- Email: [email protected]
Fractal structure of gel porosity
Anolytica Chimica Acta, 163 (1984) 17-24 Elsevier Science Publishers B.V., Amsterdam -Printed
FRACTAL
STRUCTURE
B. GELLERI
and M. SERNETZ*
in The Netherlands
OF GEL POROSITY
Znstitut fiir Biochemie und Endokrinologie, Justus Liebig-Universith’t Giessen, Frankfurter Str. 100, D-6300 Giessen (Federal Republic of Germany) (Received 17th April 1984)
SUMMARY The concept of fractals as a dimensional measure to describe random irregularity and fragmentation, applied to the structure of gels enables the distribution of pore sizes to be explained as a generalized log-logistic function. This is achieved by introducing upper and lower limits to the power function of self-similar structured organization. In contrast to previous empirical relations for gel-permeation techniques, this approach provides a theoretically based relation between the measured distribution coefficients and the size of the eluting molecules as a measure without undue assumptions. Linearization by transformation of the cumulative log-logistic distribution function is proposed for regression of experimental data. The validity of the theory is demonstrated with size-exclusion data for different gel matrices. Other systems with properties based on irregularity and fragmentation of their heterogeneous structure could be described and analyzed by the same theory.
Knowledge of the porosity of a gel, defined as the differential or cumulative frequency distribution of pore sizes or of the inner volume or surface, as a function of the size of permeating or interacting sample molecules is of special importance for the analytical description of gels. Such knowledge is also valuable in making the best practical use of inert permeation gel matrices (e.g., in gel chromatography) or of activated carrier matrices (e.g., for the immobilization of enzymes and the development of enzyme reactors). Some methods for the determination of porosity, such as mercury porosimetry, nitrogen capillary condensation or scanning-electron microscopy, are restricted to porous materials in the dry state. Therefore, they can give only indirect information on the supramolecular organization which is responsible for the peculiar properties of the swollen gel. Gel-permeation (size-exclusion) chromatography provides the actual pore-size distribution, because it enables the relative accessible volume of the gel to sample molecules of known size to be evaluated from their elution volumes. The regression of experimental data within a given range of validity has previously been based either on empirical relations such as proportionality between distribution coefficient and molecular weight or on the assumption of postulated formal distributions. Usually, normal distributions [l, 21 or 0003-2670/84/$03.00
o 1984 Elsevier Science Publishers B.V.
18
log-normal distributions [3] of pore sizes have been assumed. Both these distributions have been adopted merely for convenience and from lack of logical arguments, but both include wrong assumptions. Thus, the normal distribution with a range from -00 to += includes pores with negative diameter, whereas the log-normal distribution with range from 0 to +a allows pores of unlimited size. This is also unrealistic because the radius of the biggest pores cannot surpass, indeed cannot even reach, the size of the gel particles, and at least an upper limit of pore sizes is logically essential. A theoretical approach is described here for the general analysis of gel structure. The approach needs no prior assumptions on the kind of poresize distribution; instead, the distribution function is derived rationally as a consequence of the nature of the gel. No statement on the gel is used other than that the irregular, fragmented and isotropic structure of the network of polymer chains is fractal, and in particular self-similar within a range sufficiently far from upper and lower limits. THEORY
Fractal and self-similar structure The theoretical approach, here applied to gel porosity, starts from the interpretation of pore porosity as a cumulative pore-volume distribution. It is based on the general observation that the measurable inner volume (i.e., the dependent, measured property) increases with reduction of the size of the sample molecules (the measure) or, in other words, with the scale of resolution. The fractal concept is now widely used to describe a shape or structure as a function of the scale of resolution; for example, if the description of a fractal structure at the macroscopic level will remain valid at the microscopic level and even for large molecular aggregates, it becomes selfsimilar within that range. When a measurable property of a system is not independent, but remains a function of the measure used, or of the scale of resolution, the system under observation is considered to be of fractal structure [4]. For any scale, the dependence of the measured function y on the size of the measure x can be described by a power function Y = a .$Dr - Dr(x)l
(1)
in which & denotes the expected topological integer dimension and DF the so-called fractal dimension at the range LXof the measure. By generalization, the term dimension includes non-integer values. It becomes a measure of the gradation (degree, excess or intensity) by which a real property of a structure evades the measuring procedure with a topologically defined measure within a certain range of scale. The structured system deviates into a dimension alien to the dimension of the topological measure. The VdUe of the fractal exponent DT - &, the dimensional excess, may change in different ranges of scale, according to the changing degree of irregularity observed. Thus the fractal exponent defines a measure of the
19
degree of structured organization, given by a certain geometrical or statistical rule of substructure formation. In ranges where there is no dependence on the measure or scale, DF becomes Dr. If, however, the extent of gradation remains identical over a wide range of scales, and therefore the fractal exponent remains constant, i.e., (& - Dv) = constant = b, then the fractal system becomes self-similar. In a selfsimilar structure, the scale of resolution can no longer be perceived from the extent of structured organization in any partial frame. In the self-similar case, the fractal dependence becomes a power function with constant exponent b y=C&
(2)
which in a double logarithmic plot, log y vs. log X, is a straight line with the slope b = DT -Dr. The derivative of the power function, without limits, describes the hyperbolic density distribution (Pareto distribution) of the structural components [ 41. Limited self-similarity In contrast to the unlimited mathematical case, natural systems are selfsimilar only within certain limits and are bounded by upper and lower limits, namely physical constraints of the structural elements. Starting from the selfsimilar domain, these borders may be reached asymptotically with decreasing frequencies of the typical structures. This fractal mode of interpretation can easily be used to describe the porosity of gels and carrier matrices, as measured by gel-permeation techniques. The dependence of the inner volume y on the molecular size x can be presented in a logarithmic plot with linear approximation in the central part. With the smallest molecules, accessible to the entire inner volume and thus yielding constant elution volume V,,,, the upper limit 0, of structured organization is reached asymptotically; likewise, with molecules too big for permeation, the smallest elution volume V, mh is reached as lower limit U,,. Beyond these limits, the gel structure is scale-independent with respect to porosity. The difference 0, - CT, is the maximum volume range available for gel permeation and exclusion by molecular size. Without any limiting constraints, the gel structure would be described as self-similar, and the pore volume distribution would follow the self-similar power function (Eqn. 2). Into this power function, the upper limit 0, and lower limit U, are introduced as properties of the real natural system in such a way that the derivative of the power function, dy /do = y b/x, becomes zero on approaching these limits [ 51, i.e., dy/k
= [(O, -Y)(Y
- U,)/(O,
-
U,)l [b/xl
(3)
Thus the unlimited self-similar relation of the power function (Eqn. 2) becomes a fractal relationship within the range 0, - U,, (Eqn. 3). The solution of this differential equation [6] yields the function y = (0,Pxb
+ U,)/(l
+ Prb)
(4)
20
which is called log-logistic here, because of its relationship to the logistic function. By rearrangement, the parameter of position, P, is P = [(Yo - U,)/(G, --Yo)l
(5)
Wool b
defined by the limits of integration n o and y. and by the exponent b. Whereas in the self-similar power function (Eqn. 2), the exponent b characterizes the dimensional excess (i.e., the extent to which the structure deviates from the dimension of the topological, unstructured measure), in the loglogistic function (Eqn. 4) the exponent becomes the parameter of dispersion of a distribution function. In a double logarithmic plot, the power function of self-similarity is a straight line with slope b as dimensional excess. The corresponding loglogistic function is sigmoidal with a wide, quasilinear central range. The scaledependent dimensional excess m, i.e., the slope for any scale, can be found from the logarithmic derivative m = d log y/d log x = b(0,
- U,)Pxb/(l
+ Pxb)(O&b
+ U,)
(6)
The broader the quasi-linear fitted range of the log-logistic function, the smaller the difference between the maximum slope mi at the inflexion point i of the log-logistic function and the exponent b of the self-similar case. The log-logistic function (Eqn. 4) results from the kind of approach towards the borders chosen, i.e., equal weighting of the distances. It yields symmetry in logarithmic coordinates. In differently defined structural systems, other asymptotic transitions may be possible. Here, linear transformation of the log-logistic function (Eqn. 4) by a logit transformation [ 71 logit = log [(y - U,)/(O, -y)]
= 1ogP + b logx
(7)
is proposed for graphical regression of experimental data in a logit-log plot. If the upper and lower limits 0, and U,, are known from experiments (i.e., Ve,,, and Ve.mti in gel permeation), then the transformation provides P as parameters of position and b as dispersion. EXPERIMENTAL
Experimental results on Eupergit C (Rohm-Pharma, Darmstadt) and literature data on size exclusion for Sephadex G-100 (Pharmacia, Uppsala) [8] and Merckogel SI-100 (Merck, Darmstadt) [3] were used to prove the validity of the proposed fractal concept and to demonstrate the suitability of the log-logistic function to describe gel porosity. Experiments on Eupergit C were done with glass chromatographic columns (60 cm long, 1.6 cm i.d.) with a bed volume of about 85 ml. The oxirane groups of the gel matrix were deactivated by 2-mercaptoethanol. Adsorption was prevented by 1% sodium dodecyl sulphate (SDS). Details of the procedures are available elsewhere [6]. The substances used for size-exclusion chromatography and their molecular data are listed in Table 1.
21 TABLE 1 Molecular data of test substances used for gel permeation on deactivated Eupergit C, and experimental elution volumes V,, distribution coefficients Kd and increments of difference distribution p Substance tested 1 2 3 4 5 6 7 8 9 10 11 12
Mol. wt. (lo-’ Dalton)
Dextran blue Thyreoglobulin Ferritin BSA Ovalbumin Cbymotrypsinogen Myoglobin Cytochrome C Insulin A-chain Lactose Glucose D,O
2000 660 340 68 43.5 22.8 17 12.3 2.5 0.36 0.18 0.020
13 Latex 0.22 flm
-
Radius
f/f,,
Runs n
V, (ml)’
Ed
P
1.34 1.18 1.12 1.11
25 11 8 16 10 5 7
41.7 41.4 42.4 48.3 52.6 55.6 62.0
2.2 2.8 3.5 3.1 2.8 2.8 3.1
0.0 0.0 0.004 0.239 0.384 0.485 0.7
0.03 0.22 1.14 1.27 0.92 3.18
1.11 -
6 8 7 7 5
64.1 65.7 67.4 69.3 70.9
3.0 3.1 2.2 3.9 2.1
0.771 0.825 0.882 0.963 1.0
;‘i4
-
2
40.7
-
-
-
(nm)
37.4 8.2 6.2 3.5 2.7 2.1 1.8 1.7 1.0 0.44 0.36 0.1 110.0
0:16 0.72 o *1
‘Mean and standard deviation. RESULTS
AND DISCUSSION
The results of size-exclusion chromatography on deactivated Eupergit C are presented in Table 1. The elution volumes, V,, were normalized to a loo-ml column and conventional Kd values were derived. In Table 1, p denotes the difference distribution of pore radii, defined as p = AV,lViA
log r
(8)
which is the relative increment of pore size as a function of the molecular size. Figure 1 shows the elution volumes of Eupergit C, Sephadex G-100 and Merckogel SI-100 in a double logarithmic plot. These volumes correspond to discrete points of the actually continuous cumulative distribution of the inner volume of the gel. They can be fitted by the log-logistic function. The excellent agreement of the experimental data of three independent gels, covering the entire measuring range, demonstrates the validity of the theoretical approach based on the fractal concept. This fractal concept is characterized by the acceptance of limits to the self-similarity of gel structure. The specific access to the limits results in the log-logistic function of pore-size distribution. This seems to be the first theoretically justified approach on the descrip tion and analysis of the structure of porous gels, which avoids undue preliminary conditions. The description of gel structure becomes independent of the assumption of formal mathematical distributions, such as the Gaussian
22
%I
a
“,
60 50
t
025
0.15
005 I
.
,
,,,( 1
,
, , ,_, 10
, , . , ,,,, , 100 r[nml
Fig. 1. Elution volume V, as a function of molecular radius r with logarithmic coordinates. Experimental results for (a) Eupergit C, (b) Sephadex G-100 and (c) Merckogel SI-100 were fitted by the log-logistic function. Fig. 2. Difference distribution p of pore radii for (a) Eupergit C and (b) Merckogel SI-100 in comparison with the pore-size distribution d log y/d log x, calculated from the logarithmic derivative of the log-logistic function.
[21 or log-Gaussian distribution [ 31 as approximations. It is well known that the conventional distribution coefficients Kd can be used only in a limited range, usually 0.2 < Kd < 0.8 for approximation of Gaussian distributions, because of systematic deviations outside that range. In the present case, e.g., with Eupergit C, only five Kd values (Table 1, compounds 4-8) could have been used for linearization, whereas with the log-logistic function the entire range of molecular data shown in Table 1 can be utilized. Figure 2 shows the difference distributions p (Eqn. 8) of the pore sizes for Eupergit C and Merckogel SI-100 in comparison with the differential pore-size distributions d log y/d log x, calculated from the logarithmic derivative of the log-logistic function. These plots represent the portion of pores of a certain size over the entire pore size distribution. Compared to the smooth course of the differential distribution, a plot of the difference distribution p is of course highly sensitive to experimental noise, both with respect to elution differences A V, and with the molecular data A log r. Nevertheless, the fit of the distributions is very satisfactory. The fit to the log-logistic function seems distinctly more appropriate than the formal approximation to a log-Gaussian distribution [ 31.
23
With the distinct experimental limiting conditions 0, and U,, the loglogistic distribution provides by the parameter P = XL UJO,, the mode x, and by the parameter b the dispersion of the pore sizes of the gel. Figure 3 shows the linearization of experimental data in a logit-log plot and the estimation of the parameters. Because of its theoretical justification, this is more powerful than empirical relations like K, vs. log (m.w.), which exhibit only restricted ranges with approximately linear dependence. Thus the log-logistic distribution describes the structure of the gel by the dispersion of porosity within the natural borders 0, and U,. Shifting of these borders to infinity, namely 0, + m and U, + 0, re-establishes the unlimited self-similar case, described by the power function in Eqn. 2. In doing so, the parameter b of the log-logistic function as a measure of dispersion again becomes the exponent b of the power function and thus the fractal dimension as the measure of gradation and structured organization. This is a new, unconstrained and meaningful characterization of a peculiar structural property, here of a gel, which previously has been considered only by operational correction terms such as tortuosity or a labyrinth factor. The relation between the log-logistic dependence in a limited system and the power function in the unlimited case for self-similar structure can be transferred to a series of analogous relationships and processes. Introducing fractal or self-similar concepts into descriptions of log-logistic phenomena
I
,
,
, 1
, ,
,
,
10
r Inml
,
,, ,
,
100
Fig. 3. Linearization of the cumulative log-logistic pore-size distribution by logit transformation (logit-log plot) for (a) Eupergit C, (b) Sephadex G-100 and (c) Merckogel SI-100.
24
should allow better understanding of these systems on the basis of heterogeneity of molecular states or irregularity and fragmentation of their structure. This applies especially for all functions that can be described by the general equation y = (Axb + C)/(B + xb). Examples are the heterogeneity of the Freundlich isotherm, the cooperativity of oligomeric enzymes, the generalized antigen-antibody interaction, or the dose-response relation in pharmacology, where the exponent b of interaction represents (as dispersion) the heterogeneity of molecular states, This investigation was supported by Grant Se 315/11-6 of the Deutsche Forschungsgemeinschaft. REFERENCES 1 2 3 4 5 6 7 8
G. K. Ackers, J. Biol. Chem., 242 (1967) 3026. W. Boguth, R. Repges and M. Sernetz, Z. Anal. Chem., 243 (1968) 464. J. Halasz and K. Martin, Angew. Chem., 90 (1978) 954. B. B. Mandelbrot, Fractals: Form, Chance, Dimension, W. H. Freeman, San Francisco, 1977. J. Petroll, Staub-Reinhalt. Luft, 34 (1974) 445. B. Gelleri, Thesis, Giessen, 1982. W. D. Ashton, The Logit Transformation, Griffin, London, 1972. P. Andrews, Biochem. J., 91 (1964) 222.
FRACTAL
STRUCTURE
B. GELLERI
and M. SERNETZ*
in The Netherlands
OF GEL POROSITY
Znstitut fiir Biochemie und Endokrinologie, Justus Liebig-Universith’t Giessen, Frankfurter Str. 100, D-6300 Giessen (Federal Republic of Germany) (Received 17th April 1984)
SUMMARY The concept of fractals as a dimensional measure to describe random irregularity and fragmentation, applied to the structure of gels enables the distribution of pore sizes to be explained as a generalized log-logistic function. This is achieved by introducing upper and lower limits to the power function of self-similar structured organization. In contrast to previous empirical relations for gel-permeation techniques, this approach provides a theoretically based relation between the measured distribution coefficients and the size of the eluting molecules as a measure without undue assumptions. Linearization by transformation of the cumulative log-logistic distribution function is proposed for regression of experimental data. The validity of the theory is demonstrated with size-exclusion data for different gel matrices. Other systems with properties based on irregularity and fragmentation of their heterogeneous structure could be described and analyzed by the same theory.
Knowledge of the porosity of a gel, defined as the differential or cumulative frequency distribution of pore sizes or of the inner volume or surface, as a function of the size of permeating or interacting sample molecules is of special importance for the analytical description of gels. Such knowledge is also valuable in making the best practical use of inert permeation gel matrices (e.g., in gel chromatography) or of activated carrier matrices (e.g., for the immobilization of enzymes and the development of enzyme reactors). Some methods for the determination of porosity, such as mercury porosimetry, nitrogen capillary condensation or scanning-electron microscopy, are restricted to porous materials in the dry state. Therefore, they can give only indirect information on the supramolecular organization which is responsible for the peculiar properties of the swollen gel. Gel-permeation (size-exclusion) chromatography provides the actual pore-size distribution, because it enables the relative accessible volume of the gel to sample molecules of known size to be evaluated from their elution volumes. The regression of experimental data within a given range of validity has previously been based either on empirical relations such as proportionality between distribution coefficient and molecular weight or on the assumption of postulated formal distributions. Usually, normal distributions [l, 21 or 0003-2670/84/$03.00
o 1984 Elsevier Science Publishers B.V.
18
log-normal distributions [3] of pore sizes have been assumed. Both these distributions have been adopted merely for convenience and from lack of logical arguments, but both include wrong assumptions. Thus, the normal distribution with a range from -00 to += includes pores with negative diameter, whereas the log-normal distribution with range from 0 to +a allows pores of unlimited size. This is also unrealistic because the radius of the biggest pores cannot surpass, indeed cannot even reach, the size of the gel particles, and at least an upper limit of pore sizes is logically essential. A theoretical approach is described here for the general analysis of gel structure. The approach needs no prior assumptions on the kind of poresize distribution; instead, the distribution function is derived rationally as a consequence of the nature of the gel. No statement on the gel is used other than that the irregular, fragmented and isotropic structure of the network of polymer chains is fractal, and in particular self-similar within a range sufficiently far from upper and lower limits. THEORY
Fractal and self-similar structure The theoretical approach, here applied to gel porosity, starts from the interpretation of pore porosity as a cumulative pore-volume distribution. It is based on the general observation that the measurable inner volume (i.e., the dependent, measured property) increases with reduction of the size of the sample molecules (the measure) or, in other words, with the scale of resolution. The fractal concept is now widely used to describe a shape or structure as a function of the scale of resolution; for example, if the description of a fractal structure at the macroscopic level will remain valid at the microscopic level and even for large molecular aggregates, it becomes selfsimilar within that range. When a measurable property of a system is not independent, but remains a function of the measure used, or of the scale of resolution, the system under observation is considered to be of fractal structure [4]. For any scale, the dependence of the measured function y on the size of the measure x can be described by a power function Y = a .$Dr - Dr(x)l
(1)
in which & denotes the expected topological integer dimension and DF the so-called fractal dimension at the range LXof the measure. By generalization, the term dimension includes non-integer values. It becomes a measure of the gradation (degree, excess or intensity) by which a real property of a structure evades the measuring procedure with a topologically defined measure within a certain range of scale. The structured system deviates into a dimension alien to the dimension of the topological measure. The VdUe of the fractal exponent DT - &, the dimensional excess, may change in different ranges of scale, according to the changing degree of irregularity observed. Thus the fractal exponent defines a measure of the
19
degree of structured organization, given by a certain geometrical or statistical rule of substructure formation. In ranges where there is no dependence on the measure or scale, DF becomes Dr. If, however, the extent of gradation remains identical over a wide range of scales, and therefore the fractal exponent remains constant, i.e., (& - Dv) = constant = b, then the fractal system becomes self-similar. In a selfsimilar structure, the scale of resolution can no longer be perceived from the extent of structured organization in any partial frame. In the self-similar case, the fractal dependence becomes a power function with constant exponent b y=C&
(2)
which in a double logarithmic plot, log y vs. log X, is a straight line with the slope b = DT -Dr. The derivative of the power function, without limits, describes the hyperbolic density distribution (Pareto distribution) of the structural components [ 41. Limited self-similarity In contrast to the unlimited mathematical case, natural systems are selfsimilar only within certain limits and are bounded by upper and lower limits, namely physical constraints of the structural elements. Starting from the selfsimilar domain, these borders may be reached asymptotically with decreasing frequencies of the typical structures. This fractal mode of interpretation can easily be used to describe the porosity of gels and carrier matrices, as measured by gel-permeation techniques. The dependence of the inner volume y on the molecular size x can be presented in a logarithmic plot with linear approximation in the central part. With the smallest molecules, accessible to the entire inner volume and thus yielding constant elution volume V,,,, the upper limit 0, of structured organization is reached asymptotically; likewise, with molecules too big for permeation, the smallest elution volume V, mh is reached as lower limit U,,. Beyond these limits, the gel structure is scale-independent with respect to porosity. The difference 0, - CT, is the maximum volume range available for gel permeation and exclusion by molecular size. Without any limiting constraints, the gel structure would be described as self-similar, and the pore volume distribution would follow the self-similar power function (Eqn. 2). Into this power function, the upper limit 0, and lower limit U, are introduced as properties of the real natural system in such a way that the derivative of the power function, dy /do = y b/x, becomes zero on approaching these limits [ 51, i.e., dy/k
= [(O, -Y)(Y
- U,)/(O,
-
U,)l [b/xl
(3)
Thus the unlimited self-similar relation of the power function (Eqn. 2) becomes a fractal relationship within the range 0, - U,, (Eqn. 3). The solution of this differential equation [6] yields the function y = (0,Pxb
+ U,)/(l
+ Prb)
(4)
20
which is called log-logistic here, because of its relationship to the logistic function. By rearrangement, the parameter of position, P, is P = [(Yo - U,)/(G, --Yo)l
(5)
Wool b
defined by the limits of integration n o and y. and by the exponent b. Whereas in the self-similar power function (Eqn. 2), the exponent b characterizes the dimensional excess (i.e., the extent to which the structure deviates from the dimension of the topological, unstructured measure), in the loglogistic function (Eqn. 4) the exponent becomes the parameter of dispersion of a distribution function. In a double logarithmic plot, the power function of self-similarity is a straight line with slope b as dimensional excess. The corresponding loglogistic function is sigmoidal with a wide, quasilinear central range. The scaledependent dimensional excess m, i.e., the slope for any scale, can be found from the logarithmic derivative m = d log y/d log x = b(0,
- U,)Pxb/(l
+ Pxb)(O&b
+ U,)
(6)
The broader the quasi-linear fitted range of the log-logistic function, the smaller the difference between the maximum slope mi at the inflexion point i of the log-logistic function and the exponent b of the self-similar case. The log-logistic function (Eqn. 4) results from the kind of approach towards the borders chosen, i.e., equal weighting of the distances. It yields symmetry in logarithmic coordinates. In differently defined structural systems, other asymptotic transitions may be possible. Here, linear transformation of the log-logistic function (Eqn. 4) by a logit transformation [ 71 logit = log [(y - U,)/(O, -y)]
= 1ogP + b logx
(7)
is proposed for graphical regression of experimental data in a logit-log plot. If the upper and lower limits 0, and U,, are known from experiments (i.e., Ve,,, and Ve.mti in gel permeation), then the transformation provides P as parameters of position and b as dispersion. EXPERIMENTAL
Experimental results on Eupergit C (Rohm-Pharma, Darmstadt) and literature data on size exclusion for Sephadex G-100 (Pharmacia, Uppsala) [8] and Merckogel SI-100 (Merck, Darmstadt) [3] were used to prove the validity of the proposed fractal concept and to demonstrate the suitability of the log-logistic function to describe gel porosity. Experiments on Eupergit C were done with glass chromatographic columns (60 cm long, 1.6 cm i.d.) with a bed volume of about 85 ml. The oxirane groups of the gel matrix were deactivated by 2-mercaptoethanol. Adsorption was prevented by 1% sodium dodecyl sulphate (SDS). Details of the procedures are available elsewhere [6]. The substances used for size-exclusion chromatography and their molecular data are listed in Table 1.
21 TABLE 1 Molecular data of test substances used for gel permeation on deactivated Eupergit C, and experimental elution volumes V,, distribution coefficients Kd and increments of difference distribution p Substance tested 1 2 3 4 5 6 7 8 9 10 11 12
Mol. wt. (lo-’ Dalton)
Dextran blue Thyreoglobulin Ferritin BSA Ovalbumin Cbymotrypsinogen Myoglobin Cytochrome C Insulin A-chain Lactose Glucose D,O
2000 660 340 68 43.5 22.8 17 12.3 2.5 0.36 0.18 0.020
13 Latex 0.22 flm
-
Radius
f/f,,
Runs n
V, (ml)’
Ed
P
1.34 1.18 1.12 1.11
25 11 8 16 10 5 7
41.7 41.4 42.4 48.3 52.6 55.6 62.0
2.2 2.8 3.5 3.1 2.8 2.8 3.1
0.0 0.0 0.004 0.239 0.384 0.485 0.7
0.03 0.22 1.14 1.27 0.92 3.18
1.11 -
6 8 7 7 5
64.1 65.7 67.4 69.3 70.9
3.0 3.1 2.2 3.9 2.1
0.771 0.825 0.882 0.963 1.0
;‘i4
-
2
40.7
-
-
-
(nm)
37.4 8.2 6.2 3.5 2.7 2.1 1.8 1.7 1.0 0.44 0.36 0.1 110.0
0:16 0.72 o *1
‘Mean and standard deviation. RESULTS
AND DISCUSSION
The results of size-exclusion chromatography on deactivated Eupergit C are presented in Table 1. The elution volumes, V,, were normalized to a loo-ml column and conventional Kd values were derived. In Table 1, p denotes the difference distribution of pore radii, defined as p = AV,lViA
log r
(8)
which is the relative increment of pore size as a function of the molecular size. Figure 1 shows the elution volumes of Eupergit C, Sephadex G-100 and Merckogel SI-100 in a double logarithmic plot. These volumes correspond to discrete points of the actually continuous cumulative distribution of the inner volume of the gel. They can be fitted by the log-logistic function. The excellent agreement of the experimental data of three independent gels, covering the entire measuring range, demonstrates the validity of the theoretical approach based on the fractal concept. This fractal concept is characterized by the acceptance of limits to the self-similarity of gel structure. The specific access to the limits results in the log-logistic function of pore-size distribution. This seems to be the first theoretically justified approach on the descrip tion and analysis of the structure of porous gels, which avoids undue preliminary conditions. The description of gel structure becomes independent of the assumption of formal mathematical distributions, such as the Gaussian
22
%I
a
“,
60 50
t
025
0.15
005 I
.
,
,,,( 1
,
, , ,_, 10
, , . , ,,,, , 100 r[nml
Fig. 1. Elution volume V, as a function of molecular radius r with logarithmic coordinates. Experimental results for (a) Eupergit C, (b) Sephadex G-100 and (c) Merckogel SI-100 were fitted by the log-logistic function. Fig. 2. Difference distribution p of pore radii for (a) Eupergit C and (b) Merckogel SI-100 in comparison with the pore-size distribution d log y/d log x, calculated from the logarithmic derivative of the log-logistic function.
[21 or log-Gaussian distribution [ 31 as approximations. It is well known that the conventional distribution coefficients Kd can be used only in a limited range, usually 0.2 < Kd < 0.8 for approximation of Gaussian distributions, because of systematic deviations outside that range. In the present case, e.g., with Eupergit C, only five Kd values (Table 1, compounds 4-8) could have been used for linearization, whereas with the log-logistic function the entire range of molecular data shown in Table 1 can be utilized. Figure 2 shows the difference distributions p (Eqn. 8) of the pore sizes for Eupergit C and Merckogel SI-100 in comparison with the differential pore-size distributions d log y/d log x, calculated from the logarithmic derivative of the log-logistic function. These plots represent the portion of pores of a certain size over the entire pore size distribution. Compared to the smooth course of the differential distribution, a plot of the difference distribution p is of course highly sensitive to experimental noise, both with respect to elution differences A V, and with the molecular data A log r. Nevertheless, the fit of the distributions is very satisfactory. The fit to the log-logistic function seems distinctly more appropriate than the formal approximation to a log-Gaussian distribution [ 31.
23
With the distinct experimental limiting conditions 0, and U,, the loglogistic distribution provides by the parameter P = XL UJO,, the mode x, and by the parameter b the dispersion of the pore sizes of the gel. Figure 3 shows the linearization of experimental data in a logit-log plot and the estimation of the parameters. Because of its theoretical justification, this is more powerful than empirical relations like K, vs. log (m.w.), which exhibit only restricted ranges with approximately linear dependence. Thus the log-logistic distribution describes the structure of the gel by the dispersion of porosity within the natural borders 0, and U,. Shifting of these borders to infinity, namely 0, + m and U, + 0, re-establishes the unlimited self-similar case, described by the power function in Eqn. 2. In doing so, the parameter b of the log-logistic function as a measure of dispersion again becomes the exponent b of the power function and thus the fractal dimension as the measure of gradation and structured organization. This is a new, unconstrained and meaningful characterization of a peculiar structural property, here of a gel, which previously has been considered only by operational correction terms such as tortuosity or a labyrinth factor. The relation between the log-logistic dependence in a limited system and the power function in the unlimited case for self-similar structure can be transferred to a series of analogous relationships and processes. Introducing fractal or self-similar concepts into descriptions of log-logistic phenomena
I
,
,
, 1
, ,
,
,
10
r Inml
,
,, ,
,
100
Fig. 3. Linearization of the cumulative log-logistic pore-size distribution by logit transformation (logit-log plot) for (a) Eupergit C, (b) Sephadex G-100 and (c) Merckogel SI-100.
24
should allow better understanding of these systems on the basis of heterogeneity of molecular states or irregularity and fragmentation of their structure. This applies especially for all functions that can be described by the general equation y = (Axb + C)/(B + xb). Examples are the heterogeneity of the Freundlich isotherm, the cooperativity of oligomeric enzymes, the generalized antigen-antibody interaction, or the dose-response relation in pharmacology, where the exponent b of interaction represents (as dispersion) the heterogeneity of molecular states, This investigation was supported by Grant Se 315/11-6 of the Deutsche Forschungsgemeinschaft. REFERENCES 1 2 3 4 5 6 7 8
G. K. Ackers, J. Biol. Chem., 242 (1967) 3026. W. Boguth, R. Repges and M. Sernetz, Z. Anal. Chem., 243 (1968) 464. J. Halasz and K. Martin, Angew. Chem., 90 (1978) 954. B. B. Mandelbrot, Fractals: Form, Chance, Dimension, W. H. Freeman, San Francisco, 1977. J. Petroll, Staub-Reinhalt. Luft, 34 (1974) 445. B. Gelleri, Thesis, Giessen, 1982. W. D. Ashton, The Logit Transformation, Griffin, London, 1972. P. Andrews, Biochem. J., 91 (1964) 222.