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Fractal kinetics versus fractional derivative kinetics

This study presents a detailed comparison of the two most popular fractal theories used in the field of kinetics sorption of pollutants in porous materials: the Brouers-Sotolongo model family of kinetics based on the BurrXII statistical distribution and the fractional kinetics based on the Riemann-Liouville fractional derivative theory. Using the experimental kinetics data of several studies published recently, it can be concluded that, although these two models both yield very similar results, the Brouers-Sotolongo model is easier to use due to its simpler formal expression and because it enjoys all the properties of a well-known family of distribution functions. We use the opportunity of this study to comment on the information, in particular, the sorption strength, the half-life time, and the time dependent rate, which can be drawn from a complete analysis of measured kinetics using a fractal model. This is of importance to characterize and classify sorbent-sorbate couples for practical applications. Finally, a generalization form of the Brouers-Sotolongo equation is presented by introducing a time dependent fractal exponent. This improvement, which has a physical meaning, is necessary in some cases to obtain a good fit of the experimental data.

variables should be limited and have some clear physical interpretation. This is an old philosophical problem going back to Occam's razor principle: entia non sunt multiplicanda praeter necessitatem.
Therefore, the present work is written with the objective to reduce this gap and to present a method which takes advantage of new development in statistical and stochastic theories.
We will do that by reviewing, commenting, and comparing some recent published kinetics works.
The philosophy of our approach is to define and calculate the macroscopic quantities characterizing the sorption with mathematically well-defined functions containing a minimum of parameters. To interpret macroscopic data, we have to approach the problem with a mixture of mathematical rigor and empirical flair and to leave the belief that the use of microscopic or mesoscopic precise models, with the help of supercomputers, might be able to describe macroscopically complex systems. The reason is simply that the experience has shown that several different microscopic models can yield the same macroscopic equations. The best example is the Weibull function which can represent a number of different and unrelated physical systems with a variety of static and dynamic interactions. . They both rely on the concept of time fractality and include memory non-Markovian effects. We will compare these two methods theoretically and experimentally.
Finally, we will take the opportunity of this work to introduce a further improvement of the BSf(n, ) model, by assuming a time variation of the fractal exponent to account the change of the physical conditions during the evolution of the sorbing process.

Experimental data
In the present study, seven sets of experimental kinetic data of: tetracycline adsorption onto magnesium oxide-coated lightweight expanded clay aggregate (set 1), indigo carmine adsorption onto activated carbon without and with alternating current (set 2 and 3, respectively); cyanide onto LTA-zeolite (set 4); copper recovery from the Chilean mineral deposit by flotation (set 5); water vapor adsorption in cement (set 6), and fluorine adsorption onto clay (set 7) were analyzed. The experimental procedure details of these kinetics data are found in ( Where ( ) is the cumulative Burr XII distribution function where the variable in this context is the time ( ), is the fractional reaction order parameter; and is a characteristic time which the scale of the process.
In order to account for the initial power law behavior for small time observed in most complex systems (Kopelman, 1988), an exponent has been added to the variable of the original Burr function. It is the so-called generalized Burr XII function which due to its Alternatively, (9) and (10) can be combined as shown in Equation (11) The function ̃( ) varies slowly from to / . One has: ̃( ) → when → 0 and ̃( ) → , when → ∞.
The differential equation describes a birth and death process modulated by a quasi-hyperbolic function which expresses the irreversibility of the kinetics. Equations (9-11) can be modified to represent more complex systems as this is the case in the field of epidemiology.
The two asymptotic behaviors of the survival function are: When ( −1 ) < 1, , the Burr XII function belong to the basin of attraction of the heavy tail Lévy distributions. Equation (6) Where exp ( ) is the q-deformed exponential (1 − ) − 1 The differential eqs. (9 and 10) can be rewritten in the form of a simple first order differential equation: The proposed kinetics equation based on this theory in the field of sorption has been named ( , ) (eq. (17)) or sometimes BSW(t).
Where ( ) and are the time dependent and maximum sorbed quantities, respectively (mg/g); n is the fractional order of the reaction; is the fractal coefficient expressing macroscopically the complexity of the sorbent-sorbate couple, and is a characteristic time (min). In addition, the half time 50% (min) corresponding to ( ) =0.5 is given by: It is interesting to note that when = 1 in equation (8) the BurrXII survival function is the Tsallis density function while in eq.(7) the BurrXII density function is the Tsallis "escort" probability in the non-extensive theory (Tsallis, 2002). generalization of the first order differential equation (19) governing the exponential decay with n =1:
Whose solution is shown in equation (20) below: Where is the Riemann-Liouville fractional integral. For any > 0, this fractional integral is defined as: , is the Gamma function. For the existence of the integral (24), it is sufficient that the function ( ) be locally integrable in + and for → 0 behaves like ( − ) with a number < .
One of the most useful properties of the Riemann-Liouville fractional derivative is given by its Laplace transform as: The sign ℒ means the Laplace transform.
If we introduce a relaxation time * as a time scale, Equation (21) Which allows us to write the fractional derivative survival function or relaxation function as: Where: For = 1, one recovers an exponential (Debye) decay, (t) is given by: The rate here is given by: where * is a characteristic time corresponding to the peak in the susceptibility .
The Cole-Cole response function ( ) is given by the inverse Laplace transform of (s) : And the relaxation function: Therefore, in the framework of the fractional derivative theory, the sorption kinetics equation which has been used in sorption and to be compared with (17) is: Using the asymptotic behavior of the ML function is given by function we have: The two functions eq. (36) and eq. (17)

Results and discussion
In this study, a detailed comparison of the two methods using some recent published data was presented. The full calculations have been done with the nonlinear fitting programming of Mathematica software (version 10) which includes the computation of the ML functions. First we will proceed with a formal comparison determining, for a chosen value of in the ML and the ( , ) formulas, the value of which yields the same calculated kinetics. It appears that for a given common value of the fractional solution almost coincides numerically with the fractal solution having value of n such that 1 ≤ ≤ 2, (see Fig.1 a-c). For the comparison, to have the same asymptotic behavior for → 0, we have use for the scaling factors = 1 and * = Γ(1 + α) −1/ . As recalled earlier in sorption problems n is not easily determined, being the relevant macroscopic parameter, we can therefore anticipate that in practice both methods can give very similar representation of the experimental curve. For a common value of in ML( ) and  (1, 0.75) ; dashed: (2, 0.75) ; black solid: (1.75, 0.75)] We are now going to compare the results of both methods using seven set of data from the recent literature and some unpublished data, the results are tabulated in Tables (1-7). They will confirm the conclusion of the theoretical comparison. One can compare the results of sets 1, 2, 3 and 4, there is practically no difference between the results for the three characteristic coefficient and using ( , ) (with =1) and eq.36. The introduction of an alternating current does not change the sorption power but increases the velocity of the reaction. This is due to an increase of the fractal coefficient and the corresponding decrease of the half-time by a factor of order 3. The calculations with both methods quantize what is apparent in the data.
We have treated the data of reference (Vinnett et al., 2015) calculating the percentage of copper uptake (set 5). Here the two methods described in section 2.2.1 and 2.2.2, provide similar results. In that paper a different fractional equation has been used introducing a three parameter function different from the ML functions. The comparison shows that their parameter does not have a clear fractal or asymptotic behavior interpretation.     Recently a paper has been published on the sorption of water in cement (Saeidpour, M., Wadsö, L., 2015). The data contain more than hundred kinetics sorption points measured with great precision. In that case, it is possible to determine the exact value of n and therefore a comparison between the full BSf(n,a) equation and the ML form is more accurate (see table 6 and Fif. we have assumed a power law variation of the fractal exponent between its initial value (at = 0 ) and (at = saturation time, ) where the sorption can be assumed to be saturated.
A new parameter has been introduced which is a measure of the time evolution of the fractal coefficient . In that case, the best fit is obtained for a variation of from 0.65 to 0.47.
We have compared with the calculations of Li et al 2016 using BSf(n, ) under another confusing name. Because of the great number of data and therefore the highest precision, we have added in the table the error range (Error) in which the parameter around their more probable value could lie using results given by the Mathematica program.
In that case we have also calculated the rate (eq.16) (Fig. 3). The BSf(1.180, 0.655) rate is closer to Weibull rate. The ML rate is also between BSf(1, ) and BSf(2, ) rates and has calculated asymptotic behaviors closer to the Hill rate. The reason is that the asymptotic exponents of the survival function ( ) are the same in both cases (see eqs.12-13). The Tsallis or q-exponential kinetics rate where the fractal exponent is 1, is much lower because it neglects an important fractal factor of the kinetics process.   (Table 7).

Conclusions
The two methods can provide fits with almost the same precision. Both have a degree of empiricism. Bf(n, ) introduces an empirical fractal time and the fractional equation is the Laplace transform of the empirical Cole-Cole expression of the relaxation susceptibility in the frequency domain. We believe the BSf(n,a) is in practice more useful in that field when data are small compared with those used in econometrics or life-time studies. The reason of our preference is that, being a well-defined distribution function (BurrXII as well as its approximations Weibull and Log-logistic distributions) and knowing their statistical properties, it is easier to characterize physically and mathematically the sorption properties of a sorbent-sorbate couple. In particular, it yields simple analytical expressions for the rate equation and the half-life time. Another reason, more fundamental from a theoretical point of view is that the BSf( , ) can be derived from a stochastic description giving a physical meaning to the two coefficient and n which are related to the multiscale fractal organization and cluster architecture of internal system at the micro-and mesoscopic scales. Another practical advantage of BSf( , ) is that it is has a simple analytic form contrary to the Mittag-Leffler formula and as a consequence, it can be applied simply to other problems. For that reason the computational time is also at least five times more rapid.