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ХИМИЯ, ТЕХНОЛОГИЯ И ИСПОЛЬЗОВАНИЕ ПОЛИМЕРОВ
УДК 678.01:53
Yu. G. Medvedevskikh, O. Yu. Khavunko, L. I. Bazylyak, G. E. Zaikov
VISCOELASTIC PROPERTIES OF THE POLYSTYRENE IN CONCENTRATED SOLUTIONS AND MELTS (PART 2)
Keywords: effective viscosity, frictional and elastic components of the viscosity, m-ball, segmental motion, activation energy.
A gradient dependence of the effective viscosity n for the concentrated solutions of the polystyrene in toluene at three concentrations p = 0,4-105; 0,5-105; 0,7-105 g/m3 correspondingly for the fourth fractions of the polystyrene with the average molar weights М = 5,1104; 4,1-104; 3,3-104; 2,2-104 g/mole respectively has been experimentally investigated. For every pair of the values p and М a gradient dependence of the viscosity was studied at four temperatures: 25, 30, 35 and 40°С. An effective viscosity of the melts ofpolystyrene was studied for the same fractions, but at the temperatures 190, 200 and 210°С. The investigations have been carried out with the use of the rotary viscosimeter «Rheotest 2.1» under the different angular velocities m of the working cylinder rotation. An analysis of the dependencies n(m) permitted to mark the frictional nf and elastic ne components of the viscosity ant to study their dependence on temperature Т, concentration p and on the length of a chain N. It was determined, that the relative movement of the intertwined between themselves polymeric chains into m-ball, which includes into itself the all possible effects of the gearings, makes the main endowment into the frictional component of the viscosity. The elastic component of the viscosity ne is determined by the elastic properties of the conformational volume of the m-ball ofpolymeric chains under its shear strain. The numerical values of the characteristic time and the activation energy of the segmental movement were obtained on the basis of the experimental data. In a case of a melt the value of E and AS*/R are approximately in two times more than the same values for the diluted and concentrated solutions of the polystyrene in toluene; this means that the dynamic properties of the polymeric chains in melt are considerably near to their values in polymeric matrix than in solutions. Carried out analysis and generalization of the obtained experimental data show that as same as for low -molecular liquids the studying of the viscosity of polymeric solutions permits sufficient adequate to estimate the characteristic time of the segmental movement accordingly to which the coefficients of polymeric chains diffusion can be calculated in solutions and melt, in other words, to determine their dynamic characteristics.
Ключевые слова: эффективная вязкость, фрикционный и упругий компоненты вязкости, клубок, сегментальная
подвижность, энергия активации.
Экспериментально исследована зависимость градиента эффективной вязкости n для концентрированных растворов полистирола в толуоле при трех концентрациях p = 0,4105; 0,5105; 0,7105 г/м3 соответственно для четырех фракций полистирола со средними молярными массами М = 5,1104; 4,1104; 3,3104; 2,2-104 г/моль соответственно. Для каждой пары значений p и М зависимость градиента вязкости изучалась при четырех температурах: 25, 30, 35 и 40°С. Эффективная вязкость расплавов полистирола была изучена для тех же фракций, но при температурах 190, 200 и 210°С. Исследования были проведены с использованием ротационного вискозиметра «Rheotest 2,1» при разных угловых скоростях m вращения рабочего цилиндра. Анализ зависимостей n (m) позволяет выделить фрикционный п/ и упругий компонент ne вязкости и изучить их зависимость от температуры Т, концентрации p и длины цепи N. Было установлено, что относительное движение переплетенных между собой полимерных цепей в клубке дает основной вклад в фрикционную составляющую вязкости. Упругий компонент вязкости ne определяется упругими свойствами конформационного объема клубка полимерных цепей при деформации сдвига. На основе экспериментальных данных были получены численные значения характеристического времени и энергии активации сегментального движения. В случае расплава значение Е и AS*/R примерно в два раза больше тех же значений для разбавленных и концентрированных растворов полистирола в толуоле; это означает, что динамические свойства полимерных цепей в расплаве значительно ближе значениям этих свойств в полимерной матрице, чем в растворах. Проведенный анализ и обобщение полученных экспериментальных данных показывает, что так же, как и для низкомолекулярных жидкостей, изучение вязкости полимерных растворов позволяет достаточно адекватно оценить характеристическое время сегментальной подвижности, соответственно которому коэффициенты диффузии полимерных цепей могут быть рассчитаны в растворах и расплаве, другими словами, чтобы определить их динамические характеристики.
Introduction
The viscosity n of polymeric solutions is an object of the numerous experimental and theoretical investigations generalized in ref. [1-4]. This is explained both by the practical importance of the presented property of polymeric solutions in a number of the technological processes and by the variety of the factors having an influence on the n value, also by a
wide diapason (from 10-3 to 102 Pa-s) of the viscosity change under transition from the diluted solutions and melts to the concentrated ones. The all above said gives a great informational groundwork for the testing of different theoretical imaginations about the equilibrium and dynamic properties of the polymeric chains.
It can be marked three main peculiarities for the characteristic of the concentrated polymeric solutions viscosity, namely:
1. Measurable effective viscosity n f°r the concentrated solutions is considerable stronger than the n for the diluted solutions and depends on the velocity gradient g of the hydrodynamic flow or on the shear rate.
It can be distinguished [4] the initial no and the final n« viscosities (n0>n«), to which the extreme conditions g ^ 0 and g ^ « correspond respectively.
Due to dependence of n on g and also due to the absence of its theoretical description, the main attention of the researches [4] is paid into, so-called, the most newton (initial) viscosity n0, which is formally determined as the limited value at g^-0. Exactly this value n0 is estimated as a function of molar mass, temperature, concentration (in solutions).
The necessity of the experimentally found values of effective viscosity extrapolation to «zero» shear stress doesn't permit to obtain the reliable value of n0. This leads to the essential and far as always easy explained contradictions of the experimental results under the critical comparison of data by different authors.
2. Strong power dependence of n on the length N of a polymeric chain and on the concentration p (g/m3) of a polymer in solution exists: j ~ paNp with the indexes a
= 5 - 7, p = 3,3 - 3,5, as it was shown by authors [4].
3. It was experimentally determined by authors [1, 5] that the viscosity n and the characteristic relaxation time t* of the polymeric chains into concentrated solutions and melts are characterized by the same scaling dependence on the length of a chain:
n ~ t * ~ Np (1)
with the index p = 3,4.
Among the numerous theoretical approaches to the analysis of the polymeric solutions viscosity anomaly, i. e. the dependence of n on g, it can be marked the three main approaches. The first one connects the anomaly of the viscosity with the influence of the shear strain on the potential energy of the molecular kinetic units transition from the one equilibrium state into another one and gives the analysis of this transition from the point of view of the absolute reactions rates theory [6]. However, such approach hasn't take into account the specificity of the polymeric chains; that is why, it wasn't win recognized in the viscosity theory of the polymeric solutions. In accordance with the second approach the polymeric solutions viscosity anomaly is explained by the effect of the hydrodynamic interaction between the links of the polymeric chain; such links represent by themselves the «beads» into the «necklace» model. Accordingly to this effect the hydrodynamic flow around the presented "bead" essentially depends on the position of the other «beads» into the polymeric ball. An anomaly of the viscosity was conditioned by the anisotropy of the hydrodynamic interaction which creates the orientational effect [7, 8]. High values of the viscosity for the concentrated solutions and its strong gradient dependence cannot be explained only by the effect of the hydrodynamic interaction.
That is why the approaches integrated into the conception of the structural theory of the viscosity were generally recognized. In accordance with this theory the viscosity of the concentrated polymeric solutions is determined by the quasi-net of the linkages of twisted between themselves polymeric chains and, therefore, depends on the modulus of elasticity E of the quasi-net
and on the characteristic relaxation time t* [1-2]:
*
n = E • t (2)
It is supposed, that the E is directly proportional to the density of the linkages assemblies and is inversely proportional to the interval between them along the same chain. An anomaly of the viscosity is explained by the linkages assemblies' density decreasing at their destruction under the action of shear strain [9], or by the change of the relaxation spectrum
[10], or by the distortion of the polymer chain links distribution function relatively to its center of gravity
[11]. A gradient dependence of the viscosity is described by the expression [11]:
(n - n«)l{lo - nj = f (gt*) (3)
It was greatly recognized the universal scaling ratio [1, 5]:
n = no • f (gt') (4)
in which the dimensionless function f (gt* )= f(x) has
the asymptotesf(0) = 1,f(x)x»i = x-Y, y= 0,8.
Hence, both expressions (3) and (4) declare the gradient dependence of n by the function of the one non-dimensional parameter gt*. However, under the theoretical estimation of n and t* as a function of N there are contradictions between the experimentally determined ratio (1) and p = 3,4. Thus, the analysis of the entrainment of the surrounding chains under the movement of some separated chain by [12] leads to the dependencies j ~N3,5 but t* ~ N4 5. At the analysis [13] of the self-coordinated movement of a chain enclosing into the tube formed by the neighbouring chains it was obtained the n~N3, t* ~ N4. The approach in [14] which is based on the conception of the reptational mechanism of the polymeric chain movement gives the following dependence j ~ t*N3.
So, the index p = 3,4 in the ratio (1) from the point of view of authors [2] remains by one among the main unsolved tasks of the polymers' physics.
Summarizing the above presented short review, let us note, that the conception about the viscosity-elastic properties of the polymeric solutions accordingly to the Maxwell's equation should be signified the presence of two components of the effective viscosity, namely: the frictional one, caused by the friction forces only, and the elastic one, caused by the shear strain of the conformational volume of macromolecules. But in any among listed above theoretical approaches the shear strain of the conformational volumes of macromolecules was not taken into account. The sustained opinion by authors [3-4] that the shear strain is visualized only in
the strong hydrodynamic flows whereas it can be neglected at little g, facilitates to this fact. But in this case the inverse effect should be observed, namely an increase of n at the g enlargement.
These contradictions can be overpassed, if to take into account [15, 16], that, although at the velocity gradient of hydrodynamic flow increasing the external action leading to the shear strain of the conformational volume of polymeric chain is increased, but at the same time, the characteristic time of the external action on the rotating polymeric ball is decreased; in accordance with the kinetic reasons this leads to the decreasing but not to the increasing of the shear strain degree. Such analysis done by authors [15-17] permitted to mark the frictional and the elastic components of the viscosity and to show that exactly the elastic component of the viscosity is the gradiently dependent value. The elastic properties of the conformational volume of polymeric chains, in particular shear modulus, were described early by authors [18-19] based on the self-avoiding walks statistics (SAWS).
Here presented the experimental data concerning to the viscosity of the concentrated solutions of styrene in toluene and also of the melt and it is given their interpretation on the basis of works [15-19].
Experimental data and starting positions
In order to obtain statistically significant experimental data we have studied the gradient dependence of the viscosity for the concentrated solution of polystyrene in toluene at concentrations 0,4-105; 0,5-105 and 0,7-105 g/m3 for the four fractions of polystyrene characterizing by the apparent molar weights M = 5,1-104; M = 4,1104; M = 3,3104 and M = 2,2-104 g/mole. For each pair of values p and M the gradient dependence of the viscosity has been studied at fourth temperatures 25 °C, 30 °C, 35 °C and 40 °C.
The viscosity for the polystyrene melt were investigated using the same fractions at 210 0C. Temperature dependence of the polystyrene melt was investigated for the fraction with average molecular weight M = 2,2-104 g/mole under three temperatures, namely 190, 200 and 210 0C.
The experiments have been carried out with the use of the rotary viscometer RHEOTEST 2.1 equipped by the working cylinder having two rotary surfaces by diameters d1 = 3,4-10-2 and d2 = 3,9-10-2 m in a case of the concentrated solutions of polystyrene investigation and using the device by «cone-plate» type equipped with the working cone by 0,30 angle and radius r = 1,8-10-2 m in a case of the polystyrene melt investigation.
Polystyrene's melt
Experimental data
Typical dependencies of the melt viscosity n on the angular rate w (rotations per second) of the working cone rotation are represented on Figures 1 and 2.
In order to analyze the experimental curves of n(w), the equation (5) [20] with the same remarks as to
the numerical estimations of parameters ne, nf and b was used.
As it can be seen from the Figures 1 and 2, calculated curves of n(w) accordingly to the equation (5) [20] with the founded parameters nf, ne and b describe the experimental data very well.
Results of the numerical estimations of nf, ne Ta b on a length of the polymeric chain at 210 0C are represented in Table 1 and the temperature dependencies are represented in Table 2. Review of these data shows, that the all three parameters are the functions on the length of a chain and on temperature. But at this, ne and nf are increased at N increasing and are decreased at T increasing, whereas b parameter is changed into the opposite way.
Fig. 1 - Experimental (points) and calculated in accordance with the equation (5) [20] (curves) dependencies of the effective viscosity on the velocity of the working cone rotation. T = 210 0C
in
Fig. 2 - Experimental (points) and calculated accordance with the equation (5) [20] (curves) dependencies of the effective viscosity on the velocity of the working cone rotation. M = 2,2*104 g/mole
Table 1 - Optimization parameters n and b obtained from the experimental data at T =2100C
M-10-4, g/mole nf, Pa-s ne, Pa-s b, s-1
5.1 18.49 7.09 0.0019
4.1 10.58 3.19 0.0025
3.3 6.50 2.65 0.0096
2.2 3.69 0.55 0.0169
Table 2 - Optimization parameters nf, qe and b obtained from the experimental data for polystyrene with M = 2,2104 g/mole
T, 0C nf, Pa-s ne, Pa-s b, s-1
190 10.60 2.50 0.0045
200 7.76 1.40 0.007
210 3.69 0.55 0.0169
Frictional component of the effective viscosity of melt
Results represented in Table 1 and Table 2 show that the frictional component of the viscosity nf very strongly depends on the length of the polymeric chains and on the temperature. The whole spectrum of the dependence of nf on N and T will be considered as the superposition of the above earlier listed three forms of the motion which make the endowment into the frictional component of the viscosity of melt, namely the frictional coefficients of viscosity nsm, npm and npz [20].
Not the all listed forms of the motion make the essential endowment into nf, however for the generalization let us start from the taking into account of the all forms. So, the frictional component of the viscosity should be described by the expression (19) [20], but for the melt it is necessary to accept that ns = 0, and q> = 1.
nf =Vsm + npm +Vpz . (5)
In a case of the melts npz is determined by the equations (30) [20], but at p = p/pm = 1
\ 2.5
= nlN2
f p
vPc y
(6)
fz
Here the coefficient of the proportionality tf p
contains the ratio Lm / Lf , however, since the melt is
the ideal solution for polymer, that is why it can be assumed that Lm = 1.
By substituting of the (6) into (40) [20], we will
obtain
\2.5
nf =nsm + n pm + n°pzN :
P
Vpo y
(7)
Let's estimate an endowment of the separate components into nf. The results represented in Table 1 show, that under conditions of our experiments the
frictional component of viscosity is changed from the minimal value ~ 3,7 Pa s till the maximal one equal to ~ 18,5 Pa s. The value of the viscosity coefficient nsm, which represents the segmental motion of the polymeric chains and estimated [21] on the basis of nf for diluted solution of polystyrene in toluene is equal approximately 5 • 10-3 Pa s. Thus, it can be assumed that nsm << nf and to neglect by respective component in (7). Taking into account of this fact, the expression (7) let's rewrite as follow:
\2-5
nf = n pm +npzN
P Po
(8)
For the melts pjp0 = const, that is why the
interpretation of the experimental values of nf as the function of N2 is represented on Figure 3. It can be seen from the Figure 3, that the linear dependence corresponding to the equation (8) is observed, and the
numerical value of no p| was found upon the
U y
inclination tangent of the straight line; under the other temperatures this coefficient was found using the experimental data from Table 2. For the estimation of tfpz it was assumed that p = 1.05 • 106 g/m3 p0 was
calculated in accordance with (11) [20] at M0 = 104,15 g/mole, a = 1,86-10~l° m.
20 n
15-
0
03
c 10-
£
5-
0-
60000 120000 180000 240000 N2
Fig. 3 - Interpretation of the experimental values of nf in the coordinates of the equation (8) at T = 210 0C
Temperature dependence of rfpz into coordinates
of the Arrhenius equation is represented on Figure 4.
0,0-,
-0,30 m
pr -0,6-c
-0,9-1,2-1,5
0,00207 0,00210 0,00213 0,00216 1/T, K1
Fig. 4 - Temperature dependence of the numerical
estimations of n0 in the coordinates of the
pz
Arrhenius equation
2.5
So, an activation energy Epz, of the relative motion of intertwined polymeric chains into polystyrene's melt consists of 106 ± 35 kJ/mole.
It can be seen from the presented Figure 3 and
from the regression equation, that the values of ri°pm are
very little and are within the ranges of their estimation error; this cannot give the possibility to estimate the value of gm/g.
So, as the analysis of the experimental data showed, the main endowment into the frictional component of the effective viscosity of the polystyrene's melt has the relative motion of the intertwining between themselves into m-ball polymeric chains. Exactly this determines the dependence of % on the length of a chain (nf ~N2).
Elastic component of the effective viscosity of
melt
It can be seen from the Table 1 and Table 2, that the elastic component of the viscosity n is strongly growing function on a length of a chain N and declining function on temperature T.
The elastic component of the viscosity is described by the equation (6), but at Lm = 1.
n = < (9)
Correspondingly, instead of the (34) [20] we obtained
RT r34 n=xrN P
M 0
V3'5
P0
(10)
Using the expression (10) and the experimental values of qe (see Table 1 and Table 2) it was found the numerical values of the characteristic time of the segmental motion Tm. The results of calculation (Tm )n
are represented in Table 3 and Table 4. Despite the disagreement in numerical estimations, it is observed their dependence on T, but not on the N; this fact is confirmed by the expression (10).
Table 3 - Characteristic times of the segmental motion calculated based on the experimental values of n and b (M = 2.2104 g/mole)
T, 0C (Tm)ne-1011, s (Tm)b-1011, s Tm-1011, s
190 6.86 5.50 6.18
200 3.76 3.58 3.67
210 1.45 1.48 1.47
Table 4 - Characteristic times of the segmental motion calculated based on the experimental values of ne Ta b (T = 210 0C)
M-10-4, g/mole (Tm)ne-1011, s (Tm)b-1011, s Tm-1011, s
5.1 1.11 1.70 1.48
4.1 1.05 2.25
3.3 1.84 0.99
2.2 1.45 1.48
Parameter b
In accordance with the determination (6) [20], the b parameter is a measure of the velocity gradient of hydrodynamic flow created by the working cone
rotation, influence on characteristic time t*v of g
action on the shear strain of the m-ball and its rotation
*
movement. Own characteristic time tm of m-ball shear
and rotation accordingly to (16) [20] depends only on N
and T via Tm .
It is follows from the experimental data (see Table 1 and Table 2) that the b parameter is a function of N and T, but at this it is increased at T increasing and is decreased at N growth. In order to describe of these dependencies let's previously determine the angular rate
G)0m (s-1) of the rotation of m-ball with the effective
radius Rm, which contacts with the surface of the working cone with radius r
Cm = nrc/ Rm (11)
mm
Here n is appeared as a result of the different units of the dimention cC,(s-1) and c,(rot/s) .
Let's determine the t„ as the reverse one to the
cm
tl = Rm /nrc
(12)
In equation (12) tv is a time during which the
m-ball with the conformational radius Rm under the action of working cone rotation with radius r will be
rotated on the angle equal to the one radian. Let us note, *
that the tm was determined by authors [17] also in
calculation of the m-ball turning on the same single angle.
Thereby, t° is inversely proportional to m, so via
the constant of the device is inversely proportional to g:
t° ~ g 1. However, into m-ball as a result of the
difference in the linear rates of the links of polymeric chains under their rotation the hydrodynamic interaction is appeared, which leads to the appearance of the additional to the g local averaged upon m-ball gradient velocity of the hydrodynamic flow gm. This local gradient gm acts not on the conformational volume of m-ball, but on the monomeric frame of the polymeric chains (the inflexible Kukris wire model [23]). That is
why the endowment of g m into characteristic time tl depends on the volumetric part p of the links into the conformational volume of m-ball, i. e. t* ~ (g + gmp)-1. Into the melt q> = 1, therefore, it can be written the following:
tHtl = g/(g + gm ), (13)
that leads with taking into account of the (12), to the expression
R
<=-
nrc
1 + gm g
(14)
By combining of the (16) [20] and (14) in (6) [20], we obtained
b =
7a
4nrr
N2
V2
Vp0 y
1 +
gm
(15)
As we can see, using the experimental values of b parameter (see Table 1 and Table 2) it can be calculated (rm )b . After the substitution of the values a = 1.86-10-10 m, r = 1,8-10-2 m, we obtained
(rm)b = 3.78 • 10-7 N2
f „\2(
P0 y
1 + Mm.
b. (16)
was
Numerical value of the ratio gm/g
considered as a parameter, which selected in such a way, that calculated accordingly to (16) values of (rm )b corresponded to the calculated values (rm )n
accordingly to (10). So, the obtained value of gm/g = 39.
The results of calculations of (m )b and (rm )n , are compared in Tables 3 and 4. As the results show, (Tm )b and (rm )n is visible function on the
temperature but not on the N. That is why based on the data of Tables 3 and 4 it were calculated the averaged values of the Tm of the characteristic time of segmental motion of the macromolecule.
Temperature dependence of Tm into coordinates of the Arrhenius equation is represented on Figure 5.
-23,4 -23,7 H -24,0 - -24,3 -24,6 -24,9
0,00207 0,00210 0,00213 0,00216 1/T, K1
Fig. 5 - Temperature dependence of the averaged values of characteristic time t of the segmental motion of polymeric chain into coordinates of the Arrhenius equation
Characteristic time of the segmental motion of polystyrene in solutions and melt
The presentation of values Tm into Arrhenius'
coordinates equation (Fig. 6) [20] and (Fig. 5) permitted to obtain the expressions for concentrated solutions and melt, respectively:
ln~m =-42.23 + 5950/T, (17)
ln Tm =-58.05 +16020/T, (18)
For diluted solution of polystyrene in toluene it was early obtained [21]:
ln~ =-44.07 + 6660/ T,
(19)
On a basis of the equations (17) - (19) it was calculated the activation energies of the segmental motion of polystyrene in diluted, concentrated solutions and melt, which consists of 55,4; 49,4 and 133.0 kJ/mole, respectively.
Characteristic time t can be obtained by equation of the theory of absolute reactions rates [16]:
2h j AS
t = — exp <--
kT I R
expjAf" ^ = T>exp{E/RT}, (20)
where AH * = E is an activation energy of the
segmental motion; AS is an activation entropy of the segmental motion.
By comparing the expression (20) and experimental data (17) - (19), it was found for solutions at t = 30 °C and for melt at t = 200 °C the values of the activation entropy AS7R = 15,3; 13,5 and 28,8 respectively.
As we can see, the difference between energies and entropies of activation in diluted and concentrated solutions is little and is in a range of the error of their estimation. At the same time, indicated parameters into melt of the polystyrene is approximately in two times higher. Besides, the growth of the activation entropy does not compensate the activation energy growth; as a result, the characteristic time of the segmental motion into melt is on 2-3 orders higher, than into the solutions (at the extrapolation of t on general temperature).
Let's compare the values of the activation energies E with the evaporation heats AHevap of styrene (-43,94 kJ/mole) and toluene (-37,99 kJ/mole). So, independently on fact, which values of AHevap were taken for styrene or toluene, it is observed a general picture: ETm, Ef > AHeun. It is known [24], that for the low-molecular liquids, viscosity of which is determined by the Brownian or translational form of the molecules motion, the activation energy of the viscous flow is in 3-4 times less than the evaporation heats. This points on fact, that the segmental motion which is base of the reptation mechanism of the polymeric chains motion, is determined by their deformation-vibrational freedom degrees.
However, let's mark another circumstance. During the study of the bimolecular chains termination kinetics [25] which is limited by their diffusion, in polymeric matrixes of the dimethacrylate TGM-3 (triethylenglycole dimethacrylate), monomethacrylate GMA (2,3--epoxypropylmethacrylate) and their equimolar mixture TGM-3 : GMA = 1 : 1 in the temperature range 20 ^ 70 °C it were obtained the following values of the activation energies: 122,2, 142,3 and 131,0 kJ/mole. Since the diffusion coefficient of the macroradical is also determined by the characteristic time of the segmental motion, it can be stated that the presented above activation energies of the segmental motion in melt and polymeric matrix are good agreed between themselves. A sharp their difference from the activation energy into solutions points on: firstly, a great influence of the solvent as a factor activating the segmental motion of polymeric chain, and secondly, on fact, that the dynamic properties of the polymeric chains
g
g
in melt are very near to their dynamic properties in polymeric matrix.
Dynamic properties of polystyrene in solutions and melt
Dynamical properties of the polymeric chains are
determined by characteristic times of their translational
* *
motion (tt ) and rotation (tr) motions. As it was noted
earlier, the characteristic time of the shear strain is also
*
equal to tr. Since the monomeric links connected into a chain, the all of these types of motion are realized exceptionally in accordance with the reptation mechanism, that is via the segmental motion with the characteristic time ts. That is why, let's analyze and generalize once more the obtained experimental data of the characteristic times of the segmental motion of the chains of polystyrene in solutions and melts, which were estimated based on elastic component of the viscosity 7]e and parameter b. Besides, let's add to this
analysis the characteristic times of the segmental motion, estimated based on coefficient of the frictional component of viscosity of diluted solution (nsm),
concentrated solution and melt (n°pz).
The values Ts will be used in the sequel for the estimation of the characteristic time of the translation motion t* and of the coefficient of the diffusion D of
the polystyrene chains into solutions and melt. Accordingly to the experimental data the temperature dependence Ts, estimated based on the elastic
component of the viscosity rje and parameter b, is
described by the equations:
in diluted solution (temperature range 20 - 35 °C)
lnTs = -44.07 + 6660/T , (21)
in concentrated solution (temperature range 25 - 40 °C) lnTs = -42.23 + 5950/T , (22)
in melt (temperature range 190 - 210 °C)
ln Ts = -58.05 +16020/ T (23)
Let's write also the temperature dependencies of the coefficients of a frictional component of the viscosity: in diluted solution
ln=-29.04 + 7300/T, (24) in concentrated solution
lnn;z = -16.67 + 4800/T, (25)
in melt
ln nfpz = -27.65 +12740/T . (26)
Next, let's use the proposed earlier expression for characteristic time of the segmental motion in the following form
lnT = ln2h+ A, (27) s kT R RT
where ln2 A = -28,78 and -29,22 at T = 303 K and T =
kT
473 K correspondingly.
Using of these values and comparing (27) and (21) - (23), we will obtain the numerical estimations for the activation entropy of the segmental motion ASJR,
which represented in Table 5.
In Table 5 also the activation energies Es of the
segmental motion and the value Ts at T = 303 K and T
= 473 K are represented too. Values Ts at T = 303 K in
melt were obtained by the extrapolation of expression (20) on given temperature, at which melt is in the solid glass-like state.
Table 5 - Characteristic parameters of segmental motion of polystyrene in solutions and melt
System Es, kJ/mo le R E kJ/ mole AS ~R T = 303 K T = 473 K
Ts , s Tpz , s Ts , s Tpz , s
Diluted solutions 55,3 15,3 — — 2,5x 10-10 — — —
Concentr ated solutions 49,4 13,5 39,9 6,0 1,5x 10-10 6,0x 10-9 — 1,9x 10-11
Melt 133,1 28,8 105,9 17,0 5,6x 10-3* 1,5x 10-2* 3,1x 10-11 4,0x 10-9
Note. *Data found by the extrapolation accordingly to the equations (23) and (34) in the field of the glass-like state of melt
It can be seen from the Table 5, that the numerical values both of Ts, and the thermodynamic
characteristics (ASjR and Es) of the segmental
motion into diluted and concentrated solutions are differed only within the limits of the experimental error of their estimations. In melt these values are essentially differed. At this, the growth of the activation energy (approximately from 55 kJ/mole till 133 kJ/mole) of the segmental motion is not compensated by the growth of the activation entropy (till ASs/R « 29 ); as a result,
the values of Ts in melt are on two orders greater than
in solutions (at T = 473 K) and on six orders greater at T = 303 K.
Let's assume, that the coefficients of the frictional component of the viscosity of polymeric chains are described by as same general expression [16], as the coefficients of the viscosity of low-molecular solution. At that time it can be written:
_ _ RT
f sm v sm
o - RT n = 3-t ,
' p y pz
(28) (29)
where V is the partial-molar volume of the monomeric link of a chain.
Tsm and Tpz are per sense, the characteristic
times of the segmental motion of free polymeric chain into diluted solution and overlapping one with others polymeric chains into the concentrated solution and melt taking into account the all possible gearing effects, correspondingly.
Since the partial-molar volume V of the monomeric link of the polystyrene is unknown, then for the next calculations it can be assumed without a great error to be equal to the molar volume of the monomeric link into the melt:
V = p/M0, (30)
where p = 1.05 -106 g/m3 is a density of the polystyrene melt; M0 = 104.15 g/mole is the molar mass of styrene. Let us write (28) and (29) in general form:
lnn = ln3RTP + ln t . Mn
(31)
At this ln3RTP = 18,15 and 18,59 at T = 303
M 0
K and T = 473 K, correspondingly. Taking into account of this value and comparing the (31) and (24) and (25) and (26), we will obtain the temperature dependences
Tsm and Tpz :
for diluted solution
lnt^ =-47.15 + 7300/T,
for concentrated solution
for melt
ln tz =-34.82 + 4800/T , ln tpz =-46.24 +12740/T.
(32)
(33)
(34)
On the basis of two last ones expressions the Tpz
have been calculated at T = 303 K and T = 473 K. Taking into account a general equation (27) it has been found also the value of the activation entropy ASpzj R
(see Table 5).
Comparing the parameters of the equation (21) for Ts and (32) for Tsm, it can be seen, that the difference between them is adequately kept within the error limits of their estimation. The values of s and
Tsm at T = 303 K, equal to 2.5-10-10 s and 1.0-10-10 s correspondingly prove of this fact. Thus, it can be assumed that Ts = Tsm , and that is why the coefficient
of the frictional component of the viscosity 7]sm of the
polymeric chains can be described by as same general expression (28) as for the coefficient of the low-molecular solution. The values of pz calculated
pZ
accordingly to the expressions (34) and (35) for concentrated solution at T = 303 K and melt at T = 473 K correspondingly (see Table 5), are essentially differed from Ts: T pz > Ts, approximately on two orders. An analysis of the parameters of the equations (22), (23) and (33), (34) showed that the difference between Ts
and pz is caused by two factors, which abhorrent the
one of the other: by insignificant decreasing of the activation energy (Epz < Es), that should be decreased
the pz , and by a sharp decreasing of the activation
entropy (AS z < ASs), that increases of T .
As it was said, the coefficient of the frictional component of viscosity rjpz in concentrated solutions
and melt caused by the motion of the overlapping between themselves polymeric chains relatively the one of the other and characterizes the efficiency of the all possible gearings. However, the mechanism of this motion is also reptational that is realized via the segmental motion. Correspondingly, between the times Ts and T pz the some relationship should be existing.
Let's assume the thermodynamical approach for the determination of this relationship as a one among the all possible.
Let's determine the notion «gearing» as the thermodynamical state of a monomeric link of the chain, at which its segmental motion is frozen. This means, that under the relative motion of the intertwining between themselves polymeric chains the reptational mechanism of the transfer at the expense of the segmental motion takes place, but under condition that the part of the monomeric links of a chain is frozen.
Let the AGz° is a standard free energy of the monomeric link transfer from a free state into the frozen one. Then the probability of the frozen states formation or their part should be proportional to the value exp{-AG°JRT}. That is why, if the ks is a constant
rate of the free segmental transfer, and kpz is the rate
constant of the frozen segmental transfer, then between themselves the relationship should be existing:
kpZ = ks exp{- AGz°/RT}. (35) Then kpz, additionally to ks, has a free
activation energy equal to the standard free defrosting energy of the frozen state.
Since ks = T-1, kpz = Tpl, we obtained
Tpz =T- exp{A G;/RT}. (36)
By assigning
AGz =AH°z - TAS: (37)
and taking into account the experimentally determined ratios t > t , E < E and AS < AS , we conclude,
pz S y pz s pz s y y
that in (37) AGz° > 0, AHz° < 0 and ASZ° < 0, and
besides the entropy factor TAS°Z should be more upon
the absolute value than the enthalpy factor AHz°. These ratios per the physical sense are sufficiently probable. A contact of the links under the gearing can activates a weak exothermal effect (AH°Z < 0) at the expense of the
intermolecular forces of interaction, and the frosting of the segmental movement activates a sharp decrease of
the entropy of monomeric link AS°z < 0, but at this |TAS;| >|ahz| . Let's rewrite the (36) with taking into account of (37) in a form
lnr = ln t +-z---1
pz s RT R
Comparing the expressions (22) and (33), (23) and (35) and taking into account (38) we obtained: for concentrated solution
AGz° = 9.0 kJ/mole, AH°Z = -9.6 kJ/mole,
AS;/R = -7.4, for melt
AG" = 15.0 kJ/mole, AH" =-31.4 kJ/mole,
;;
as;/r = -11.8.
In connection with carried out analysis the next question is appeared: why in the concentrated solutions and melt the gearing effect hasn't an influence on the
elastic component of viscosity rj°e, and determined
based on this value characteristic time of the segmental
motion is Ts; at the same time, the gearings effect
strongly influences on the frictional component of
viscosity, on the basis of which the Tp; is estimated.
Probably, the answer on this question consists in fact that the elastic component of the viscosity is determined by the characteristic time of the shear which is equal to the characteristic time of rotation. Accordingly to the superposition principle the rotation motion of the m-ball of the intertwining between themselves polymeric chains can be considered independently on their mutual relocation, that is as the rotation with the frozen conformation. As a result, the gearings effects have not an influence on the characteristic time of the rotation motion. Free segmental motion gives a contribution in a frictional component of viscosity, but it is very little and is visible only in the diluted solutions. That is why even a little gearing effect is determining for the frictional component of viscosity in concentrated solutions and melts.
Let's use the obtained numerical values of the characteristic times of the segmental movement Ts for
the estimation of dynamical properties of the polystyrene chains that is their characteristic time of the
translational movement t* and coefficient of diffusion
D into solutions and melt. Accordingly to [26], the
*
values tt and D are determined by the expressions: in diluted solutions
t* = N 8/V
D =-N -
2t
in concentrated solutions and melt
t;=N 3
Po
D = —
N 2
(39)
(40)
(41)
(42)
2tJ Ia>,
In order to illustrate the dynamic properties of the polystyrene in solutions and melt in Table 6 are
given the numerical estimations of the characteristic
*
times of segmental Ts and translational tt motions of the polystyrene and diffusion coefficients D. It was
assumed for the calculations a = 1.86 •10-10m, N = 103 and p = 0.5-106g/m3 for concentrated solution and melt correspondingly. As we can see, the characteristic time of the translational motion t* of the polystyrene
chains is on 4 and 6 orders higher than the characteristic
time of their segmental motion; this is explained by a
*
strong dependence of tt on the length of a chain. The coefficients of diffusion weakly depend on the length of a chain, that is why their values into solutions is on 2-3 order less, than the coefficients of diffusion of low-molecular substances, which are characterized by the order 10-9 m2/s.
A special attention should be paid into a value of the diffusion coefficient at T = 303 K in a field of the glass-like state of melt D = 7 -10-22 m2/s. Let's compare of this value D with the diffusion coefficients of the macroradicals in polymeric matrixes TGM-3, TGM-3-GMA and GMA which estimated experimentally [25] based on the kinetics of macroradicals decay, which under the given temperature consist of 10-21 +10-22 m2/s.
Table 6 - Dynamic characteristics of polystyrene in solutions and melt
T = 303 K T = 473 K
System
; D, m2/sc ; D, m2/s
T , s tt, s T , s tt, s
Diluted 2,0x 1,3x 1,4x
solutions io-10 10-6 10-12
Concentrated 2,0x 2,9x 1,0x
solutions io-10 10-4 10-13
Melt 5,0x 10-3* 7,2x 10-3* 7,3x 10-22* 3,0x 10-11 4,3x 10-5 1,2x 10-13
Note. *Data found by the extrapolation in a field of the glass-like state of melt.
Thus, carried out analysis shows, that the studies of the viscosity of polymeric solutions permits sufficiently accurately to estimate the characteristic times of the segmental and translational movements, on the basis of which the coefficients of diffusion of polymeric chains into solutions can be calculated.
Conclusions
Investigations of a gradient dependence of the effective viscosity of polystyrene melt permitted to mark its factional nf and elastic ne components and to study of their dependence on a length of a polymeric chain N, on concentration of polymer p in solution and on temperature T. It was determined that the main endowment into the frictional component of the viscosity has the relative motion of the intertwined between themselves in m-ball polymeric chains. An efficiency of the all possible gearings is determined by the ratio of the characteristic times of the rotation motion of intertwined between themselves polymeric
chains in m-ball fm and Flory ball tf . This lead to the
dependence of the factional component of viscosity in a
2
3/5
t
S
2.5
form nf ~ N2 for melt, which is agreed with the
experimental data.
It was experimentally confirmed the determined earlier theoretical dependence of the elastic component of viscosity for the melt r ~ N3 4, that is lead to the
well-known ratio r ~ t* ~ N3 4, which is true,
I e m y y
however, only for the elastic component of the viscosity. On a basis of the experimental data of ne and b it were obtained the numerical values of the characteristic time Tm of the segmental motion of polymeric chains in melt. As the results showed, Tm doesn't depend on N, but only on temperature. The activation energies and entropies of the segmental motion were found based on the average values of Tm.
In a case of a melt the value of E and AS*/R is approximately in twice higher than the same values for diluted and concentrated solutions of polystyrene in toluene; that points on a great activation action of the solvent on the segmental motion of the polymeric chain, and also notes the fact that the dynamical properties of the polymeric chains in melt is considerably near to their values in polymeric matrixes, than in the solutions.
An analysis which has been done and also the generalization of obtained experimental data show, that as same as in a case of the low-molecular liquids, an investigation of the viscosity of polymeric solutions permits sufficiently accurately to estimate the characteristic time of the segmental motion on the basis of which the diffusion coefficients of the polymeric chains in melt can be calculated; in other words, to determine their dynamical characteristics.
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© Уи. С. МеиуеиеузЫкЬ - сотр. отдела физической химии горючих ископаемых Института физико-органической химии и углехимии имени Л.М. Литвиненко Национальной академии наук Украины; О. Уи. КЬауипко - сотр. отдела физической химии горючих ископаемых Института физико-органической химии и углехимии имени Л.М. Литвиненко Национальной академии наук Украины; Ь. 1 Ба/у1уак - сотр. отдела физической химии горючих ископаемых Института физико-органической химии и углехимии имени Л.М. Литвиненко Национальной академии наук Украины; С. Е. 2а1коу -д.х.н., проф. каф. технологии пластических масс КНИТУ, ov_stoyanov@mail.ru.
