V. Mayagoitia et al. /Journal of Non-Crystalline Solids 192&193 (1995) 111-115 [9] J.L. Riccardo, V. Pereyra, G. Zgrablich, V. Mayagoitia, F. Rojas and I. Komhauser, Langmuir 9 (1993) 2730. [10] V. Mayagoitia, A. Domlnguez and F. Rojas, J. Non-Cryst. Solids 147&148 (1992) 183. [11] V. Mayagoitia, F. Rojas and I. Kornhauser, J. Sol-Gel Sci. Technol. 2 (1994) 259.
115
[12] G.B. Kuznetsova, V. Mayagoitia and I. Kornhauser, J. Polym. Mater. 19 (1993) 19. [13] R.J. Faccio, G. Zgrablich and V. Mayagoitia, J. Phys.: Condens. Matter 5 (1993) 1823.