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51
G. V. Kozlov, G. E. Zaikov, E. Klodzinska,
Richert Jozef
THE FRACTAL KINETICS OF POLYMERIZATION CATALYZED BY NANOFILLERS (PART 2)
Keywords: polymerization, kinetics, catalyst, nanofiller, fractal analysis, strange diffusion.
The fractal analysis of polymerization kinetics in nanofiller presence was performed. The influence of catalyst structural features on chemical reaction course was shown. The notions of strange (anomalous) diffusion conception was applied for polymerization reactions description.
Ключевые слова: полимеризация, кинетика, катализатор, нанонаполнитель, фрактальный анализ, аномальная диффузия.
Был выполнен фрактальный анализ кинетики полимеризации в присутствии нанонаполнителя. Было показано влияние структурных особенностей катализатора на ход химической реакции. Понятие странной (аномальной) диффузии было использовано для описания реакций полимеризации.
Introduction
By Sergeev’s definition the nanochemistry is a science field connected with obtaining and studying of physical-chemical properties of particles having sizes of nanometer scale. Let’s note that according to this definition polymers synthesis is automatically a nanochemistry part as far as according to the Melikhov’s classification polymeric macromolecules (more precisely macromolecular coils) belong to nanoparticles and polymeric sols and gels - to nanosystems. Catalysis on nanoparticles is one of the most important sections of nanochemistry.
The majority of catalytic systems are nanosystems. At heterogeneous catalysis the active substance is tried to deposit on carrier in nanoparticles form in order to increase their specific surface. At homogeneous catalysis active substance molecules have often in them selves nanometer sizes. The most favourable conditions for homogeneous catalysis are created when reagent molecules are adsorbed rapidly by nanoparticles and are desorbed slowly but have high surface mobility and, consequently, high reaction rate on the surface and at the reaction molecules of such structure are formed at which desorption rate is increased sharply. If these conditions are realized in nanosystem with larger probability than in macrosystem, then nanocatalyst has the raising activity that was observed for many systems. In the connection such questions arise as adsorption and desorption rate, surface mobility of molecules and characteristics frequency of reagents interaction acts depend on the size, molecular relief and composition of nanoparticles and the carrier.
The presence paper purpose it the application of fractal analysis for description of polymerization kinetics in nanofiller presence.
Results and Discussion
In previous part of the article the solid-state imidization reactions were represented [1]. Let’s consider the interfacial interactions problem of PI forming macromolecular coil and Na+-montmorillonite on nanofiller surface. As Pfeifer shows [2], a macromolecular coil on hard surface changes its
configuration (structure), which can be characterized by its fractal dimension Df. This change is described with the help of the following equation [2]:
i .Df
surf f Df
sol
■ = d°
rf
(1)
where dsurf and d°surf are fractal dimensions of nanofiller surface in nanocomposite and in initial state, respectively, Dfso1 and Df are fractal dimensions of PI macromolecular coil in solution (the blending of PAA and Na+-montmorillonite was carried out in N,N-dimethylacetamide solution [3]) and in solid-phase state on nanofiller surface, respectively.
Let’s consider the estimation of the parameters including in the equation (1). As it was shown in paper [4], a polymer chain, possessing by finite rigidity and consisting of statistical segments of finite length, was not capable to reproduce growing surface roughness at d°surf increase and at d°surf >2.5 the value dsurf is determined as follows [5]:
d f = 5 - d0 .. (2)
surf surf
For Na+-montmorillonite the value d°surf is determined experimentally and equal to 2.78 [6]. The value Dfso1 can be accepted in the first approximation equal to macromolecular coil dimension in a good solvent (Dfso1 =1.667 [7]). Then the estimation according to the equation (1) gives Df =1.33. It is obvious, that this dimension of the macromolecular coil, stretched on Na+ -montmorillonite surface will be designated further as Df .
The calculation of real values of macromolecular coil fractal dimension Df for the first order reaction, which is solid-state imidization [3], can be fulfilled with the help of the equation:
^i1 Q) (3)
In other words the calculation according to this equation shows that for the studied imidization reactions the condition Df°<Df is fulfilled. Such relation allows to assume that only part of PI macromolecular coils interacts with Na+-montmorillonite surface. This is confirmed by the data of Fig. 1, where the difference ADf=Df- Df° is plotted on the graph as a function of nanofiller contents Wc for four imidization temperatures. As it follows from this Figure plots, the
c
AOf value decreases at Wc growth or Of ^ Of0 and these plots extrapolation shows, that at Wc »17.5 mass. % Of = Of0 or AOf=0. Let’s note, that the indicated value Wc is true only for exfoliated (nonaggregated) nanofiller.
The AOf decrease at Wc growth assumes the interacting phase fraction фМ increase in the imidization process. The value фІП can be determined according to the mixtures law from the equation [8]:
’ Df =фий D0f +(1 -Фint )f, (4)
where Of is the macromolecular coil fractal dimension in nanofiller absence.
m
Fig. 1 - The dependences of the fractal dimension difference ADf=Df - Df0 on nanofiller contents Wc for nanocomposites polyimide/Na+-montmorillonite at imidization temperatures: 423 (1), 473 (2), 503 (3) and 523K (4) [8]
In Fig. 2 the dependence <p¡nt(Wc) for Г,=423К is adduced. As one can see, this correlation is linear, passes through coordinates origin and is described analytically by the following empirical equation [8] :
9int = 0.0575W,, (5)
where Wc is given in mass. %.
Pint
Fig. 2 - The dependence of interacting phase relative fraction 9int on nanofiller contents Wc for nanocomposites polyimide/Na+-montmorillonite at imidization temperature 423K [8]
lt is obvious, that at Wc =17.5 mass.%, obtained by plots of Fig. 1 extrapolation, the value фи =1.0, i.e. in an imidization reaction the entire
reactionary system PAA-Na+-montmorillonite is
influenced.
In Fig. 3 the dependence of reaction rate constant k1 on interacting phase relative fraction <pint, is adduced which turns out approximately linear and shows k1 growth at <p¡nt increase. This allows to assume the direct dependence of solid-phase imidization rate on interfacial interactions level in the reactionary system [8]. ' '
k\, mm"1
<Pmi
Fig. 3 - The dependence of the first order reaction rate constant kl on relative fraction of interacting phase 9¡nt for nanocomposites polyimide/Na+-montmorillonite at imidization temperature 423K [8]
In paper [3] the reduction of imidization process activation energy Eact at Wc increase was found out - from 66 up to 51 kj/mole within the range Wc =0-7 mass.%. Earlier the authors [9] offered the following dependence of Eact on Df in case of polyarylate thermooxidative degradation:
Eact = 16.6Df - 2.8Df. (6)
In table 1 the comparison of experimental Eact and calculated according to the equation (6) Eactth activation energy values of solid-state imidization is adduced. As one can see, a good correspondence between the indicated values of activation energy is obtained (the average discrepancy of Eact and Eactth makes less than 5%). This means, that association energy (imidization reaction) and dissociation one (thermooxidative degradation) are approximately equal, that was to expected.
Table 1 - The comparison of experimental Eact and calculated according to the equation (6) Eactth values of solid-state imidization process activation energy [8]
Wc, mass. % Eact, kj/mole Eactth, kj/mole
0 бб.0 б8.б
2 57.5 бЗ.4
5 54.0 52.8
7 51.5 49.З
Hence, the results obtained above have demonstrated again that the cause of imidization process acceleration at nanofiller contents growth is
macromolecular coil structure change owing to its interfacial interactions with Na+-montmorillonite surface. The interacting phase relative fraction increases at nanofiller contents raising and at its content about 17.5 mass.% this phase ocuppies the entire reactionary
(7)
system. The imidization process activation energy reduction at nanofiller contents increase is also due to structural factors, namely, to a macromolecular coil fractal dimension decrease.
The authors [3] have found out that the kinetic curves Q(t) have typical shape for polymerization reactions with autodeceleration showing imidization rate reduction as time is passing [1]. As it is known [10], such curves Q(t) are specific for the reaction course in heterogeneous medium and are described by a simple relationship [10]:
dQ ~ t h, dt
where h is heterogeneity exponent (0<h<1), turning into zero for homogeneous (Euclidean) mediums; incidentally the behavious is classical: dQ/dt = const.
The mentioned relationship supposes strong effect of this heterogeneity degree characterized by exponent h on reaction rate. Therefore the authors [9, 11] undertake an attempt of clarification of the reactionary medium heterogeneity physical significance in case of PAA solid-phase imidization and the factors defining the medium heterogeneity exponent value.
The solid-phase imidization reactions were shown earlier [1]. The obtained dependences are linear and according to their slope the value of spectral dimension ds characterized reactionary medium connectivity can be obtained. T) increase within the range 423-523K results to substantial growth of ds: from 0.42 up to 1.68. Let’s note that such ds increase occurs without reactionary mixture composition change. This means, that the energetic restrictions result to the appearance of fractal space, in which instead of the value ds an effective spectral dimension ds’ must be used, reflecting the existence of the restrictions
mentioned above and connected with ds by the equation [10]:
ds =p d, (8)
where p;- is the parameter, characterizing distribution of reagents “jumps” (displacements) times.
From the dependence h(T) [1] one can see that fast decrease h or reduction of reactionary medium heterogeneity at T raising follows. At T «540K exponent h=0, i.e. reactive medium becomes
homogeneous. Since for PI Tg«533K, then, as it was expected [12], that the reactionary medium in case of solid-phase imidization became homogeneous
(Euclidean) at glass transition. The shape of the curve h(T) [1], i.e. h goes to zero at temperature raising, assumes, that the fractal-like effects, namely, ds’ variation, are connected with energetic disorder [10]. In such case the energetic state of polymer structure can be characterized by an excess energy localization regions dimension Dfe [13]. The value Dfe can be estimated according to the following equation [14]:
4nj (9)
Def =
ln(l/. f.
where fg is a relative fluctuational free volume.
In Fig. 4 the dependence h(Dfe) is adduced, from which the expected result: follows polymer structure energetic excitation degree raising, due to
thermal energy “pumping” at T increase, results to h reduction. At Dfe «6.3 the reactionary medium becomes homogeneous (/?=()).
Fig. 4 - The dependence of heterogeneity exponent h of reactionary medium on excess energy localization regions dimension Dfe for PAA solid-phase imidization at Na+-montmorillonite contents Wc: 0 (1), 2 (2) and 5 (3) mass.% [11]
Therefore, the data of Fig. 4 and the curve h(T) [1] give the answer to the question, at what conditions h=0, i.e. when the reactionary medium becomes homogeneous. Nevertheless, the physics of this process remains vague. The glass transition gives singularities neither in fg behavior nor in Dfe behavior. Therefore for the explanation of heterogeneous^homogeneous medium transition let’s use representations of the conception of fractal (local) free volume fgfr. According to this conception free volume microvoid is necessary to simulate not by three-dimensional sphere, as it was accepted in classical polymer physics [15], but by Dfe -
fr
dimensional sphere with the volume vh . The value vh can be estimated as follows [14]:
v,, =
Tm
(10)
where percolation index v was accepted equal to 0.85 [16].
Further from geometrical considerations in the assumption of three-dimensional microvoid of free volume its radius rh can be estimated and then vhfr can be calculated according to the equation [11]:
vh =
De, /2 D‘
n rn
{/)■ /2)!
(11)
where rn is radius of free volume microvoid.
In Fig. 5 the dependence h(fgfr) is adduced where value fgfr was calculated according to the equations (9), (10) and
(12)
where relative fraction of fluctuational free volume fg can be accepted equal to 0.060 for solid-phase polymers [12].
As it follows from the data of this Figure, value h=0 or reactionary medium homogeneity at fgfr =0.34 is achieved. Let’s remind that the mentioned value fgfr corresponds to percolation threshold for overlapping
h
fr
spheres [17]. In other words, at fg =0.34 fluctuational free volume microvoids, simulated by Df - dimensional sphere, form continuous percolation network or continuous diffusion channels [18]. Therefore, between heterogeneous and homogeneous reactionary medium, at any rate, in case of solid-phase imidization, qualitative difference exists. For heterogeneous reactionary medium dehydration product (water molecule), which is in a free volume microvoid, is forced to expect the opening of overlapping it neighbouring microvoid, after that it makes “jump” from the first to the second and further the process repeates. For homogeneous reactionary medium such process of “expectation” is not required by virtue of the existence of through percolation channels of free volume. Let’s note, that the mentioned processes of “jumps” are realized on local level. The indicated effect is the cause of diffusive processes intensification in solid-phase imidization course, which was mentioned above.
h
ft
Fig. 5 - The dependence of heterogeneity exponent h of reactionary medium on relative fractal free volume fgfr for PAA solid-phase imidization. The notation is the same, that in Fig. 4 [11]
And lastly, in Fig. 6 the dependence of coefficient Py in the equation (2) on fgfr is adduced. Again the value Py reaches its limiting magnitude Py =1 (i.e. ds=ds) at fgfr =0.34. The relationship between Py and fgfr is given by the simple empirical equation [11]:
P , = 2.94 . (13)
The plot of Fig. 6 demonstrates that the energetic restriction, defining transition from ds to ds', is the necessity of “jumps” of reaction product or reagents between free volume microvoids. It is clear, that T raising decreases “jump” expectation time and the formation of through percolation channels of free volume microvoids cancels these restrictions.
Hence, the results considered above demonstrated that the notion of reactionary medium heterogeneity in case of solid-phase imidization was connected with free volume representations that were expected for diffusion-limited solid-phase reactions. If free volume microvoids are not connected with one another, then medium is heterogeneous, and in case of formation of overlapping percolation network of such microvoids it’s homogeneous. To obtain such definition
is possible only within the framework of the fractal free volume conception.
P,
f.
Fig. 6 - The dependence of the coefficient Py in the equation (8) on relative fractal free volume fgfr for PAA solid-phase imidization. The notation is the same that in Fig. 4 [11]
As it was shown in paper [3], the temperature imidization T raising within the range 423-523K and the nanofiller contents Wc increase within the range 0-7 mass.% results to essential imidization kinetics change expressed by two aspects: by an essential increase of reaction rate (reaction rate constant of the first order k1 increases almost on two orders) and by raising of conversion (imidization) limiting degree Qlim almost: from 0.25 for imidization reaction without nanofiller at T =423K up to 1.0 at Na+-montmorillonite content 7 mass. % and T =523K. Let’s also remind, that all kinetic curves Q(t) for the indicated imidization reactions have typical shape of curves with autodeceleration [1], characteristic for fractal reactions, i.e. either fractal objects reactions, or reactions in fractal spaces [19]. In other words, the indicated imidization reaction aspects in sufficient degree have general character. If for the first effect (k1 increase) the authors [3] offered probable chemical treatment considering nanofiller as a catalyst, then the second effect (Qlim raising) did not obtain any explanation, although its theoretical and practical significance is obvious. Therefore the authors [20] offered structural treatment of limiting conversion degree in solid-phase imidization process based on the general principles of fractal analysis [19].
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© G. V. Kozlov - ст. науч. сотр. каф. высокомолекулярных соединений, Кабардино-Балкарский госуд. ун-тет G. E. Zaikov -д.х.н., проф., Институт биохимической физики им. Н.М. Эмануэля РАН, проф. каф.ТПМ КНИТУ, ov_stoyanov@mail.ru; E. Klodzinska - проф., Institute for Engineering of Polymer Materials and Dyes, Torun, Poland; Richert Jozef - проф. Institute for Engineering of Polymer Materials and Dyes, Torun, Poland.
