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Dynamics of polymer θ - solutions in the dilute - semidilute transition region
A.N. Semenov
To cite this version:
A.N. Semenov. Dynamics of polymerθ- solutions in the dilute - semidilute transition region. Journal de Physique, 1988, 49 (8), pp.1353-1363. �10.1051/jphys:019880049080135300�. �jpa-00210816�
Dynamics of polymer 03B8 2014 solutions in the dilute 2014 semidilute transition
region
A. N. Semenov
Physics Department, Moscow State University, Moscow, 117234, U.S.S.R.
(Requ le 23 décembre 1987, accepté sous forme définitive le 11 avril 1988)
Résumé. 2014 Nous considérons les propriétés rhéologiques et de diffusion des solutions 03B8 de polymères à partir
d’une approche théorique nouvelle. Nous étudions l’effet des contraintes topologiques dans la région de
transition diluée-semidiluée. Nous montrons en particulier que (i) la viscosité relative ~r de la solution augmente selon la loi ~r~N1,5(C/C*-1)6,0>C/C*-1~1 où N est le nombre de maillons dans une chaîne de polymères, C*, la concentration critique au-delà de laquelle les chaînes se recouvrent mutuelle- ment ; (ii) la constante de diffusion D des macromolécules décroît exponentiellement avec la concentration D ~ exp (- C/C *),1 C/C * 1,5 In N (iii) le mouvement des macromolécules peut être décrit par une loi de simple diffusion avec une seule constante seulement à une très grande échelle q-1 ~ Rc où q est le nombre
vecteur d’onde caractéristique et Rc ~ N1,25a ; Rc est donc beaucoup plus grand que la taille des macromolécules R = N1/2a. Nous comparons les résultats théoriques à l’expérience.
Abstract. 2014 Rheological and diffusional properties of a polymer theta-solution are considered on the basis of novel theoretical approach. The effect of topological constraints on the dynamics of the system in the dilute- semidilute transition region is studied. In particular it is shown that (i) the relative viscosity, ~r of the solution increases according to the law ~r ~ N1.5 (C/C* - 1)6, 0 C/C* - 1 ~ 1, where N is the number of links in
a polymer chain, C * is the critical concentration of order of the concentration of overlapping ; (ii) the self-
diffusion constant of macromolecules, D decreases exponentially with the concentration :
D ~ exp (-C/C*), 1 C/C* 1.5 ln N ; (iii) the motion of macromolecules can be described by the simple single-constant diffusion law only on very large scales q-1 > Rc (q is a characteristic wave vector), where
Rc ~ N1.25a, i.e. Rc essentially exceeds the size of macromolecules, R = N1/2a. Theoretical results are
compared with experimental data in detail.
Classification
Physics Abstracts
05.90 - 81.20S
1. Introduction.
In a previous paper [1], devoted to the dynamics of
flexible macromolecules in a theta-solution, it was
shown that the effect of entanglements (uncrossing restriction) results in an appearance of very slow conformational relaxation processes with the charac- teristic topological time
where T 1 is the relaxation time for one isolated link and N is the number of links in a polymer chain (note that according to the terminology adopted in [1] a link is some large enough segment of a macromolecule which can be entangled with another segment of the same length with a rather high probability of order 1/2).
Let us consider the dilute solution : CR3 1,
where C is the mean number of polymer chains in a
unit volume and R = aN 1/2 is the radius of gyration
of macromolecules (a is the size of a link). Different
clusters consisting of mutually entangled macromol-
ecules can be distinguished in the system at each
moment of time. The clusters can disintegrate and
arise again, the characteristic life time of a cluster
being obvioulsy of order rtop. On the other hand the relaxation time of stress (for example after rapid
shear deformation) in a small cluster (or in an
isolated coil) is of order of the hydrodynamic time [2]
which is much shorter than Ttop (in the limit
N > 1). Thus, a stress in a dilute solution relaxes with the characteristic time of order Th Ttop.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080135300
The situation changes in the dilute-semidilute transition region where an infinite cluster appears in the system at the critical concentration [1]
The relaxation time of the stress in the infinite cluster is very slow : it is of order of the topological
time Ttop > Th- It was shown in [1] that, as a
consequence of this slow relaxation, the viscosity of
the solution increases significantly in the vicinity of
the point C = C *.
The present paper is devoted to the investigation
of some rheological and relaxational properties of
theta-solutions in the dilute-semidilute transition
region. The theoretical approach suggested in [1] is
used. In the next section the dependence of zero-
shear viscosity, q, on the concentration is considered in the region C - C *. In section 3 the precise
numerical factor in the right-hand side of equation (1.2) is calculated. The self-diffusion con- stant of polymer chains is considered in section 4 and the coherent scattering function of macromolecules
- in section 5. Some theoretical results are compare
semi-quantitatively with experimental data in sec-
tion 6.
2. Zero-shear viscosity in the region C - C *.
It was shown in [1] that the dependence of the zero-
shear viscosity of theta solutions on the concen-
tration must be anomalous in the vicinity of
C *. Formally this dependence can be represented in
the form :
where qr = n] /n s is the relative viscosity of the polymer solution (n s is the viscosity of the solvent).
It is natural to assume that there is some narrow
transition regime conjugating the asymptotics (2.1)
and (2.2).
Let us firstly consider the region C > C * in the
vicinity of the critical point :
The part of the viscosity due to the infinite cluster in
the region can be written as
where Goo (0) == Goo is the shear elastic modulus of the infinite cluster and Goo (t) is the relaxation
function of the stress in the cluster. Therefore,
within an order of magnitude,
where T * is the characteristic time of a disintegration
of the initial infinite cluster.
Let us estimate the time T *. The density of the
infinite cluster is rather low in the region 6 « 1.
Therefore its disintegration does not assume a break
of all bonds (entanglements) of the cluster or even of half of these bonds. It is sufficient to tear a small part q - 6 of the bonds since this process obviously corresponds to the effective decrease of the conver-
sion below the critical point (note that the conversion
is approximately proportional to the concentration
C). Thus
The function q(t) (i.e. the probability that two initially entangled macromolecules will disentangle during the time of order t) is calculated in appendix (see Eqs. (Al), (A15)) :
Substituting equation (2.6) into equation (2.5) we
get
where z = 3 [1].
The elastic modulus of the infinite cluster, Goo, depends on 5 according to the power law :
where const can be easily estimated using the
obvious relation
At C - C * each polymer chain has a lot of
neighbours which it can be entangled with. Therefore
it seems reasonable to use the classical (mean-field)
value T = 3 [2] for the elastic exponent [3, 4]. Using equations (2.8), (2.9) we obtain
Substituting equations (2.7) and (2.10) into equation (2.4) and using equations (l.l), (1.2) we
get
A comparison of equations (2.11) and (2.2) reveals
the following asymptotic behaviour of the function
Q1; :
Thus, in the region C > C * the viscosity of the
solution is nearly proportional to (C - C * )6.
An analogous consideration (using the classical theory of sol-gel transition) in the region C C *
leads to the conclusion that the function cpo(C/C*) (see Eq. (2.1)) tends to a constant of order of unity
in the limit C/C * --> 1, i.e.
In view of equations (2.1, 2.2, 2.12, 2.13) it is
clear that the topological effects become significant
not exactly at C = C* but at C>-C*(1+5c),
where
The region I C / C * - 11 :5 5, obviously corresponds
to a transition between asymptotics (2.1) and (2.2).
At this stage the general form of the dependence of 17 r on the concentration and the molecular weight
can be suggested. The most appropriate form, which
is in agreement with the asymptotics (2.1) and (2.2),
is
where cp 1 (C /C * ) =- 0 for C /C * 1. The first term
in equation (2.15) corresponds to the usual hydro- dynamic effects ; the second term - to the topologi-
cal effects. Note that Qo and Q 1 are universal functions of reduced concentration, C /C *, N being
the number of topological links in a polymer chain (see Sect. b.1).
Fig. 1. - The schematic dependence of the relative
viscosity, q , of a theta-solution on the reduced concen-
tration C/C *. Chain curves correspond to the asympto- tics : (1) Tlr = CPo(C/C*); (2) Tlr = N1.5 cp(C/C*);
dc ~ N - 0-25 is the relative width of the transition region.
Thus in the region 0 C /C * 1 the viscosity of
the solution increases by a factor of order of unity, whereas, for example, in the region 1 C / C * 2
the increase of the viscosity is much more pro-
nounced, by a factor of N 1.5. The schematic plot of
the viscosity vs. concentration is shown in figure 1.
3. The critical concentration C *.
The critical value C *R 3 is calculated in this section.
An E-expansion method is used (e = 4 - d, d is the dimensionality of the space).
Let rn be the mean number of polymer chains entangled with a given chain. This quantity is equal
to
where
p (r) is the probability of an entangled state of two
macromolecules (r is the distance between the centres of mass of the chains). Integral probability,
B, was calculated in [1] :
Here
and u is the dimensionless parameter characterizing
the probability of a local entanglement of two links.
In the limit N > 1 the parameter u tends to the fixed value [1]
The function p(r) is essentially nonzero in the region r 5 R, therefore its characteristic value is
Thus, assuming that E - 1, we conclude that the
probability of an entanglement of the macromol-
ecules, po, is also small, po 1. Therefore the
probability of closed loops (consisting of mutually entangled macromolecules) is negligible [8], so that
the critical concentration in the main s-order can be obtained using the classical theory (the approxi-
mation of trees) [2].
Let x be the probability that a given chain belongs
to a finite cluster. In the approximation of trees this quantity satisfies an equation
The appearance of an infinite cluster corresponds to
Substituting equation (3.8) into equation (3.7) we
get m =1, i.e.
Using equations (3.3), (3.5) and (3.4) we obtain in
the main e-order
The universal (for Gaussian chains) quantity C * R 3 was obtained in [9] using computer simu- lation. The result [9]
agrees quite well with the analytical estimate (3.10).
4. Self-diffusion constant of macromolecules in theta-solution.
In a semidilute solution (C > C * ) at each moment
of time there are : (1) macromolecules belonging to
finite clusters or macromolecules which are not
entangled at all with any other chains (such macro-
molecules will be called free) and (2) macromolecules
belonging to an infinite cluster (bounded macromol- ecules). Let us obtain the probabilities, pl and pz, for a macromolecule to belong to each of these classes. Using the approximation of trees which is
valid in the main E-order (see Sect. 3) we get
where x is determined by equation (3.7). Taking into
account that m = C /C * we obtain for C > C *
It can be easily shown that the concentration
Cn of n-clusters is proportional to xn :
where x is assumed to be small, x 1. Therefore the first macromolecular class consists mainly of free chains, the role of larger finite clusters being negli- gible.
The self-diffusion constant of free chains, Di, is approximately equal to the diffusion constant of a
chain in a dilute solution, Do :
The self-diffusion constant of bounded macromol-
ecules, D2, is appreciably smaller [10] :
The mean self-diffusion constant, D, is equal to
Thus for
where
The second term in the right-hand side of equation (4.6) is negligible provided C/C * : 1.5 In N.
Therefore in the region
the mean diffusion constant of macromolecules decreases exponentially as concentration increases :
The rapid exponential decrease of D ceases in more
concentrated region,
The sharp, nearly exponential, dependence of the
diffusion constant D on the concentration is observed
experimentally in the whole semidilute (and concen- trated) region C > C * [11, 12] rather than solely in
the regime (4.7). The sharp dependence of D on C in
the regime (4.9) can be explained in the following
way : for the systems encountered in experiments
the region (4.9) is probably so concentrated that the effective change of the solvent viscosity due to the
presence of polymer links becomes essential. This
change results in a considerable decrease of the
mobility of polymer links [13], and thus to the
decrease of the diffusion constant D.
5. Coherent scattering function.
Let us consider the structure factor of macromol- ecules (the coherent scattering function)
where indices n, m run over all links of a given macromolecule, and q is a wave vector. In the limit q - 0 the structure factor has the simple form :
The dependence (5.2) corresponds to the simple
diffusion motion of a macromolecule as a whole. It is
generally assumed that equation (5.2) is valid for
A =- 2 7T Iq R, i. e. for q 1 /R. It is shown below in this section that actually this is not the fact, provided that topological constraints are taken into account : the transition to the simple exponential
form (5.2) occurs at A a Rc (q:5 1/Rc), where the
characteristic scale Rc essentially exceeds the size of
a macromolecule R, Rc >> R.
Let y 1 be the frequency of the macromolecular transitions from a free state (1) to a bounded state (2) and Y2 be the frequency of the inverse transitions.
Obviously
The transitions between states 1 and 2 are governed by the following equations :
Equation (5.5) is a consequence of equation (5.4)
due to the obvious condition
Substituting the stationary probabilities (4.1) into equation (5.4) we get
taking into account that x = e - c /C - .-, 1 we obtain
Let Pi (r ) d3r be the probability for a macromol-
ecule to be in the state i (i = 1, 2 ), and for its centre
of mass, r, to be within the small volume d3r.
Obviously
Taking into account the transitions between the states 1 and 2 (see Eqs. (5.4), (5.5)) and the
diffusion of macromolecules we get the following
kinetic equations (in the Fourier representation) for
the functions p 1 and p 2 (it is assumed that the characteristic scale A = 2 7T Iq is large, A > R) :
Let us assume that at the initial moment t = 0
some macromolecule was near the point r = 0. Thus
the distribution functions p 1 and p 2 for this macro-
molecule at t = 0 are (the space dimensions of the macromolecule can be neglected due to the condition q .: 1/R) :
The structure factor S(q, t) is connected with the functions p 1 and p (provided q -- 1/ R) :
Substituting the solution of the system of equa- tions (5.9), (5.10) with the initial condition (5.11)
into equation (5.12) we get (the inequalities D2 D1, Y2 y 1 are taken into account) :
In the region (4.7) the first term in the right-hand
side of equation (5.13) can be neglected. Thus
where D==Diexp(-C/C*) is the self-diffusion constant of macromolecules (see Eq. (4.6)), and
Substituting equations (4.3) and (5.3) into equation (5.15) we get
Thus the transition to the simple diffusion regime
occurs at the scales
Note that the characteristic scale Rc is difficult to
interpret in pure geometrical terms since the depen-
dence Rc on N,
is even more rapid than that for the total contour
length of a macromolecule.
An appearance of a large characteristic scale
Rc > R in the dilute-semidilute transition region,
C > C *, was reported in [12], where the diffusion of
high-molecular weight polystyrene was studied by
the impulse NMR method. The dependence (5.13) is
in a qualitative agreement with the experimental
results [12].
6. Comparison with experiments.
6.1 THE SIZE OF A LINK. - The term link has been
initially introduced in [1]. A link is a long enough part of a macromolecule which can be (with rather high probability) in an entangled state with another part of the same length. Let us put this definition on a quantitative basis.
An important topological characteristic of a link is the quantity v (see [1])
where cp (r) is the probability of an entangled state of
two links, r being the distance between their centres of mass.
The term entangled state for two unclosed parts of
a macromolecule as itself was determined in [1]. In
the first approximation the function Q (r ) can be regarded as the probability of a true (in mathematical
sense) entanglement of two formally closed links.
The dimensionless parameter
(a is the size (radius of gyration) of a link, Kd = 21- d 7T - d/2/ r (d/2) is the numerical constant, and d is the dimensionality of space) tends to the
fixed point (see [1])
as the molecular weight of a link increases (as a
result of renormalizational transformations).
It is natural to adopt the following definition of a
topological link (taking into account that the par- ameter u increases with the molecular weight of the link) : a topological link is a part of a macromolecule characterized by the value
Substituting equations (6.1)-(6.3) into equation (6.4)
we get the following criterion (for d = 3) :
In order to obtain the values u for the links of different lengths, the data [14] were used. The probability of entanglement of two polymer Gaus-
sian rings was studied in [14] by computer simulation.
The dependence of the parameter u on the number g of Kuhn segments in a ring is shown in figure 2.
The value u = u */2 = 3/8 corresponds to
Fig. 2. - The dependence of the parameter u characteriz-
ing the probability of an entanglement of two polymer rings vs. g- 0.25 , where g is the number of Kuhn segments in each ring. The dependence of u on g- 0.25 must be
approximately linear in the limit g - oo (see the text).
Therefore a topological link consists of about 32 Kuhn segments.
In the following subsections some experimental
data on dynamics of polystyrene (PS) and poly(a- methylstyrene) (PMS) theta-solutions are con-
sidered. In this view let us calculate the molecular
weight, Mo, of the (topological) link of the polymers.
Taking into account that the molecular weight of the
Kuhn segment of PS and PMS is mo = 550 we get
[15]
Therefore the number of links in ultra-high molecu-
lar weight PS with M = 107 is ,
Thus, in the dilute-semidilute transition regime (4.7)
for PS or PMS with M = 107, the concentration of the solution changes by a factor of 10.
6.2 THE CRITICAL CONCENTRATION. - As a meas- ure of the concentration in rheological experiments
the quantity
( [n ] is the intrinsic viscosity of the solution) is often
used. In the high-molecular-weight limit, N - oo, this quantity is equal to [17]
where 0 . =-- 2.8 x 1023 and NA is the Avogadro
number.
Let us consider the dependence of the relative
viscosity, q,, on the reduced concentration, V and
on the molecular weight, M in order to obtain the
critical concentration C * (or, equivalently, the criti-
cal value V * = C * [T/]) from experimental data.
Equations (2.1), (2.2) or (2.15) suggest that at fixed V V * the viscosity nearly does not depend on the
molecular weight, whereas at V > V * an additional
dependence of q, on M becomes essential. Taking
into account equation (2.13) we thus conclude that the difference between relative viscosities for two molecular weights, åT/r== T/r(M2) - T/r(M1) tends
to zero as V decreases to the value V *, the dependence of Aq r on AV = V - V * being govern- ed by the 6th power law: å T/ r oc AV 6. Therefore
(An r )1/6 must tend to zero almost linearly.
Some rheological experimental data [18] for PMS
theta-solutions (M = 0.44 x 106, 1.65 x 106) are pre- sented in figure 3 [19]. The plot of (åT/r)1I6vs. V
obtained using the data of figure 3 is shown in
figure 4. After a linear extrapolation (see Fig. 4) we
get
Fig. 3. - The plots of the relative viscosity of PMS theta- solutions vs. reduced concentration V for the molecular
weights M1 = 0.44 x 106 (o), M2 = 1.65 x 106 (x) [18].
Fig. 4. - The plot of (A-q,p)"6 vs. reduced concentration V. Here A7y = n,sp (M,) - llsp(M1) is the relative viscosity
difference. The chain part of the curve is due to a nearly
linear extrapolation.
The result of computer simulation, equation (3.11), can be rewritten using equation (6.9) as
Thus, the critical values which were independently
obtained from experimental data [18] and by com- puter simulation [9] agree well with each other (and
also with the analytical result, Eq. (3.10)).
6.3 THE TOPOLOGICAL EXPONENT z. - The ex-
ponent z corresponds, in particular, to the depen-
dence of relative viscosity of a theta-solution on the molecular weight M at fixed reduced concentration
V (V > V * ). In the asymptotic limit M --> oo the
dependence is: n, OC M, - 1.5 (see Eq. (2.2)). It
should be stressed, however, that this true asymp- totic behaviour is not available in practice (see below) since the molecular weights of the polymers
are not large enough. Thus the corrections to the
asymptotic scaling laws must be taken into account.
The generalization of the theoretical dependence of
11r on N = MIMO at V > V * has the form (compare
with Eq. (2.2)) :
where (- y ) is the largest inessential exponent [21].
The exponent (- y ) determines the relative order of corrections to all scalling dependences. In particular,
we can write :
The RG equation for parameter u was obtained in
[1] (see Eq. (A. 17) in [1]). Solving this RG equation
and comparing the solution with equation (6.14) we
get
For d = 3 we thus obtain
From equation (6.12) we have
Consequently, the dependence of § -
(õ In 7J r/ Õ In N ) on X = N - Y must be approximately
linear in the limit N >- oo, and, besides,
The dependence of ’onX = (M/Mo )- °w at fixed
V = 7 is shown in figure 5 [22]. The plot is based on
Fig. 5. - The plot of , = a log ’r1 ria log M vs. X =
(Mol M)O.25 at fixed V = 7. The bar A is obtained using
the data shown in figure 3 ; the bar B - using the data [23]
for PS solutions, M1 = 6.77 x 106 and M2 = 20.6 x 106.
The chain lines represent the boundaries of an uncertainty.
