CC BY
12
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 1)
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; 2,2104 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 q 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 q value, also by a wide diapason (from 10-3 to 102 Pas) 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 q for the concentrated solutions is considerable stronger than the q 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 70 and the final 7» viscosities (70» 7»), 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 7 on the length N of a polymeric chain and on the concentration p (g/m3) of a polymer in solution exists: 7 ~ 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 7 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:
7~ 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 7 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]:
*
7 = 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]:
(7 - 7„)/(7o - 7») = f g*) (3)
It was greatly recognized the universal scaling ratio [1, 5]:
7 = 7o • f (gt*) (4)
in which the dimensionless function f (gt* )= f(x) has
the asymptotesf(0) = 1, f(x)x>>1 = x-r, r= 0,8.
Hence, both expressions (3) and (4) declare the gradient dependence of 7 by the function of the one non-dimensional parameter gt*. However, under the theoretical estimation of 7 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 7 ~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 7~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 7 ~ 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 7 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 = 4,1-104; M = 3,3-104 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 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.
Results and discussion: concentrated solutions
Initial statements
Typical dependences of viscosity n of solution on the angular velocity с (turns/s) of the working cylinder rotation are represented on Fig. 1-3. Generally it was obtained the 48 curves of r(c).
For the analysis of the experimental curves of Г (с) it was used the expression [15, 20]:
Г = Vf +re (1 - exp{- Ъ/с})/(1 + exp{- Ъ/с}) (5)
in which n is the measured viscosity of the solution at given value ю of the working cylinder velocity rate; n, and ne are frictional and elastic components of n;
b lc = t * /1 *
(6)
where t*m is the characteristic time of the shear strain of
the conformational volume for m-ball of intertwined
polymeric chains; t* is the characteristic time of the
external action of gradient rate of the hydrodynamic flow on the m-ball.
The notion about the m-ball of the intertwined polymeric chains will be considered later.
The shear strain of the conformational volume of m-ball and its rotation is realized in accordance with
the reptational mechanism presented in ref. [2], i. e. via
the segmental movement of the polymeric chain, that is *
why t m is also the characteristic time of the own, i. e. without the action g, rotation of m-ball [17].
0,36 0,33 £ 0,30 çr 0,27 0,24 0,21
T=25°C
^=0.19310.00331 Па*с r^=0.73±0.15424 Па*с b=0.00337±0.0266 с1
1 2 3 со, об/с
0,27 0,24
и
[= 0,21 ^ 0,18 0,15 0,12
Т=35°С
Л=0.133±0.00253 Па*с ^=0.39410.81594 Па*с Ь=0.00578±0.01261 с1
0,30 0,27
£
",0,24 0,21 0,18 0,15
0,20 0,18 0,16
Г
0,14 0,12
Т=30°С
rlf=0.169±0.00185 Па*с 1^=0.5710.43608 Па*с Ь=0.0043±0.01119 с"1
1 2 3 со, об/с
Т=40°С
Л=0.11939±00019Па*с ^=0.18885+0.55364 Па*с Ь=0.00664±0.01657 с1
2
), об/с
2
о. об/с
Fig. 1 □ Experimental (points) and calculated in accordance with the equation (5) (curves) dependencies of the effective viscosity on the rotation velocity of the working cylinder: p = 4.0*105 g/m3, M = 4.1104 g/mole, T = 25 - 40 0C
The expression (5) leads to the two asymptotes: Г = Г/ + re at blСО» 1
Г = Г/ at b l С « 1
So, it is observed a general regularity of the effective viscosity dependence on the rotation velocity w of the working cylinder for diluted, concentrated solutions and melts. Under condition, that blc>> 1, that is at с 0, the effective viscosity is equal to a sum of the frictional and elastic components of the viscosity, and under condition С — ю the measurable viscosity is determined only by a frictional component of the viscosity.
In accordance with eq. (5) the effective viscosity r(c) is a function on three parameters, namely Г^f, re and b. They can be found on a basis of the
experimental values of r(c) via the optimization method in program ORIGIN 5.0. As an analysis showed,
the numerical values of Г f are easy determined upon a
plateau on the curves r(c) accordingly to the condition b/ С « 1 (see Figures 1-3). However, the optimization method gave not always the correct values of re and b. There are two reasons for this. Firstly, in a
field of the с — 0 the uncertainty of r(c) measurement is sharply increased since the moment of force registered by a device is a small. Secondly, in very important field of the curve transition r(c) from the strong dependence of r on с to the weak one the parameters re and b are interflowed into a composition
reb, i. e. they are by one parameter. Really, at the
1 57
condition b/ c « 1 decomposing the exponents into (5) and limiting by two terms of the row expi- b 1«1 - b, we will obtained q = qf + qeb/ 2-I cj C
Due to the above-mentioned reasons the optimization method gives the values of qe and b depending
between themselves but doesn't giving the global minimum of the errors functional. That is why at the estimation of qe and b parameters it was necessary
sometimes to supplement the optimization method with the «manual» method of the global minimum search
varying mainly by the numerical estimation of qe .
M=5.1*104 r/Monb ^=1.11210.01048 lla*c r|e=2.50±0.02 ria*c b=0.00166±0.00005 c1
! 0,78 ' 0,76 ' 0,74 0,72 0.70
M=4.1*10* r/Monb 1^=0.69910.00303 na*c 1^=1.07410.07741 na*c b=0.00102±0.0004 c1
0,0 0,1
0,2 0,3 co. o6/c
0,4 0,5
0,3 0,6 0,9 CD, o6/c
^ 0,6 0,5 0,4
M=3.3*104 r/Monb i>=0.43113±0.007 ria*c
b=0.00291±0.0019 c1
0,0 0,5 1,0 1,5 2,0 2,5 co, o6/c
0,60 0,55
o
ro 0,50 ? 0,45 0,40 0,35 0,30
M=2.2*104 r/Monb i>=0.360±0.00511a*c r|e=0.352±0.02 I1a*c b=0.00731 ±0.0004 c1
0,0 0,3 0,6 0,9 co, o6/c
1,2 1,5
Fig. 2 - Experimental (points) and calculated in accordance with the equation (5) (curves) dependencies of the effective viscosity on the rotation velocity of the working cylinder: p = 5.0-105 g/m3, M = 5.1 -2.2104g/mole, T=25°C
0,24 0,22 0,20
1= ^ 0,18
0,16'
0,14
p=4.0*105r/M3 i>=0.159±0.00133 I1a*c r|e=0.32789t1.36192 I1a*c b=0.00418+0.0157 c1
0,9 0,8 0,7
ro
= M
0,5 04
p=5.0*105r/M3 i>=0.43113±0.007 ria*c 1^=0.86665+0.49 ria*c b=0 00291+0 0019 c1
1 2 CO. o6/c
0,0
0,5 1,0 1,5 2,0 2,5 co. o6/c
4,20 3,85 £ 3,50 P 3,15 2,80 2.45
p=7.0*106r/M3 r^=2.644±0.033 ria*c 1^=2.365^.06411a*c b=0 0015+0 00056 c1
0,00
0,07 ' 0,14 co o6/c
0,21 0,28
Fig. 3 - Experimental (points) and calculated in accordance with the equation (5) (curves) dependencies of the effective viscosity on the rotation velocity of the working cylinder: p = 4.0-105 + 7.0-105 g/m3, M = 3.3-104 g/mole, T = 25 °C
As we can see from the Figures 1-3, calculated curves accordingly to the equation (5) and found in such a way parameters nf, ne and b, are described the experimental values very well.
The results of nf, ne and b numerical estimations for the all 48 experimental curves q(w) are represented in Table 1. The mean-square standard deviations of the n, ne and b calculations indicated on the Figures.
A review of these data shows that the all three parameters are the functions on the concentration of polymer into solution, on the length of a chain and on the temperature. But at this, the qe and q^ are
increased at the p and M increasing and are decreased at the T increasing whereas the b parameter is changed into the opposite way. The analysis of these dependencies will be represented further. Here let us present the all needed for this analysis determinations, notifications and information concerning to the concentrated polymeric solutions.
Investigated solutions of the polystyrene in toluene were concentrated; since the following condition was performing for them:
p>p , (7)
where p* is a critical density of the solution per
polymer corresponding to the starting of the polymeric chains conformational volumes overlapping having into
diluted solution p < p* j the conformation of Flory
ball by the radius
Rf = aN3/5, (8)
here a is a length of the chain's link. It's followed from
the determination of p
p = M / NaR 3 = M0 N / NaR 3
(9)
where M0 is the molar weigh of the link of a chain. Taking into account the eq. eq. (8) and (9) we have:
p =Po N-4/5, (10)
where
Po = Mo/ a3 Na (11)
can be called as the density into volume of the monomeric link.
In accordance with the SARWS [19] the conformational radius Rm of the polymeric chain into concentrated solutions is greater than into diluted ones and is increased at the polymer concentration p increasing. Moreover, not one, but m macromolecules with the same conformational radius are present into the conformational volume R3 . This leads to the notion of
twisted polymeric chains m-ball for which the conformational volume Rm3 is general and equally
accessible. Since the m-ball is not localized with the concrete polymeric chain, it is the virtual, i. e. by the mathematical notion.
It is followed from the SARWS [19]:
Rm = Rf
m
1/5
m15 =p/p*)/2 at p> p thus, it can be written
Rm = aN (pip, f
(12)
(13)
(14)
Table 1 - Optimization parameters qfi qe and b in equation (5)
p-10-5, g/m3 4,0 5,0 7,0
T, 0C M-104 g/mole 5,1 4,1 3,3 2,2 5,1 4,1 3,3 2,2 5,1 4,1 3,3 2,2
25 tff, Pas 0,35 0,19 0,16 0,06 1,11 0,69 0,43 0,36 6,50 2,66 2,64 0,86
ne, Pa s 1,40 0,73 0,33 0,09 2,50 1,10 0,87 0,35 7,60 3,75 2,37 1,50
M03, s-1 1,15 3,37 4,20 32,3 1,66 1,02 2,91 7,31 0,36 0,76 1,50 2,44
30 nf, Pa s 0,31 0,17 0,14 0,05 1,00 0,62 0,36 0,24 4,95 2,11 2,03 0,68
ne, Pa s 0,95 0,57 0,25 0,06 1,30 0,76 0,52 0,32 4,05 2,21 1,86 1,00
M03, s-1 1,38 4,30 5,90 35,0 2,23 1,80 3,14 8,69 0,72 0,83 1,70 2,65
35 nf, Pas 0,19 0,13 0,11 0,04 0,68 0,50 0,26 0,19 4,07 1,85 1,45 0,43
ne, Pa s 0,60 0,39 0,21 0,05 0,90 0,35 0,23 0,22 3,50 1,80 1,59 0,79
M03, s-1 3,67 5,80 6,37 49,0 2,41 3,56 4,60 9,10 0,88 0,96 1,93 3,20
40 nf, Pas 0,17 0,12 0,10 0,04 0,56 0,42 0,22 0,17 2,91 1,46 0,98 0,27
ne, Pa s 0,40 0,19 0,13 0,03 0,65 0,29 0,15 0,12 2,01 1,39 1,19 0,57
M03, s-1 5,35 6,60 6,90 73,9 2,67 5,60 5,60 16,8 1,33 1,41 2,27 4,24
The shear modulus j for the m-ball was determined by the expression [19]:
RT
H = 1.36
NAa3
(15)
0 J
and, as it can be seen, doesn't depend on the length of a
chain into the concentrated solutions.
*
Characteristic time tm of the rotary movement
of the m-ball and, respectively its shear, in accordance with the prior work [17] is equal to
C= 4 N34
(P?
,Po
L t
m m
(16)
Let us compare the tm with the characteristic time t*f of the rotary movement of Flory ball into diluted solution [17]:
t :
= 4 N14 Lt
7
•rf '
In these expressions t„
and
lf
(17)
are
characteristic times of the segmental movement of the
polymeric chains and Lm and Lf are their form factors
m j
into concentrated and diluted solutions respectively. Let us note also, that the expressions (16) and (17) are self-coordinated since at p = p the expression (16) transforms into the eq. (17). The form factors Lm and Lj are determined by a fact how much strong the
conformational volume of the polymeric chain is strained into the ellipsoid of rotation, flattened or elongated one as it was shown by author [21].
Frictional component of the effective viscosity In accordance with the data of Table 1 the frictional component of the viscosity rjf strongly
depends on a length of the polymeric chains, on their concentration and on the temperature. The all spectrum of 7f dependence on N, p and T we will be
considered as the superposition of the fourth movement forms giving the endowment into the frictional component of the solution viscosity. For the solvent such movement form is the Brownian movement of the molecules, i. e. their translation freedom degree: the solvent viscosity coefficient 7s will be corresponding to
this translation freedom degree. The analogue of the Brownian movement of the solvent molecules is the segmental movement of the polymeric chain which is responsible for its translation and rotation movements and also for the shear strain. The viscosity coefficient 7 will be corresponding to this segmental movement
' sm
of the polymeric chain.
Under the action of a velocity gradient g of the hydrodynamic flow the polymeric m-ball performs the rotary movement also giving the endowment into the frictional component of the viscosity. In accordance with the superposition principle the segmental movement and the external rotary movement of the polymeric chains will be considered as the independent ones. In this case the external rotary movement of the polymeric chains without taking into account the segmental one is similar to the rotation of m-ball with the frozen equilibrium conformation of the all m polymeric chains represented into m-ball. This corresponds to the inflexible Kuhn's wire model [22]. The viscosity coefficient 7m will be corresponding to
the external rotating movement of the m-ball under the action of g. The all listed movement forms are enough in order to describe the diluted solutions. However, in a case of the concentrated solutions it is necessary to embed one more movement form, namely, the transference of the twisted between themselves
polymeric chain one respectively another in m-ball. Exactly such relative movement of the polymeric chains contents into itself the all possible linkages effects. Accordingly to the superposition principle the polymeric chains movement does not depend on the above-listed movement forms if it doesn't change the equilibrium conformation of the polymeric chains in m-ball. The endowment of such movement form into „
instead of the eq. (19) we will write
let us note via
V
pz-
Not all the listed movement forms give the
essential endowment into the
Vf ■
however for the
generality let us start from the taking into account of the all forms. In such a case the frictional component of a viscosity should be described by the expression:
qf =qs (1 - p) + q + qpm + qPz p, (18)
or
Vf
-Vs - {Vs,
+ V + V
I pm I p
-Vs V'
(19)
here p is the volumetric part of the polymer into solution. It is equal to the volumetric part of the monomeric links into m-ball; that is why it can be determined by the ratio:
p = VN/nR, (20)
in which V is the partial-molar volume of the monomeric link into solution.
Combining the eq. eq. (9)-(14) and eq. (20) we will obtain:
p = VpM0 . (21)
The ratio of M0/- should be near to the density pm of the liquid monomer. Assuming of this approximation, M0 /V ~ pm we have:
P = p/pm . (22)
At the rotation of m-ball under the action of g the angular rotation rate for any polymeric chain is the same but their links depending on the remoteness from the rotation center will have different linear movement rates. Consequently, in m-ball there are local velocity
gradients of the hydrodynamic flow. Let gm
represents the averaged upon m-ball local velocity gradient of the hydrodynamic flow additional to g. Then, the tangential or strain shear J formed by these
gradients gm and g at the rotation movement of m-
ball in the medium of a solvent will be equal to:
J = qs g + gm). (23)
However, the measurable strain shear correlates with the well-known external gradient g that gives another effective viscosity coefficient:
J = qpmg (24)
Comparing the eq. (23) and eq. (24) we will
obtain
(25)
(26)
Vpm Vs Vsgm / g •
Noting
Vf -Vs -\Vsm +Vpm +Vpz
V
(27)
The endowment of the relative movement of twisted polymeric chains in m-ball into the frictional component of the viscosity should be in general case depending on a number of the contacts between monomeric links independently to which polymeric chain these links belong. That is why we assume:
VVz ~ V •
(28)
The efficiency of these contacts or linkages let us
estimate comparing the characteristic times of the
*
rotation (shear) of m-ball into concentrated solution tm and polymeric ball into diluted solution tf determined
by the expressions (16) and (17).
Let's note that in accordance with the
determination done by author [17] t *m is the
characteristic time not only for m-ball rotation, but also
*
for each polymeric chain in it. Consequently, t m is the
characteristic time of the rotation of polymeric chain
twisted with others chains whereas t* is the
f
characteristic time of free polymeric chain rotation. The above-said permits to assume the ratio t*m / t*f as a
measure of the polymeric chains contacts or linkages efficiency and to write the following in accordance with the (16) and (17):
q„: ~C/ f = N2 pp )2 5 gm/Lf ) (29)
Taking into account the (22) and combining the (28) and (29) into one expression we will obtain:
v 2ID If
2.5 / \2
(30)
Here the coefficient of proportionality qQpz includes the ratio LmTm /Lf tf, which should considerably weaker depends on p and N that the value qpZ.
Substituting the (30) into (27) with taking into account the (22) we have:
Vf-VS
Vmm V + V^ | f J [ ff
f (31)
Pm
V pm Vs gm / g
Let us estimate the endowment of the separate terms in eq. (31) into qj . In accordance with Table 1
under conditions of our experiments the frictional component of the viscosity is changed from the minimal value « 4-10-2 Pas to the maximal one « 6,5 Pas. Accordingly to the reference data the viscosity coefficient qs of the toluene has the order 5-10 4 Pa-s.
The value of the viscosity coefficient qSm representing
the segmental movement of the polymeric chains estimated by us upon q^ of the diluted solution of
polystyrene in toluene consists of the value by 5-10-3 Pas order. Thus, it can be assumed qsm, qs <<qand
it can be neglected the respective terms in eq. (31). With
2.5
2
taking into account of this fact, the eq. (31) can be rewritten in a form convenient for the graphical test:
rt pp = r°fm 2
m 25
Uo )
Pm
\2
(32)
On Fig. 4 it is presented the interpretation of the experimental values of r f into coordinates of the
equation (32).
^ 4 2
3 6 9 12 N2(p/pJ2(p/p0)25
2
Y=-0.07-t0.421X
T=35°C
o p=0,4*105r/M3 y
■ p=0,5*10W
- P=0,7*10V/M3
3
3 0 3 6 9 12 15 18
N^f/pj'W25
Y=0.001-K).302X
T=40°C
o |=0,4*10W y
■ p=0,5*10V/M3
* i=0,7*10W
-3 0
3 6 9 12 N^Pn/if/Po)25
15 18
-3 0
3 6 9 12
nVpJ2^)2'5
15 18
Fig. 4 - An interpretation of the experimental data of nf in coordinates of the equation (32)
At that, it were assumed the following values:
.—10
m under (11) and
M = 104,15 g/mole, a = 1,8610 determination of p0 accordingly to eq.
pm = 0,906 • 106 g/m3 for liquid styrene. As we can, the
linear dependence is observed corresponding to eq. (32) at each temperature; based on the tangent of these straight lines inclination (see the regression equations on Fig. 4) it were found the numerical values of r°pz, the
temperature dependence of which is shown on Fig. 5 into the Arrhenius' coordinates.
It is follows from these data, that the activation energy Epz regarding to the movement of twisted
polymeric chains in toluene is equal to 39,9 kJ/mole.
It can be seen from the Fig. 4 and from the represented regression equations on them, that the values J1°pm are so little (probably, r/°pm « 0,1 Pas)
that they are located within the limits of their estimation error. This, in particular, didn't permit us to found the numerical values of the ratio gm/ g.
So, the analysis of experimental data, which has been done by us, showed that the main endowment into the frictional component of the effective viscosity of the concentrated solutions "polystyrene in toluene" has the separate movement of the twisted between themselves into m-ball polymeric chains. Exactly this determines a strong dependence of the rf on
concentration of polymer into solution ^ ~ p5'5) and on the length of a chain ^ ~ N2 ).
0,00320 0,00325 0,00330 0,00335 1/T, K1
Fig. 5 - Temperature dependence of the viscosity coefficient r0 in coordinates of the Arrhenius
equation
Elastic component of the effective viscosity
It is follows from the data of Table 1, that the elastic component of viscosity re is a strong increasing
function on polymer concentration p, on a length of a chain N and a diminishing function on a temperature T.
The elastic properties of the conformational state of the m-ball of polymeric chains are appeared in a form of the resistance to the conformational volume deformation under the action of the external forces. In particular, the resistance to the shear is determined by the shear modulus JU, which for the m-ball was determined by the expression (15). As it was shown by author [17], the elastic component of the viscosity is equal to:
re = UCLm . (33)
The factor of form Lm depends on the
deformation degree of the conformational volume of a ball [17, 21].
Combining the (15) and (16) into (33) and
4 • 136 « 1 we will obtain
7 ' ~
assuming 4 • ^ ~ 1
RT
Ve = ^N34 p M 0
p0
LmTm .
(34)
Comparing the (16) and (34) we can see, that the known from the reference data ratio re ~ t*m ~ N3'4
is performed but only for the elastic component of a viscosity.
It is follows from the expression (34), that the parameters Lm and rm are inseparable; so, based on
the experimental values of re (see Table 1) it can be
found the numerical values only for the composition Lm •rm . The results of (LmTm )r calculations are
m „m. The results of (Lmrm )r calculations
represented in Table 2. In spite of these numerical estimations scattering it is overlooked their clear dependence on T, but not on p and N. Table 2 - Calculated values Lt, t/L, t and L based on the experimental magnitudes ne and b
161
p-10-5, g/m3 4,0 5,0 7,0 T-1010, s L
T, 0C M104, g/mole 5,1 4,1 3,3 2,2 5,1 4,1 3,3 2,2 5,1 4,1 3,3 2,2
25 (LT)m-101l>, s 2,63 3,14 2,72 2,99 1,71 1,72 2,61 4,25 1,15 1,29 1,57 4,00
(t/L)b-10l°, s 3,25 1,81 2,54 0,89 1,17 3,43 1,91 2,06 1,98 1,86 1,38 2,29
T-1010, s 2,92 2,38 2,63 1,63 1,41 2,43 2,23 2,96 1,51 1,61 1,47 3,03 2,19
L 0,90 1,32 1,03 1,83 1,21 0,71 1,17 1,44 0,76 0,86 1,07 1,32 1,13
30 (LT)„e-10W, S 1,75 2,41 2,03 1,96 0,88 1,17 1,54 3,83 0,60 0,75 1,21 2,63
(t/L)b-10l°, s 2,10 1,56 1,81 0,82 0,87 1,94 1,39 1,73 1,00 1,62 1,22 2,11
r-1010, s 2,17 1,94 1,92 1,27 0,88 1,51 1,46 2,57 0,78 0,98 1,21 2,56 1,59
L 0,81 1,24 1,00 1,55 1,00 0,78 1,05 1,49 0,78 0,60 1,00 1,12 1,04
35 L)„e-1010, s 1,09 1,62 1,67 1,61 0,60 0,53 0,67 2,58 0,51 0,60 1,02 2,04
(t/L)b-10l°, s 1,01 1,16 1,67 0,59 0,79 0,98 1,21 1,65 0,81 1,35 1,09 1,75
T-1010, s 1,05 1,37 1,67 0,97 0,70 0,72 0,90 2,06 0,64 0,90 1,05 1,89 1,16
L 1,04 1,18 1,00 1,65 0,87 0,73 0,74 1,25 0,79 0,67 0,97 1,08 1,00
40 L)„e-1010, s 0,72 0,78 1,03 0,96 0,43 0,44 0,43 1,40 0,29 0,46 0,75 1,46
(t/L)b-1010, s 0,70 1,01 1,54 0,39 0,73 0,62 1,00 0,90 0,54 0,92 0,91 1,31
T-1010, s 0,71 0,89 1,26 0,61 0,56 0,52 0,66 1,12 0,40 0,65 0,83 1,38 0,80
L 1,01 0,88 0,82 1,57 0,77 0,84 0,66 1,25 0,73 0,71 0,91 1,06 0,93
Parameter b
In accordance with the determination (6), the b parameter is a measure of the velocity gradient of
hydrodynamic flow created by the working cylinder
*
rotation, influence on characteristic time tv 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) depends only on p, N and T via Tm .
It is follows from the experimental data (see Table 1) that the b parameter is a function on the all three variables p , N and T, but, at that, is increased at T increasing and is decreased at p and N increasing. In order to describe these dependences let us previously determine the angular rate ( (s-1) of the strained m-
ball rotation with the effective radius r l of the
m m
working cylinder by diameter d contracting with the surface:
ml = ndml RmLm (35)
Here 7 is appeared due to the difference in the dimensionalities of oI and O.
Let us determine the tV as the reverse one
O
o .
t0 = RL I ndo
(o .
(36)
Accordingly to (36) tv is a time during which
the m-ball with the effective radius Rm Lm under the
action of working cylinder by diameter d rotation 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.
Since in our experiments the working cylinder had two rotating surfaces with the diameters d1 and d2, the value oI was averaged out in accordance with the
condition d = (d1 + d2)I2; so, respectively, the value tv
was averaged out too:
tV = 2RmLm l7(d + d2)O. (37)
So, t0 is in inverse proportion to O; therefore
through the constant device it is in inverse proportion to
g : t0 ~ g_1. However, as it was noted, in m-ball due
to the difference in linear rates of the polymeric chains links it is appeared the hydrodynamic interaction which leads to the appearance of the additional to g local
averaged upon m-ball velocity gradient of the hydrodynamic flow gm. This local gradient gm acts
not on the conformational volume of the m-ball but on the monomeric framework of the polymeric chains (the inflexible Kuhn's wire model [22]). That is why the
endowment of g m into characteristic time tv* depends
on the volumetric part p of the links into the
conformational volume of m-ball, i. e.
C ~ (g + gmP\X.
Therefore, it can be written the following:
tv
V _
0 _ "
v
g
, (38)
tV g + gm(
that with taking into account of eq. (37) leads to the expression
"l + M* p\. (39)
. = 2RmLm
tv "
n(d1 + d2)°/ { g pm Combining the (16) and (39) into (6) we will
obtain
b =
7 a
. N 24 |P| |1 + gm . (40)
2n(d1 + d2) Tm! \ p(i ) \ g Pm
As we can see, here the parameters Lm and Tm are also inseparable and can not be found
independently one from another. That is why based on the experimental data presented in Table 1 it can be found only the numerical values of the ratio (Tm / Lm )b.
After the substitution of values a = 1,86-10-10 m, d1= 3,4-10-2 m, d2= 3,3-10-2 m we have
2
N241 L
1 + g^ L
\
b
(41)
2.84 • 10
P0 A g Pm As it was marked, we could not estimate the numerical value of gm/ g due to the smallness of the
value rQpm lying in the error limits of its measuring. That is why, we will be consider the ratio g m / g as the
fitting parameter starting from the consideration that the concentrated solution for polymeric chains is more ideal than the diluted one and, moreover, the m-ball is less strained than the single polymeric ball. That is why, g m / g was selected in such a manner that the factor of
form Lm was near to the 1. This lead to the value
gm / g =25.
The calculations results of Cr / L ),
accordingly to equation (41) with the use of experimental values from Table 1 and also the values gm / g =25 are represented in Table 2. They mean that
the (m / Lm ) is a visible function on a temperature but not on a p and N.
On a basis of the independent estimations of (r / r ) and (r / L ) it was found the values of
V m m r Vm'^m/b
rm and Lm, which also presented in Table 2. An
analysis of these data shows that with taking into of their estimation error it is discovered the clear
dependence of rm and L on T, but not on p and N.
Especially clear temperature dependence is visualized
for the values rm, obtained via the averaging of rm at giving temperature for the all values of p and N (Table
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 t*m and Flory ball tf . This
lead to the dependence of the frictional component of
viscosity in a form r f ~ N2p5 5 for concentrated
solutions, which is agreed with the experimental data.
It was experimentally confirmed the determined earlier theoretical dependence of the elastic component
of viscosity for concentrated solutions r ~ N3 4p4 5,
that is lead to the well-known ratio f] ~ t
N
-3.4
which is true, 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 concentrated solutions. 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.
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 solutions can be calculated; in other words, to determine their dynamical characteristics.
2). The temperature dependence of rm into the coordinates of the Arrhenius' equation is presented on
Figure 6.
-22,2 -22,5 -22,8 -23,1 -23.4
Y=-42,233+5954X
0,00320 0,00325 0,00330 0,00335 1/T, K1
Fig. 6 - Temperature dependence of the average values of the characteristic time t of the segmental movement of polymeric chain in coordinates of the Arrhenius equation
Conclusions
Investigations of a gradient dependence of the effective viscosity of concentrated solutions of polystyrene permitted to mark its frictional 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
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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, [email protected].
