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Dynamical correlation function of polymer density fluctuations in concentrated solutions
Physica A 166 (1990) North-Holland
263-287
DYNAMICAL CORRELATION FUNCTION OF POLYMER DENSITY FLUCTUATIONS IN CONCENTRATED SOLUTIONS A.N. SEMENOV Physics Department,
Received
Moscow State University, 117234 Moscow,
6 December
USSR
1989
The time-dependent correlation function of concentration fluctuations in a polymer solution in the regime of entanglements is studied theoretically. Three stages of relaxation of concentration fluctuations are predicted. The first stage is due to cooperative deformation of entangelement network; its characteristic rate is 7,’ a q2No (here q is the wave vector and N is the number of links per polymer chain). The second stage corresponds to a Rouse-type relaxation of macromolecular tension along the “tube”. The last stage, which is due to reptation motion of macromolecules, is characterized by a spectrum of relaxation times between T,,~, and T,,,,,. Here r,,, 0~N3 is the longest reptation time, and the time 7,,pr depends on q and N in the following way: 7,Cpt a N3q4 for b-’ + q ti (Rb)-I”; T,_, E N2qo for (Rb)-“‘+ q + R-l; Treptm N’q-’ for R-’ % q % b/R’ (here R = N”*a is the size of a macromolecule, b is the effective tube diameter).
1. Introduction A theoretical investigation of dynamics of concentration fluctuations in homogeneous polymer solutions is a problem of considerable interest. An important dynamical characteristic of the system is a time-dependent correlation function, S,(q) which can be measured in experiments using quasielastic light scattering or small angle neutron scattering [l-5]. Dynamics of fluctuations and spinodal decomposition in polymer mixtures on one hand, and in concentrated solutions on the other hand, are closely connected. The dynamical behavior of mixtures of very long macromolecules was studied theoretically in refs. [6-81 on the basis of the reptational model [9, lo]. This model assumes that a set of polymer chains surrounding a given macromolecule create an effective tube of diameter 6 for it (fig. 1) so that the principal motion of the macromolecule on large scales (greater than b) must be a creeping (reptation) along the tube axis. The tube can be alternatively represented as a sequence of “entanglements” of the macromolecule with surrounding chains. The characteristic tube diameter is b - Na’2a, where a is 0378-4371/90/$03.50
0
1990 - Elsevier
Science
Publishers
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264
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
Fig. 1. Polymer chains (shown by points) create an effective tube (dashed lines) for a given macromolecule (solid line); b is the diameter of the tube. AB is the macromolecular conformation at t = to, A’B’ at f = t, + At; s is the curvilinear coordinate along the tube.
the size of a polymer link and N, is the mean number of links per entanglement (for the strict definition of N, see below, eq. (1.4)). The following main equations governing a relaxation (or a growth) of fluctuations of polymer concentration, 4A, were used in refs. [6-81 (the characteristic spatial scale 4-l is assumed to be large, q-l % b):
a+,/at+V*J=O, J = - (A/T)Vp
(1.1) .
(1.2)
Here +A = 4,(r) is the volume concentration of polymer links of type A (the mixture is assumed to be incompressible, 4, + & = l), J is the current of A-links, A = A(q) is the transport coefficient depending on the wave vector q, and p(r) = SF/S4,(r) is the effective chemical potential of A-link at the point r (F is the free energy of the system as a functional of the distribution 4,(r)). A solution of eqs. (1. l), (1.2) in a linearized form is
W,(
4, t> =
W,( 4,O)exp(- t/Tq),
where S4,(q, t) = +A(q, t), q # 0, is the concentration fluctuation in the q2A( q) B(q) / T, and B(q) is the coefficient Fourier representation. Here r,’ = which is connecting small concentration fluctuations with small changes of the chemical potential:
SP(4)= B(q) W(q). Thus, we conclude that the purely exponential relaxation law for small fluctuations is due to the particular “structure” of eq. (1.2) (note that this “structure” was assumed). On the basis of a more general approach to polymer dynamics it was shown [ll] that this assumption strictly speaking is incorrect:
A. N. Semenov I Polymer density fluctuations in concentrated solutions
265
the relaxation of (small) concentration fluctuations may be essentially nonexponential. The aim of the present paper is to study theoretically the time-dependent correlation function of concentration fluctuations S,(q) = 1 d3r( c(r, t) ~(0,0) - ci) exp(-iq
- I-)
(1.3)
in a concentrated (or semidilute) polymer solution without using an assumption of exponential relaxation. Here C(T, t) is the local (number) concentration of polymer links at the point r and at the moment t, c0 = (c(r, t)) is the mean concentration. It is assumed that polymer chains are long enough so that they are highly entangled (the number of links per chain N S NJ. It is well known [12, 131 that during small periods entangled systems reveal elastic behavior, the characteristic shear modulus (elastic plateau modulus of the entanglement network) being N%N,.
G, = c,, Tl N, ,
(1.4)
Note that eq. (1.4) can be treated as a definition of N,. Note also that N, depends on the concentration c0 (and on temperature). The following empirical relation is known [12] for a good solution: c,N, = const .
During sufficiently long periods t b T,,, (here T,,,~~is the longest time of conformational relaxation) the system behaves as a liquid with high zero-shear viscosity [ 131, 77-
Gcl~nl,, .
The scattering function S,(q) in a semidilute theta-solution was studied theoretically in ref. [14]. Two relaxational modes corresponding to the two types of dynamical behavior of the system (gel-like and liquid-like) were predicted (for q > qg, where qg is a small threshold, qg m N,NP3”): the “fast” gel mode with the characteristic time TV =
1/D,q2,
and the slow hydrodynamic 72 =
r,,,,,D,/ D,
.
mode with the time
266
A.N.
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I Polymer density fluctuations in concentrated solutions
Here D, is the effective diffusion constant which is proportional to the elastic modulus of the entanglement network and D, is the cooperative diffusion constant. In ref. [14] it was assumed that the relaxation of concentration fluctuations is a consequence of elastic deformations of the entanglement network, the direct influence of the reptation motion of polymer chains on the relaxation being neglected. Note that, on the other hand, the entanglement network was assumed to be undeformable in refs. [6-81, so that the relaxation of fluctuations was considered to be solely due to the reptational motion. This last assumption (undeformable network) proves to be correct for a (symmetric) polymer mixture; but for a polymer solution both types of motion (deformation of entanglement network and reptations) are important and should be taken into account. This is performed below#‘. The following two sections are devoted to description of the model of a polymer solution and the general method for a calculation of the scattering function S,(q). The specific results obtained for short-wave (q S R-‘) and long-wave (q + R ‘) regimes are considered in sections 4 and 5 (R is the mean size of a polymer chain in the solution). In particular, it is shown that the relaxation of fluctuations with q b (Nu) -’ is characterized by three distinct times: the “cooperative” time rc = l/D,q’, the “Rouse” time TV (TVcc qe4 for qR % 1 and rR 0~N* for qR < 1) and the reptational time crept (7, < TV< r,,rt). The longest time r,,rt depends on q in the following way: T,_,~0~N3q4 for b e q-’ G (* (note that the relaxation time decreases as the wavelength is increased in this regime); T,,_,~0~N2qo for t* s q-l 4 R and rFeptcxNq-’ for Req-’ + Na. The characteristic dynamical length t* is of order of the geometrical mean of the tube diameter and the size of a polymer coil: t* - (bR)“‘; a discussion of this length see at the end of section 4. The obtained results are compared with the results of ref. [14], and (when it is possible) refs. [6-81. The main conclusions are gathered in section 6, where a qualitative comparison of theoretical results with experimental data is presented as well.
2. The general method of accounting for volume interactions Let us consider a concentrated solution of homopolymer chains (N 9 N,). Conformations of the chains in such a system are governed by the Gaussian
this is not at all xl Note that the “reptational” law for T,,,,, was assumed in ref. [14]. Nevertheless equivalent to a direct consideration of the reptational motion (so that some important effects had been missed).
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
statistics [13]; in particular, equal to
the r.m.s. gyration radius of macromolecule,
R = N”*a ,
267
R, is
(2-l)
where a is the characteristic size of a link. Eq. (2.1) remains vaid for a semidilute solution as well, provided that a is treated as an effective renormalized size depending on the concentration. It is this renormalized size that is denoted by a in the following. Volume interactions between polymer links can be characterized by an important parameter, IJ =
(l/C,T)(anlac,)~
= (l/T)(a/.&/C?c,),
)
(2.2)
where II = H(c,, T) is the osmotic pressure of the polymer and p = &co, T) is the chemical potential of a link in a homogeneous solution with concentration cO (it is assumed that ideal-gas contributions to I;Tand p are negligible). The static correlation function (for simultaneous fluctuations) for a concentrated solution is well known [15]; in the most interesting case c,uN % 1 (which is considered below) it is equal to S,(q) = (u + a2q2/2c,)-’
= ~~'(1 + q*[$l
is the static where &, = al(2c,u)“* Let us consider the dynamics of large wavelength A = 2nlq, which length 5, and the tube diameter inequalities c,Nu%l,
450+1>
,
(2.3)
correlation length. concentration fluctuations with sufficiently is greater than both the static correlation b = Nt’*a. Thus, we are limited by the
NIN,%l,
qbel.
(2.4)
Note that as a rule the first two inequalities are the direct consequences of the second two (since usually cOuNCis larger than 1). There are two important types of interactions in a polymer system: the topological interactions which correspond to impossibility for chain segments to intersect each other (this interaction can be formally related to extremely rigid and short-range forces between the segments), and the so-called volume interactions corresponding to more or less long-range forces between the links of different polymer chains or (distant along the chain) links of the same chain”*. These last interactions can be characterized by a mean molecular X2Note that though topological of linear chains, these interactions
interactions play an important role for the dynamics of a system do not affect equilibrium properties of the system.
A.N.
268
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I Polymer
density jluctuations
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solutions
fieldx3
(25)
#L(r) = SFin:,,,lSc(r) )
where Fin, = Fin,[c(r)] is the free energy of volume interactions as a functional of c(r). Nonlocal (gradient) terms in Fin, can be neglected due to inequality q& -% 1; therefore we can write
(2.6) The total free energy density in a homogeneous f(c) = Ant(c) + (CT/N) ln(cleN)
system is
,
(2.7)
where c/N is the number concentration of macromolecules term in the r.h.s. is of ideal-gas nature. Thus, k(r) = aL,,!&z = &c(r),
T) - (TIN) ln(c(r)lN)
.
and the second
(2.8)
Assuming that St(r) = c(r) - co 4 co and taking into account that c,uN %=1 we have h(r) = G(r) - r_L(c,,T) = TV SC(r) .
(2.9)
The effect of volume interactions (i.e. of the molecular field h(r)) can be easily taken into account by the general method which was proposed in ref. [ll]. Let us consider this method briefly. Let Si”‘( 4) be the scattering function of the system without volume interactions (more precisely, S(O) corresponds to the system without “large-scale” part of volume interactions (see footnote 3), but certainly with topological ones). Consider a weak external field hext(r, t) acting on the links of polymer chains at t > 0 (it is assumed that at t < 0 the system was at equilibrium). The linear response of the system can be represented as follows: x3 It is well-known that the volume interactions in a semidilute solution generally cannot be accounted for by a mean-field approach [13]. Nevertheless, a mean-field approximation is valid for scales which are larger than the static correlation length 5,. In the following the term “volume interactions” is attributed just to these large-scale effects.
A.N.
Semenov
(Sc(r, t)) = -
I Polymer density fluctuations in concentrated
1d7
d3r’ K(O)@- r’, r) h,,t(r’, t - r)/T
solutions
,
269
(2.10)
where K(O) is the generalized susceptibility and (. . -) denotes averaging over an ensemble. After the Fourier-Laplace transformation eq. (2.10) can be written as
(sC(%P)) = -
K
‘“‘(4YP>L(q, PIIT
(2.11)
9
where
fc(‘)( q, p) = [ dt 1 d3r
~(‘)(r,
t)
eXp(-pt
-
iq
* r)
,
etc.
(2.12)
0
As a consequence
of the fluctuation-dissipation
theorem
K(‘)(q,P) = $?(q) - Ps’“‘(%P) >
[16] we have (2.13)
where S”‘( q, p) is the Laplace image of the function Si”‘( q). Let us return to the original system (with volume interactions) and calculate its general susceptibility K( q, p). In the mean-field approximation the molecular field h(r, t) simply superposes the external field Iz,,~(T, t). Thus, eq. (2.11) remains valid if h,,, is substituted by h,,, = h,,, + h:
(sC(q> P)) = -K(“)(4,p)h,,,(q, PI/T.
(2.14)
Taking into account that (see eq. (2.9))
Ok PI) = u@c(q,PI) and neglecting fluctuations from eq. (2.14)
(i.e. a difference
between h and (Iz)#~) we find
(sC(q,P)) = -K(4, P) h,,,(q, P)/T,
(2.15)
where K-l(q,
p)]-’ + U
p) = [K(')(q,
*4 The conditions which considered in ref. [17].
are
required
(2.16)
.
in order
for
these
fluctuations
to be neglected
are
A. N. Semenov
270
i Polymer
density fluctuations
in concentrated
solutions
Note that in the “static” case (p = 0) eq. (2.16) is in accordance with Edwards’ results [ 151. Using the fluctuation theorem once more we get the correlation function for the original system:
s(q>
(2.17)
= [‘d4,0) - K(‘L P)l/P .
Eqs. (2.13), (2.16), (2.17) determine the required relation between S and SC”! Thus the problem reduces to a calculation of S(O) (or K(O)) for the system of polymer chains without volume interactions.
3. The general method of calculation of K(O) First of all, let us note that the external field h,,, provokes two principal different processes in the system. The first is the induced (by the field) reptational motion of polymer chains along their tubes (see fig. 1). The second is a deformation of the entanglement network, i.e. an elongation or compression of its parts caused by the field. The corresponding mean concentration changes (due to these processes) will be denoted by SC, and &,,, (3.1)
SC = SC, + SC,,
Reptational motions of different macromolecules are independent (since volume interactions are turned off). Therefore the concentration change 6c, is completely determined by the susceptibility CYof one polymer chain:
%(%
P> = -
The susceptibility reptating chain,
K,(q) = N-l
‘d9,
P> kx,(q, PI/T
>
‘d4,
cx can be related to the coherent
ntzl (exp{iq-[r,(t) - r,(O>l>),
P> = Co”(4,
P> .
(3.2)
structure factor of a single
(3.3)
where r,(t) is the position of the nth link of the chain at the moment t. Using the fluctuation theorem we get (compare with eq. (2.13))
4%
P> = K,(q)
where K(q,
p)
-
PK(q7 P)
>
is the Laplace image of K,(q)
(3.4)
A.N.
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I Polymer density fluctuations in concentrated solutions
271
The reptational motion of a macromolecule can be considered as a onedimensional diffusion along the tube axis. Let s be the curvilinear coordinate along this axis (see fig. 1). It is convenient to choose the corresponding scale so that the (mean) “length” of the macromolecule be equal to N, i.e. s,,, - s0 2: N (here S, is the coordinate of the nth link, n = 0, 1, . . . , N). During a short time At the chain can creep along the tube some distance As so that the coordinate of a link changes: s,(t + At) = s,(t) +
NsN,
D,, = W,N,IN,
(3.6)
( W, is some microscopic frequency). The longest time of conformational relaxation is of order of the time needed for a macromolecule in order to creep out of the initial tube, i.e. r,,,,, - N2/Dll. As a result of a more precise calculation one gets [lo] 7 max
= Y2N2/D,, = T-*W,‘N3/Ne
The structure corresponding
.
(3.7)
factor K,(q) was also obtained in ref. [lo]; substituting the result into eq. (3.4) we get (details can be found in ref. [ 111) N
a(q, P) = 2 o- -u
1 1 + cth u -
(3.8)
where u = Nq*a*l2 = q2R212,
cr = ( pt*)l’*
)
t* = N214D,, = (IT*/~)T,,, . (3.9)
Let us consider the second contribution, SC,,, which is due to a deformation of the entanglement network. Let us assume that h,,, CC exp(iq - r), the vector q being directed along the z-axis. The external field would induce a displacement
*5 Here we assume that the chain in the tube is intensile. elongation or compression of macromolecular segments below (as a part of the second contribution SC,,).
The faster motions corresponding to an along the tube are taken into account
A. N. Semenov
272
I Polymer density fluctuations in concentrated solutions
of the network “points” in the z-direction: r-+
r +
where l= [(z, t). The process is governed by the following dynamic equation:
(3.10)
where POis the sedimentation constant (the effective mobility of a polymer link which can depend on concentration), gz,, is the component of the stress tensor of the entanglement network. The first term in parentheses represents the force due to the external field (acting on a link), the second is the force due to elastic stresses. For the large-scale deformations (A = 2nlq 9 b) the stres a,, is proportional to the deformation E = ag/az: UzIz=E-E,
(3.11)
where E is the longitudinal elastic modulus of the entanglement network. Eq. (3.11) is valid for sufficiently short times (t 6 T,,,,,), in the general case a possibility of relaxation of entanglements must be taken into account: I
fl,z(t>=
E(t
-
t’) dc(t’) ,
or, after the Laplace transformation,
U,,(P)= E(P)~P)
(3.12)
7
where _ E(P) = P I E(f) ew(-pt)
dt
(3.13)
is the elastic modulus at the frequency o = -ip. Substituting eq. (3.12) into eq. (3.10), solving this equation and using an obvious relation 6c,, = -co&, which is a consequence of the continuity equation, we obtain
A. N. Semenov
273
I Polymer density fluctuations in concentrated solutions
%I = - ‘h(4, I’) &x,(4>P)lT
q*coT 7
Kn = p/p0 +
q2E(p)/co
.
(3.14)
Let us consider the dependence E(t), which determines the relaxation of stress after a small step-like deformation d.s. The initial value E(0) = E, obviously must be of order of the corresponding shear modulus G, = coTIN,: (3.15)
E, = QC”TIN,,
where V, is some numerical factor. This factor can be obtained theoretically, but its value is not universal and depends on a particular model of the “tube”. So V, is considered below as a phenomenological coefficient. On larger time scales the relaxation of stress is related to two processes. The first one can be described as follows. As a result of the initial deformation some parts of macromolecules appear to be stretched and some to be compressed along the tube axis. Therefore the macromolecular tension (along the tube) would not be uniform so that the macromolecules would tend to redistribute their links along the tubes in order to smooth out the tension. This, the so-called first relaxation process [lo], is characterized by the following “Rouse” time [ 181: 2-Jq4a4 ,
qR%-1, (3.16)
rR = 1 7,N2/4 ,
qR+l,
where r0 - IV,’ is the relaxation time for one link. As a result of this first stage of the relaxation the effective elastic modulus would decrease to the value
.!?= YE,, ,
(3.17)
where r”can depend on q. Using the tube model considered in ref. [lo] one can obtain s
qR+l,
8
qR-%l.
6 7
r”=
(3.18) 15 >
A calculation of E(t) during the first “Rouse” analogy with ref. [18]; the result is E(t) = J? + (E, - i)
f(thR)
,
t=srR,
stage can be performed
in
(3.19)
274
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
where
(3.20)
and
1 + z) ,
dz exp(-xz)z”2/(
xeu2, (3.21) x9u2,
p(x) = (S/T*) In eq. (3.22)
C i2 n
(3.22)
exp(-n2n2x/4).
n runs over all odd positive
numbers.
The second stage of relaxation (for t S TV) is due to reptational motions of macromolecules along their tubes. The corresponding dependence E(t) was obtained in ref. [lo] on the basis of the assumption that the stress is proportional to the share of macromolecular links which had not left the initial tube at the moment#6: E(t) = &t/t*)
t%rR,
)
(3.23)
where t* = N2/4D,, = 7r2~,,,/4. Using eqs. (3.19), (3.23) and taking
into account
(3.16),
(E, - E)cp( q4a4t/7,)
+ &(t/t*)
)
qRs1,
(E, - i)p(4thoN2)
+ s!?p(tlt*)
,
qRe1.
we get finally
(3.24)
E(t) =
Taking
into account
K(‘)( 4, p) =
KI
eqs. (3.1), +
K,,
=
cods
(3.2), PI
(3.14) +
q2coWdPo
we obtain +
q2%Wol
2
(3.25)
where E(p) is determined by eqs. (3.24), (3.13). Thus, the correlation function S,(q) can be calculated now with the help of general equations (2.16), (2.17), (3.25). The specific results obtained on the basis of these equations are considered in the following sections. x6 Another assumption that the stresses are proportional to the square of this may be more appropriate; if this assumption is so-called double reptation [19,20]) &t/t*) in eq. (3.23) should be replaced by p*(tit*). Note that this substitution does considerably the specific results mentioned in the following sections (except that an substitution must be performed in eq. (5.9)).
share (the used, then not affect analogous
A. N. Semenov
4. The relaxation
I Polymer density fluctuations in concentrated solutions
of concentration
fluctuations
275
with q S R-’
Using the results of previous sections we obtain after some transformations (for b + q-l 4 R)
S,(q)= ~-‘L%gc(~) + A.&x(t) + 4ep&&)l
(4.1)
)
where the three terms in square brackets correspond to three relaxation stages, cooperative (the fastest), “Rouse”-like and reptational. Here A,, AR, A rept are the reduced amplitudes of the stages (A, + AR + A rept= l), and g,, g,, g rept are dimensionless functions satisfying the conditions
g,(O)= &(O) = &qm = 1 . Let us consider firstly the cooperative A, = l/(1 +
S) ,
g,(t) =
stage:
exp[-(1+ ~P,q*~l
(4.2)
,
where D, = Tc,uj3, = PO aIll$,
is the cooperative S = E,l(c,
(4.3)
diffusion constant of macromolecules
[13]#‘,
tUI/dc,) = u,,/c,,uN,
(4.4)
is an important dimensionless parameter which is equal to the ratio of the elastic modulus of the entanglement network E, and the osmotic modulus K = c0 IFIII/&z,. Usually S is small, 6 5 0.1; an exceptional case of semidilute theta-solution at low concentrations (where 6 b 1) is not considered below. Thus A, is close to unity, i.e. the lion’s share of the initial fluctuation relaxes at the cooperative stage, the relaxation being governed by the simple diffusion law with effective diffusion constant (1 + 6)D, 2: D,. The physical meaning of these results is quite simple: Here the relaxation is due to a deformation of the entanglement network. The corresponding mechanical stresses are rather weak since the elastic modulus E,, is small in comparison with the osmotic one, K. Consequently, the relaxation process must be very similar to that for a system of unconnected polymer links (i.e. for a low-molecular solution). *’ Note that & is an effective sedimentation
constant which can depend on concentration.
276
A. N. Semenov
I Polymer
At t P 7c = l/[( 1 + 6)D,q2] stage characterized by
density fluctuations
the relaxation
A.=E,I(E,+K)-El(~+K)2:S-6”,
in concentrated
solutions
passes to the second “Rouse”
grc(t)= (P(t/TFA 3
(4.5)
where s” _6IK = 63, rR = r0/q4a4 and the function cp is determined by eq. (3.21). Note that the amplitude A, is very small: A, = 6/6+ 1. The qualitative meaning of the result is the following: the rate of smoothing of the initial inhomogeneity (of concentration fluctuation) is primarily governed by the osmotic modulus K which is “counteracted” by the elastic modulus of the entangelement network, E. During the “Rouse” stage this modulus effectively slightly decreases from E, to i = FE0 giving a possibility of some reduction of the residual fluctuation (which remains after the cooperative stage). The amplitude of the last, reptational stage, A
rept
=
i?l(i + K)
2:
s”,
(4.6)
is smaller than A,, but higher than AR. The dependence essentially determined by two large parameters u and r: u = q2R212,
r=c,rx,
of grept on t is
(4.7)
where 77s G/(1 + s”)
(4.8)
(note that r + u due to inequalities
(2.4)). If r -G u’, then
(P(t/Tl)
t-et”,
7
&&
= i (2ulf)
exp(-m’tlt*)
(4.9) ,
tat*,
where 71 =
(Uq2t*
.
(4.10)
Thus, grept- 1 for t S T~; in the region or =St =St” the function decreases according to the power law grept 0~t-“2, and the exponential decrease is predicted at t b t*. If I’ % u”, the function g ,,,t(t) is characterized by the following asymptotics:
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
(1 - 2u*/r) exp(-t/r*) f3rep
+ 2(~*/r)(p(tlr~)
,
277
t+t*,
=
(4.11) i (2ulT) exp(-n2tlt*)
,
tBt*,
where r2
2t*
=
/r
7,
)
=
t*
/4u2
(4.12)
.
Thus, in this regime an almost purely exponential relaxation with the characteristic time r2 is changed (at t P T*) by a long-lived “tail” with a small relative amplitude of order u*/lY Note that the longest relaxation time is always of order t* m N3. Nevertheless, the relaxation may be crudely characterized by a shorter time r,,rt corresponding to a decrease of grept from 1 to, say, l/e. Eqs. (4.9)-(4.12) suggest that (Lgg4t*,
bsq-‘s5*,
t*/r
,$*sq-kR,
7rept ‘-
(4.13) )
where (*
=
r-l’*R
2:
(bR)“2
= aN;‘4N”4
(4.14)
(it is taken into account that $4 1 and r = r”v,NIN, 2: N/N,). Consequently, for q 9 1 /t* the time crept varies according to an anomalous law: r,,rt m q4; whereas for longer waves, l/t* B q B 1 lR, the time tends to a constant value 7
rept = 2t* /r - (N2/D,,)(N,IN)
- W;‘N2 )
(4.15)
which is comparable with the maximum “Rouse” time. The characteristic scale ,$* dividing these two regimes is of order of the geometrical mean of tube diameter and of macromolecular size. Let us consider the physical meaning of the obtained results (for the reptational stage). Note that during the first two stages of relaxation the initial concentration fluctuation 6c(r, Oj would decrease by a factor of u/u”= (1 + s”) /g. Therefore, the molecular field h(r) would also decrease (see eq. (2.9)). This can be alternatively represented as an effective decrease of the parameter of volume interactions, u, by the same factor u/u”, i.e. as a renormalization u + u”. This renormalization effectively accounts for the coupling between a deformation of the entanglement network and the reptational motion of polymer chains. Thus we can consider the reptational stage assuming
A. N. Semenov
278
I Polymer density fluctuations in concentrated solutions
that the network is undeformable; the omitted effects would be taken into account at the very end by the simple substitution u + u”in all final equations. Let h(r, 0) be the molecular field corresponding to the initial concentration fluctuation 6c(r, 0) via eq. (2.9). Consider a macromolecule with end links r0 and rN (see fig. 2). Under the influence of the molecular field the macromolecule would creep along the tube (created by other polymer chains) toward that end, where the potential h is lower, since in this case the macromolecular potential energy would decrease. The velocity of the motion, As/At, must satisfy the following relation: (As/At) = D,,[h(r,,
t) - h(r,,
(4.16)
t)]/T
(compare with eqs. (3.5)). Thus, if h(r,) < h(rN), then the macromolecular links would “flow” into vicinity of the point r0 and raise the local concentration there. The corresponding rate of the concentration change is
[(h(rNJ t>) - h(r,,
h(r,, t>, where 2c,lN taken second h(r,, Using
is the mean concentration into account that the mean end, (h(r,, small almost uncorrelated. h(r,, t) (4.17) and we get
t) =
Fig.
(4.17)
2. The links
0) exp(-2c,uD,,tlN)
move the tube be accumulated
ends. In eq. (4.17) of the molecular field (for qR % 1) the quantities
.
towards
the end
rO if point r,,.
< h(r,).
Thus,
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
279
After the substitution u 4 u”we obtain the relaxation time r = N/2c,v”D,, which coincides with r,,rt = r2 (see eq. (4.15)). So far we neglect random motions of a polymer chain along the tube axis. The characteristic value of the random displacement, As during the relaxation time r is (see eq. (3.5)) As - (D,, ~)l’~ - (N/c,@)“* .
(4.19)
This displacement can be neglected only if the spatial size of the polymer chain segment consisting of As links is smaller than the characteristic scale q-l, i.e. (4.20)
(As)“~u 4 1 /q . Substituting 1 /q
eq. (4.19) into (4.20) we get
P a(Nlc,v”)“4
= [*
.
(4.21)
Note that this inequality coincides with the condition of applicability of eqs. (4.11), (4.12) (r P u’). Ob viously eq. (4.18) is in accordance with eqs. (4.11), (4.12) apart from small corrections of order of u*/r. The considered relaxation process is characterized by an important feature: the initial inhomogeneity Sc(r, 0) relaxes primarily due to a redistribution of the end polymer links or of the links which are close to the ends. As a result the fluctuation of the total concentration decreases considerably during the time of order crept = N/2c,v”D,, . Nevertheless the system remains in an essentially nonequilibrium state since the spatial distributions of the end and the middle polymer links appear to be considerably different. This difference relaxes much more slowly, during the time of order t* % T,,+ giving rise to the long-lived “tail” which was mentioned above (see eq. (4.11)). Let us pass to a more short-wave region, q-l -+ a(Nl~,u”)“~ 2: t*. Here we meet an unusual situation: random shifts of a macromolecule along its tube slow down the relaxation rather than accelerate it. In fact, these random movements tend to smooth out the difference between the distributions of end links and of middle links, which has appeared as a result of the “rapid” relaxation of the total concentration fluctuation (during the time T,,~~). Consequently, these movements hinder the relaxation. Obviously, this effect becomes more pronounced as the scale of inhomogeneity q-’ is decreased (in comparison with the scale of random motions -a(As)‘12 - a(D,, ~)l’~). As a result we have an anomalous dependence of the characteristic time r,,+ on the wavelength 2rrlq: rEeptincreases as q-l is decreased. One may perform a more
A. N. Semenov
280
/ Polymer density fluctuations in concentrated solutions
detailed qualitative analysis and obtain the law: r,,rt CCq4 for q-l G &*, in accordance with eq. (4.13). The above consideration suggests the following interpretation of the “dynamical” length 5 * : This length is of order of the lower limit of the (average) displacement of a macromolecular end during the characteristic relaxation time crept. Some comments about the relaxation of composition in polymer mixtures which was considered in refs. [6-81. Let us consider a symmetrical (NA = NB = N; uA = a, = a; cA = cg = c,/2) and incompressible (c, + cg = const) mixture of homopolymers A and B. Obviously, the cooperative stage of the relaxation which is connected with a “longitudinal” deformation of the entanglement network, is forbidden for such a system (since a longitudinal deformation contradicts the incompressibility)#‘. On the other hand, the reptational stage of the relaxation must be qualitatively the same for a mixture and for a solution#‘. Thus, the results obtained in this section for the reptational stage can be compared with the corresponding results of refs. [7,8]. The relaxation time rs of fluctuations with wave-vector q (see eq. (4.1) of ref. [7]) in our designations and in the region limited by inequalities (2.4) can be represented as rs = const(l\rlc,uD,,)l[l
- exp(-q2Na2)]
,
(4.22)
where const is some numerical factor. This result is in accordance with our eq. (4.13) which is valid in the region q s 1 /t*; nevertheless in a more short-wave region b =Sq-l < [* a coincidence is lacking. This discrepancy is connected with the fact that the relaxation of a (harmonical) inhomogeneity was assumed in ref. [7] to be purely exponential (see section 1) whereas actually the relaxation is essentially nonexponential (see eqs. (4.9)). It is for this reason that the dynamical length [* had not been obtained in ref. [7]. Note also that the method used in ref. [7] cannot “catch” the long-lived tails with characteristic time of order t*. Using eqs. (4.9), (4.11) we obtain the behavior of the concentration fluctuation at the longest times t %-t*: the amplitude 6c decreases as 6c
K
exp(-yt)
,
tst*,
(4.23)
*’ Even if a longitudinal deformation would be allowed, it could not result in a change of the composition since the A and B polymer chains form a common entanglement network. H The only difference of minor importance is that one should assume that r = c,uN rather than r = c,u”N in eqs. (4.9)-(4.12).
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
281
where b+q-‘+R,
y = r2/t* - W,N,IN3 , i.e. the relaxation The relaxation was investigated regime and under y - c,uD,,lN
(4.24)
rate is much slower here than at shorter (of composition) in the mixtures on the theoretically in ref. [8]. The result (for the conditions (2.4)) can be represented = c,uW,,N,lN2
times. longest time-scales the entanglement as
bGq-‘
,
(4.25)
in contradiction with eq. (4.24). I comment that the result of ref. [8] (eq. (4.25)) is incorrect due to two independent reasons. Firstly, a one to one correspondence (at a fixed q) was assumed [8] between the current of polymer links, J, and the gradient of the chemical potential, Vp (a relation of type (1.2)). However, such a relation is lacking even in the region t P t* (J depends not only on Vp but also on conformational distribution of macromolecules). Secondly, an incorrect expression for the coherent structure factor of a single reptating chain, K,(q) was used in ref. [8] (in the region t S t*, qR 4 1): K,(
4) m exp(-q*D,t)
D, - R2/t* ,
,
instead of the correct one [lo]: K,(q) a exp(-n2t/4t*)
.
5. The results for the long-wave
fluctuations,
qR G 1
Let us consider a relaxation of fluctuations (or, equivalently, correlation function) for q-l P R. Let us start with the region R G q-’ <
= X, ,
the dynamical
(5.1)
where the relaxation is characterized by three stages as before (see eq. (4.1)). The characteristics of the first, cooperative stage remain the same (see eqs. (4.2), (4.3)). The amplitude and the relaxation law for the second, “Rouse” stage are A,=-
Eli E,+K
E -y-2:--_ E+K
’
&x(t) = l-4%)
2
(5.2)
282
A. N. Semenov
where
s”= F76= AS,
teristic
time
rR = r,,N2/4
of the “Rouse”
time for an isolated The amplitude sponding
I Polymer density Jfuctuations in concentrated solutions
Rouse
(see eqs.
relaxation
A rept = EI(K
stage#“,
,
+ (ru))“‘+(rutlt*)
exp(-m2tlt*) r is defined
Thus,
the charac-
with the maximum + E)
-
s”;
the corre-
law is
exp(-rut/t*)
where
(3.16)). coincides
coil.
of the reptational
relaxational
(3.18),
roughly
by eq. (4.7)
+ $
exp(-20.2tlt*),
tG t* ,
(5.3a)
t%- t* ,
(5.3b)
and
a
n-‘-f(epzx -
q(x) =
epx)( 1 -
z)-~z~‘~dz
(5.4)
.
0
The second term in eq. (5.3a) is small since TU %-1. Thus, reptational stage) is initially governed by the exponential (5.3a)) with characteristic time 7
rep
t*lTu.
=
the relaxation (at the law (the first term in
(5.5)
The exponential relaxation is changed by a long-lived “tail” (the second term in (5.3a)), the “tail” being “cut off” exponentially at t b t*. Note that eq. (5.5) is in accordance with the result of Pincus, eq. (4.22). The time r,,rt can be represented as crept = 1/D,,,,q2, where D
rest
=
TR212t*
is much smaller
(5.6)
QCN-'
than the cooperative
diffusion
constant
D, = Tc,u~, m No but is
much larger than the self-diffusion constant, D, = R2/2t* m Ne2. The physical meaning of these results can be clarified as follows. The (reptational) relaxation in this regime is governed by the translational diffusion of macromolecules (since the scale l/q is larger than R). If the volume interactions were absent, the rate of the relaxation would be controlled by the self-diffusion constant D,. The volume interactions increase the rate of the process by a factor of order Fi,,lFi, - couN, where Fin, and Fid are the characteristic values of interactional and ideal-gas contributions to the free energy of the system. Thus we obtain the effective diffusion constant Drept c,uN. D,. *lo This stage
was considered
previously
[21].
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
283
Let us consider now a more long-wave region, e q-l e X2 = N”‘X,
X, = N,“‘Nu
.
(5.7)
Note that the characteristic scale A, is of order of the (total) length of the tube axis (which can be represented as N/N, steps of the length b = UN:“). The cooperative and “Rouse” stages proceed in this regime as in the previous case provided that q-l G Nu. If on the contrary q -I 9 Nu, then the time of the cooperative stage, T,, appears to be longer than the “Rouse” time TV. Thus, the “Rouse” stage superposes the cooperative stage so that the former becomes almost indistinguishable against a background of the latter:
The characteristics A
rept
=
of the reptational
EI(K + i) = 5,
stage are
‘!Lept(4 = /-4tlt*) f
(5.9)
The dependence grept( t ) can be qualitatively understood in the following way: a complete relaxation cannot be achieved at the cooperative stage because of the elasticity of the entanglement network; in time the effective elastic modulus decreases proportionally to p(tlt*) making possible more and more complete relaxation. Let us compare the obtained results with the results of ref. [14]. Note that a (rather specific) case was considered in this reference: it was assumed that elastic modulus of the entanglement network, E,, exceeds the bulk modulus K (E,IK B 1, i.e. cOuNe 4 1). On the other hand, eqs. (5.8), (5.9) were obtained in the opposite limiting case cOuN, % 1. Nevertheless the general approach [14] can be easily applied to the case cOuN, % 1 so that the following result may be obtained (in our designations):
S,(q)=u?
4 -l
(
&
=sxx,.
0
exP(- Qq2t) + $$-
exp(-v2t/4t*) 0
, 1
(5.10)
Eq. (5.10) is in qualitative agreement with our results for the long-wave region (5.7), see eqs. (5.8), (5.9), (4.2). On the other hand, for shorter waves, the results of ref. [14] become incorrect: in this region 4 -’ G X, = NuIN:“,
284
A N. Semenov
! Polymer density fluctuations in concentrated solutions
direct reptational motions of polymer chains (which were not taken into account in ref. [14]) play an important role for the relaxation process#“. In the most long-wave region, q-l %- X,, the cooperative relaxation time becomes longer than the reptation time: rc = l/ D,q2 P t*. Thus, the topological constraints (entanglements) do not affect appreciably the relaxation process. Consequently the relaxation must be exponential: S,(q)-u-’
exp(- D,q2t) ,
q-l % X2 = N3’2N,1’2u
.
(5.11)
6. Conclusions and comparison with experiments The main results of previous sections can be summarized as follows: (1) A relaxation of concentration fluctuations in a semidilute or concentrated polymer solution proceeds in three stages with considerably different times rc, provided that the scale of inhomogeneity ‘R and ‘rept h=q-‘ N,“* N3’2a the relaxation is almost singleexponential with the time rc. The dependences of rc, TV, T,_ on X = q-’ are schematically shown in fig. 3. As X is increased, the time r, increases also
Fig. 3. A schematic (double logarithmic) plot of cooperative (T,), “Rouse” (TV) and reptational microscopic (T,.,,) relaxation times vs. the scale of inhomogeneity X E 4-l; 7Ois the characteristic time.
#I’ Note that this type of motion implies the directed tube axis under influence of the molecular field.
reptation
of a macromolecule
along
the
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
285
according to the law rC = X “/DC. The time TV0: X 4 at X c R, and it tends to a constant value rR ccN2 in the region R 6 11 s Nu. The third time 7,ept is of order of the longest reptation time, T,,, 0: N3, if X - b (b is the tube diameter); in the region b =SX s (Rb)“’ the time 7,,pt decreases (as X increases) according to an anomalous law: T,,~~0: x -4; then (in the region (Rb)“2 s 11 s R) it tends to a plateau value 7,,pt 0: N2; in the region R 5 X i N,1’2Na it increases, 7rept o( NA ‘9 and finally it again tends to another constant of order of T,,,~~CCN3 for the most long waves, X b N,“2Na. (2) As a rule the amplitude of the first, cooperative stage, A,, is the largest, and the amplitude of the second, “Rouse” stage is the smallest. (The case of a theta solution near the dilute-semidilute transition region is an exception: here A, may be smaller than A,_.) (3) The relaxation on the reptational stage is not single exponential: it is characterized by a spectrum of times between T,,,,~ (see eq. (4.13)) and 7maxCCN3. A long-lived “tail” of the relaxation usually appears as a consequence, the “tail” being cut off at t - T,,,~~. (4) The rate of the (reptational) relaxation is determined primarily by the time 7maXand by the dimensionless parameter r (see eq. (4.7)) which can be represented as r = (QJE + Lri,JK)-l, where II,,, = c,TIN is the ideal-gas part of the osmotic pressure, K is the bulk osmotic modulus and ,?? is the elastic modulus of the entanglement network. Some comments concerning the experimental results obtained in refs. [l-5] using the methods of dynamical scattering of light and neutrons for qR b 1. Semidilute solutions of polystyrene in good [l, 21 and theta [3-51 solvents were studied. In all cases a number of relaxation modes were revealed, the fastest mode (f) being always of cooperative nature: 7;’ = D,q2, where the diffusion constant D, almost does not depend on the molecular weight of the polymer, M. A second, slow mode (s) with the inverse time T,’ = D,q2 was also revealed [ 11, the corresponding diffusion constant being weakly M-dependent. Another law for the slow mode, T,’ 0~q”Mp3.’ was also reported [3]. Three modes: fast, intermediate (with the inverse time 7,’ which almost does not depend on q) and slow (Q-,’ = A + Bq2, where A and B are constants) were observed in other experiments [2]. A somewhat different behavior of the inverse times was reported in refs. [4,5]: T,’ 2: D,q2, where D, weakly depends on M for qR31 and D,,,ccM-“, n = 0.5 to 0.6, for qR < 1; 7,’ CCq”Mey, y = 3 to 3.8. Thus, the experimental data are not in a complete agreement with each other. Doubtless that there are a lot of relaxation modes, and this observation is obviously in agreement with the theoretical predictions of the paper. Moreover, the fastest mode can be identified with the cooperative stage which has been considered above. The relative amplitude of this stage is (see eq. (4.2)) A, = K/(E, + K). Taking into account that (for a good semidilute
A.N.
286
Semenov
I Polymer
0
density fluctuations
I
I
I
I
2
4
6
8
in concentrated
Ib
solutions
c(% 1
Fig. 4. The dependence of the relative amplitude of the fastest mode, 2, - A,/(A, + A,) on the concentration c (in weight %) of PS solution. The solid line corresponds to the theory (eq. (6.1), K = 0.95), the circles to experimental data [l].
polymer solution) the osmotic modulus K CCc2.25 [6] and the elastic modulus of the entanglement network, E, 0: c2 [22], we get A, = ~c’.~?(l
+ ~.(7’.~~),
K =
const .
(6.1)
The theoretical dependence (6.1) is compared with experimental data (concerning the amplitude of the fastest mode) [l] in fig. 4. A more complete verification of the theory requires experimental data for the largest possible N/N,, i.e. for concentrated solutions of ultra-high molecular weight polymers.
Acknowledgment
I am grateful to I.Ya. Erukhimovitch
for valuable discussions.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
W. Brown, Macromolecules 18 (1985) 1713. P. Stepanek, J. Jakes, C. Konak, R. Johnsen and W. Brown, Polym. Bull. 18 (1987) 175. M. Adam and M. Delsanti, Macromolecules 18 (1985) 1760. W. Brown, Macromolecules 19 (1986) 387, 3006. W. Brown, R. Johnsen, P. Stepanek and J. Jakes, Macromolecules 21 (1988) 2859. P.G. de Gennes, J. Chem. Phys. 72 (1980) 4756. P. Pincus, J. Chem. Phys. 75 (1981) 1996. K. Binder, J. Chem. Phys. 79 (1983) 6327. P.G. de Gennes, J. Chem. Phys. 55 (1971) 572. M. Doi and S.F. Edwards, J. Chem. Sot. Faraday Trans. II 74 (1978) 1789, 1802,.1818. I.Ya. Erukhimovitch and A.N. Semenov, Sov. Phys. JETP 63 (1986) 149. W.W. Graessley, Adv. Polym. Sci. 16 (1974) 1.
A. N. Semenov
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I Polymer density fluctuations in concentrated solutions
287
P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1979). F. Brochard, J. Phys. (Paris) 44 (1983) 39. S.F. Edwards, Proc. Phys. Sot. London 88 (1966) 265. L.D. Landau and E.M. Lifshitz, Statistical Physics, part 1 (Pergamon, Oxford, 1980). A.N. Semenov, Physica A, submitted. P.G. de Gennes, J. Phys. (Paris) 42 (1981) 735. M. Rubinstein, E. Helfand and D.S. Pearson, Macromolecules 20 (1987) 822. J. des Cloizeaux, Europhys. Lett. 5 (1988) 437. A.N. Semenov, Sov. Phys. JETP 63 (1986) 717. M. Adam and M. Delsanti, J. Phys. (Paris) 43 (1982) 549.
263-287
DYNAMICAL CORRELATION FUNCTION OF POLYMER DENSITY FLUCTUATIONS IN CONCENTRATED SOLUTIONS A.N. SEMENOV Physics Department,
Received
Moscow State University, 117234 Moscow,
6 December
USSR
1989
The time-dependent correlation function of concentration fluctuations in a polymer solution in the regime of entanglements is studied theoretically. Three stages of relaxation of concentration fluctuations are predicted. The first stage is due to cooperative deformation of entangelement network; its characteristic rate is 7,’ a q2No (here q is the wave vector and N is the number of links per polymer chain). The second stage corresponds to a Rouse-type relaxation of macromolecular tension along the “tube”. The last stage, which is due to reptation motion of macromolecules, is characterized by a spectrum of relaxation times between T,,~, and T,,,,,. Here r,,, 0~N3 is the longest reptation time, and the time 7,,pr depends on q and N in the following way: 7,Cpt a N3q4 for b-’ + q ti (Rb)-I”; T,_, E N2qo for (Rb)-“‘+ q + R-l; Treptm N’q-’ for R-’ % q % b/R’ (here R = N”*a is the size of a macromolecule, b is the effective tube diameter).
1. Introduction A theoretical investigation of dynamics of concentration fluctuations in homogeneous polymer solutions is a problem of considerable interest. An important dynamical characteristic of the system is a time-dependent correlation function, S,(q) which can be measured in experiments using quasielastic light scattering or small angle neutron scattering [l-5]. Dynamics of fluctuations and spinodal decomposition in polymer mixtures on one hand, and in concentrated solutions on the other hand, are closely connected. The dynamical behavior of mixtures of very long macromolecules was studied theoretically in refs. [6-81 on the basis of the reptational model [9, lo]. This model assumes that a set of polymer chains surrounding a given macromolecule create an effective tube of diameter 6 for it (fig. 1) so that the principal motion of the macromolecule on large scales (greater than b) must be a creeping (reptation) along the tube axis. The tube can be alternatively represented as a sequence of “entanglements” of the macromolecule with surrounding chains. The characteristic tube diameter is b - Na’2a, where a is 0378-4371/90/$03.50
0
1990 - Elsevier
Science
Publishers
B.V. (North-Holland)
264
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
Fig. 1. Polymer chains (shown by points) create an effective tube (dashed lines) for a given macromolecule (solid line); b is the diameter of the tube. AB is the macromolecular conformation at t = to, A’B’ at f = t, + At; s is the curvilinear coordinate along the tube.
the size of a polymer link and N, is the mean number of links per entanglement (for the strict definition of N, see below, eq. (1.4)). The following main equations governing a relaxation (or a growth) of fluctuations of polymer concentration, 4A, were used in refs. [6-81 (the characteristic spatial scale 4-l is assumed to be large, q-l % b):
a+,/at+V*J=O, J = - (A/T)Vp
(1.1) .
(1.2)
Here +A = 4,(r) is the volume concentration of polymer links of type A (the mixture is assumed to be incompressible, 4, + & = l), J is the current of A-links, A = A(q) is the transport coefficient depending on the wave vector q, and p(r) = SF/S4,(r) is the effective chemical potential of A-link at the point r (F is the free energy of the system as a functional of the distribution 4,(r)). A solution of eqs. (1. l), (1.2) in a linearized form is
W,(
4, t> =
W,( 4,O)exp(- t/Tq),
where S4,(q, t) = +A(q, t), q # 0, is the concentration fluctuation in the q2A( q) B(q) / T, and B(q) is the coefficient Fourier representation. Here r,’ = which is connecting small concentration fluctuations with small changes of the chemical potential:
SP(4)= B(q) W(q). Thus, we conclude that the purely exponential relaxation law for small fluctuations is due to the particular “structure” of eq. (1.2) (note that this “structure” was assumed). On the basis of a more general approach to polymer dynamics it was shown [ll] that this assumption strictly speaking is incorrect:
A. N. Semenov I Polymer density fluctuations in concentrated solutions
265
the relaxation of (small) concentration fluctuations may be essentially nonexponential. The aim of the present paper is to study theoretically the time-dependent correlation function of concentration fluctuations S,(q) = 1 d3r( c(r, t) ~(0,0) - ci) exp(-iq
- I-)
(1.3)
in a concentrated (or semidilute) polymer solution without using an assumption of exponential relaxation. Here C(T, t) is the local (number) concentration of polymer links at the point r and at the moment t, c0 = (c(r, t)) is the mean concentration. It is assumed that polymer chains are long enough so that they are highly entangled (the number of links per chain N S NJ. It is well known [12, 131 that during small periods entangled systems reveal elastic behavior, the characteristic shear modulus (elastic plateau modulus of the entanglement network) being N%N,.
G, = c,, Tl N, ,
(1.4)
Note that eq. (1.4) can be treated as a definition of N,. Note also that N, depends on the concentration c0 (and on temperature). The following empirical relation is known [12] for a good solution: c,N, = const .
During sufficiently long periods t b T,,, (here T,,,~~is the longest time of conformational relaxation) the system behaves as a liquid with high zero-shear viscosity [ 131, 77-
Gcl~nl,, .
The scattering function S,(q) in a semidilute theta-solution was studied theoretically in ref. [14]. Two relaxational modes corresponding to the two types of dynamical behavior of the system (gel-like and liquid-like) were predicted (for q > qg, where qg is a small threshold, qg m N,NP3”): the “fast” gel mode with the characteristic time TV =
1/D,q2,
and the slow hydrodynamic 72 =
r,,,,,D,/ D,
.
mode with the time
266
A.N.
Semenov
I Polymer density fluctuations in concentrated solutions
Here D, is the effective diffusion constant which is proportional to the elastic modulus of the entanglement network and D, is the cooperative diffusion constant. In ref. [14] it was assumed that the relaxation of concentration fluctuations is a consequence of elastic deformations of the entanglement network, the direct influence of the reptation motion of polymer chains on the relaxation being neglected. Note that, on the other hand, the entanglement network was assumed to be undeformable in refs. [6-81, so that the relaxation of fluctuations was considered to be solely due to the reptational motion. This last assumption (undeformable network) proves to be correct for a (symmetric) polymer mixture; but for a polymer solution both types of motion (deformation of entanglement network and reptations) are important and should be taken into account. This is performed below#‘. The following two sections are devoted to description of the model of a polymer solution and the general method for a calculation of the scattering function S,(q). The specific results obtained for short-wave (q S R-‘) and long-wave (q + R ‘) regimes are considered in sections 4 and 5 (R is the mean size of a polymer chain in the solution). In particular, it is shown that the relaxation of fluctuations with q b (Nu) -’ is characterized by three distinct times: the “cooperative” time rc = l/D,q’, the “Rouse” time TV (TVcc qe4 for qR % 1 and rR 0~N* for qR < 1) and the reptational time crept (7, < TV< r,,rt). The longest time r,,rt depends on q in the following way: T,_,~0~N3q4 for b e q-’ G (* (note that the relaxation time decreases as the wavelength is increased in this regime); T,,_,~0~N2qo for t* s q-l 4 R and rFeptcxNq-’ for Req-’ + Na. The characteristic dynamical length t* is of order of the geometrical mean of the tube diameter and the size of a polymer coil: t* - (bR)“‘; a discussion of this length see at the end of section 4. The obtained results are compared with the results of ref. [14], and (when it is possible) refs. [6-81. The main conclusions are gathered in section 6, where a qualitative comparison of theoretical results with experimental data is presented as well.
2. The general method of accounting for volume interactions Let us consider a concentrated solution of homopolymer chains (N 9 N,). Conformations of the chains in such a system are governed by the Gaussian
this is not at all xl Note that the “reptational” law for T,,,,, was assumed in ref. [14]. Nevertheless equivalent to a direct consideration of the reptational motion (so that some important effects had been missed).
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
statistics [13]; in particular, equal to
the r.m.s. gyration radius of macromolecule,
R = N”*a ,
267
R, is
(2-l)
where a is the characteristic size of a link. Eq. (2.1) remains vaid for a semidilute solution as well, provided that a is treated as an effective renormalized size depending on the concentration. It is this renormalized size that is denoted by a in the following. Volume interactions between polymer links can be characterized by an important parameter, IJ =
(l/C,T)(anlac,)~
= (l/T)(a/.&/C?c,),
)
(2.2)
where II = H(c,, T) is the osmotic pressure of the polymer and p = &co, T) is the chemical potential of a link in a homogeneous solution with concentration cO (it is assumed that ideal-gas contributions to I;Tand p are negligible). The static correlation function (for simultaneous fluctuations) for a concentrated solution is well known [15]; in the most interesting case c,uN % 1 (which is considered below) it is equal to S,(q) = (u + a2q2/2c,)-’
= ~~'(1 + q*[$l
is the static where &, = al(2c,u)“* Let us consider the dynamics of large wavelength A = 2nlq, which length 5, and the tube diameter inequalities c,Nu%l,
450+1>
,
(2.3)
correlation length. concentration fluctuations with sufficiently is greater than both the static correlation b = Nt’*a. Thus, we are limited by the
NIN,%l,
qbel.
(2.4)
Note that as a rule the first two inequalities are the direct consequences of the second two (since usually cOuNCis larger than 1). There are two important types of interactions in a polymer system: the topological interactions which correspond to impossibility for chain segments to intersect each other (this interaction can be formally related to extremely rigid and short-range forces between the segments), and the so-called volume interactions corresponding to more or less long-range forces between the links of different polymer chains or (distant along the chain) links of the same chain”*. These last interactions can be characterized by a mean molecular X2Note that though topological of linear chains, these interactions
interactions play an important role for the dynamics of a system do not affect equilibrium properties of the system.
A.N.
268
Semenov
I Polymer
density jluctuations
in concentrated
solutions
fieldx3
(25)
#L(r) = SFin:,,,lSc(r) )
where Fin, = Fin,[c(r)] is the free energy of volume interactions as a functional of c(r). Nonlocal (gradient) terms in Fin, can be neglected due to inequality q& -% 1; therefore we can write
(2.6) The total free energy density in a homogeneous f(c) = Ant(c) + (CT/N) ln(cleN)
system is
,
(2.7)
where c/N is the number concentration of macromolecules term in the r.h.s. is of ideal-gas nature. Thus, k(r) = aL,,!&z = &c(r),
T) - (TIN) ln(c(r)lN)
.
and the second
(2.8)
Assuming that St(r) = c(r) - co 4 co and taking into account that c,uN %=1 we have h(r) = G(r) - r_L(c,,T) = TV SC(r) .
(2.9)
The effect of volume interactions (i.e. of the molecular field h(r)) can be easily taken into account by the general method which was proposed in ref. [ll]. Let us consider this method briefly. Let Si”‘( 4) be the scattering function of the system without volume interactions (more precisely, S(O) corresponds to the system without “large-scale” part of volume interactions (see footnote 3), but certainly with topological ones). Consider a weak external field hext(r, t) acting on the links of polymer chains at t > 0 (it is assumed that at t < 0 the system was at equilibrium). The linear response of the system can be represented as follows: x3 It is well-known that the volume interactions in a semidilute solution generally cannot be accounted for by a mean-field approach [13]. Nevertheless, a mean-field approximation is valid for scales which are larger than the static correlation length 5,. In the following the term “volume interactions” is attributed just to these large-scale effects.
A.N.
Semenov
(Sc(r, t)) = -
I Polymer density fluctuations in concentrated
1d7
d3r’ K(O)@- r’, r) h,,t(r’, t - r)/T
solutions
,
269
(2.10)
where K(O) is the generalized susceptibility and (. . -) denotes averaging over an ensemble. After the Fourier-Laplace transformation eq. (2.10) can be written as
(sC(%P)) = -
K
‘“‘(4YP>L(q, PIIT
(2.11)
9
where
fc(‘)( q, p) = [ dt 1 d3r
~(‘)(r,
t)
eXp(-pt
-
iq
* r)
,
etc.
(2.12)
0
As a consequence
of the fluctuation-dissipation
theorem
K(‘)(q,P) = $?(q) - Ps’“‘(%P) >
[16] we have (2.13)
where S”‘( q, p) is the Laplace image of the function Si”‘( q). Let us return to the original system (with volume interactions) and calculate its general susceptibility K( q, p). In the mean-field approximation the molecular field h(r, t) simply superposes the external field Iz,,~(T, t). Thus, eq. (2.11) remains valid if h,,, is substituted by h,,, = h,,, + h:
(sC(q> P)) = -K(“)(4,p)h,,,(q, PI/T.
(2.14)
Taking into account that (see eq. (2.9))
Ok PI) = u@c(q,PI) and neglecting fluctuations from eq. (2.14)
(i.e. a difference
between h and (Iz)#~) we find
(sC(q,P)) = -K(4, P) h,,,(q, P)/T,
(2.15)
where K-l(q,
p)]-’ + U
p) = [K(')(q,
*4 The conditions which considered in ref. [17].
are
required
(2.16)
.
in order
for
these
fluctuations
to be neglected
are
A. N. Semenov
270
i Polymer
density fluctuations
in concentrated
solutions
Note that in the “static” case (p = 0) eq. (2.16) is in accordance with Edwards’ results [ 151. Using the fluctuation theorem once more we get the correlation function for the original system:
s(q>
(2.17)
= [‘d4,0) - K(‘L P)l/P .
Eqs. (2.13), (2.16), (2.17) determine the required relation between S and SC”! Thus the problem reduces to a calculation of S(O) (or K(O)) for the system of polymer chains without volume interactions.
3. The general method of calculation of K(O) First of all, let us note that the external field h,,, provokes two principal different processes in the system. The first is the induced (by the field) reptational motion of polymer chains along their tubes (see fig. 1). The second is a deformation of the entanglement network, i.e. an elongation or compression of its parts caused by the field. The corresponding mean concentration changes (due to these processes) will be denoted by SC, and &,,, (3.1)
SC = SC, + SC,,
Reptational motions of different macromolecules are independent (since volume interactions are turned off). Therefore the concentration change 6c, is completely determined by the susceptibility CYof one polymer chain:
%(%
P> = -
The susceptibility reptating chain,
K,(q) = N-l
‘d9,
P> kx,(q, PI/T
>
‘d4,
cx can be related to the coherent
ntzl (exp{iq-[r,(t) - r,(O>l>),
P> = Co”(4,
P> .
(3.2)
structure factor of a single
(3.3)
where r,(t) is the position of the nth link of the chain at the moment t. Using the fluctuation theorem we get (compare with eq. (2.13))
4%
P> = K,(q)
where K(q,
p)
-
PK(q7 P)
>
is the Laplace image of K,(q)
(3.4)
A.N.
Semenov
I Polymer density fluctuations in concentrated solutions
271
The reptational motion of a macromolecule can be considered as a onedimensional diffusion along the tube axis. Let s be the curvilinear coordinate along this axis (see fig. 1). It is convenient to choose the corresponding scale so that the (mean) “length” of the macromolecule be equal to N, i.e. s,,, - s0 2: N (here S, is the coordinate of the nth link, n = 0, 1, . . . , N). During a short time At the chain can creep along the tube some distance As so that the coordinate of a link changes: s,(t + At) = s,(t) +
NsN,
D,, = W,N,IN,
(3.6)
( W, is some microscopic frequency). The longest time of conformational relaxation is of order of the time needed for a macromolecule in order to creep out of the initial tube, i.e. r,,,,, - N2/Dll. As a result of a more precise calculation one gets [lo] 7 max
= Y2N2/D,, = T-*W,‘N3/Ne
The structure corresponding
.
(3.7)
factor K,(q) was also obtained in ref. [lo]; substituting the result into eq. (3.4) we get (details can be found in ref. [ 111) N
a(q, P) = 2 o- -u
1 1 + cth u -
(3.8)
where u = Nq*a*l2 = q2R212,
cr = ( pt*)l’*
)
t* = N214D,, = (IT*/~)T,,, . (3.9)
Let us consider the second contribution, SC,,, which is due to a deformation of the entanglement network. Let us assume that h,,, CC exp(iq - r), the vector q being directed along the z-axis. The external field would induce a displacement
*5 Here we assume that the chain in the tube is intensile. elongation or compression of macromolecular segments below (as a part of the second contribution SC,,).
The faster motions corresponding to an along the tube are taken into account
A. N. Semenov
272
I Polymer density fluctuations in concentrated solutions
of the network “points” in the z-direction: r-+
r +
where l= [(z, t). The process is governed by the following dynamic equation:
(3.10)
where POis the sedimentation constant (the effective mobility of a polymer link which can depend on concentration), gz,, is the component of the stress tensor of the entanglement network. The first term in parentheses represents the force due to the external field (acting on a link), the second is the force due to elastic stresses. For the large-scale deformations (A = 2nlq 9 b) the stres a,, is proportional to the deformation E = ag/az: UzIz=E-E,
(3.11)
where E is the longitudinal elastic modulus of the entanglement network. Eq. (3.11) is valid for sufficiently short times (t 6 T,,,,,), in the general case a possibility of relaxation of entanglements must be taken into account: I
fl,z(t>=
E(t
-
t’) dc(t’) ,
or, after the Laplace transformation,
U,,(P)= E(P)~P)
(3.12)
7
where _ E(P) = P I E(f) ew(-pt)
dt
(3.13)
is the elastic modulus at the frequency o = -ip. Substituting eq. (3.12) into eq. (3.10), solving this equation and using an obvious relation 6c,, = -co&, which is a consequence of the continuity equation, we obtain
A. N. Semenov
273
I Polymer density fluctuations in concentrated solutions
%I = - ‘h(4, I’) &x,(4>P)lT
q*coT 7
Kn = p/p0 +
q2E(p)/co
.
(3.14)
Let us consider the dependence E(t), which determines the relaxation of stress after a small step-like deformation d.s. The initial value E(0) = E, obviously must be of order of the corresponding shear modulus G, = coTIN,: (3.15)
E, = QC”TIN,,
where V, is some numerical factor. This factor can be obtained theoretically, but its value is not universal and depends on a particular model of the “tube”. So V, is considered below as a phenomenological coefficient. On larger time scales the relaxation of stress is related to two processes. The first one can be described as follows. As a result of the initial deformation some parts of macromolecules appear to be stretched and some to be compressed along the tube axis. Therefore the macromolecular tension (along the tube) would not be uniform so that the macromolecules would tend to redistribute their links along the tubes in order to smooth out the tension. This, the so-called first relaxation process [lo], is characterized by the following “Rouse” time [ 181: 2-Jq4a4 ,
qR%-1, (3.16)
rR = 1 7,N2/4 ,
qR+l,
where r0 - IV,’ is the relaxation time for one link. As a result of this first stage of the relaxation the effective elastic modulus would decrease to the value
.!?= YE,, ,
(3.17)
where r”can depend on q. Using the tube model considered in ref. [lo] one can obtain s
qR+l,
8
qR-%l.
6 7
r”=
(3.18) 15 >
A calculation of E(t) during the first “Rouse” analogy with ref. [18]; the result is E(t) = J? + (E, - i)
f(thR)
,
t=srR,
stage can be performed
in
(3.19)
274
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
where
(3.20)
and
1 + z) ,
dz exp(-xz)z”2/(
xeu2, (3.21) x9u2,
p(x) = (S/T*) In eq. (3.22)
C i2 n
(3.22)
exp(-n2n2x/4).
n runs over all odd positive
numbers.
The second stage of relaxation (for t S TV) is due to reptational motions of macromolecules along their tubes. The corresponding dependence E(t) was obtained in ref. [lo] on the basis of the assumption that the stress is proportional to the share of macromolecular links which had not left the initial tube at the moment#6: E(t) = &t/t*)
t%rR,
)
(3.23)
where t* = N2/4D,, = 7r2~,,,/4. Using eqs. (3.19), (3.23) and taking
into account
(3.16),
(E, - E)cp( q4a4t/7,)
+ &(t/t*)
)
qRs1,
(E, - i)p(4thoN2)
+ s!?p(tlt*)
,
qRe1.
we get finally
(3.24)
E(t) =
Taking
into account
K(‘)( 4, p) =
KI
eqs. (3.1), +
K,,
=
cods
(3.2), PI
(3.14) +
q2coWdPo
we obtain +
q2%Wol
2
(3.25)
where E(p) is determined by eqs. (3.24), (3.13). Thus, the correlation function S,(q) can be calculated now with the help of general equations (2.16), (2.17), (3.25). The specific results obtained on the basis of these equations are considered in the following sections. x6 Another assumption that the stresses are proportional to the square of this may be more appropriate; if this assumption is so-called double reptation [19,20]) &t/t*) in eq. (3.23) should be replaced by p*(tit*). Note that this substitution does considerably the specific results mentioned in the following sections (except that an substitution must be performed in eq. (5.9)).
share (the used, then not affect analogous
A. N. Semenov
4. The relaxation
I Polymer density fluctuations in concentrated solutions
of concentration
fluctuations
275
with q S R-’
Using the results of previous sections we obtain after some transformations (for b + q-l 4 R)
S,(q)= ~-‘L%gc(~) + A.&x(t) + 4ep&&)l
(4.1)
)
where the three terms in square brackets correspond to three relaxation stages, cooperative (the fastest), “Rouse”-like and reptational. Here A,, AR, A rept are the reduced amplitudes of the stages (A, + AR + A rept= l), and g,, g,, g rept are dimensionless functions satisfying the conditions
g,(O)= &(O) = &qm = 1 . Let us consider firstly the cooperative A, = l/(1 +
S) ,
g,(t) =
stage:
exp[-(1+ ~P,q*~l
(4.2)
,
where D, = Tc,uj3, = PO aIll$,
is the cooperative S = E,l(c,
(4.3)
diffusion constant of macromolecules
[13]#‘,
tUI/dc,) = u,,/c,,uN,
(4.4)
is an important dimensionless parameter which is equal to the ratio of the elastic modulus of the entanglement network E, and the osmotic modulus K = c0 IFIII/&z,. Usually S is small, 6 5 0.1; an exceptional case of semidilute theta-solution at low concentrations (where 6 b 1) is not considered below. Thus A, is close to unity, i.e. the lion’s share of the initial fluctuation relaxes at the cooperative stage, the relaxation being governed by the simple diffusion law with effective diffusion constant (1 + 6)D, 2: D,. The physical meaning of these results is quite simple: Here the relaxation is due to a deformation of the entanglement network. The corresponding mechanical stresses are rather weak since the elastic modulus E,, is small in comparison with the osmotic one, K. Consequently, the relaxation process must be very similar to that for a system of unconnected polymer links (i.e. for a low-molecular solution). *’ Note that & is an effective sedimentation
constant which can depend on concentration.
276
A. N. Semenov
I Polymer
At t P 7c = l/[( 1 + 6)D,q2] stage characterized by
density fluctuations
the relaxation
A.=E,I(E,+K)-El(~+K)2:S-6”,
in concentrated
solutions
passes to the second “Rouse”
grc(t)= (P(t/TFA 3
(4.5)
where s” _6IK = 63, rR = r0/q4a4 and the function cp is determined by eq. (3.21). Note that the amplitude A, is very small: A, = 6/6+ 1. The qualitative meaning of the result is the following: the rate of smoothing of the initial inhomogeneity (of concentration fluctuation) is primarily governed by the osmotic modulus K which is “counteracted” by the elastic modulus of the entangelement network, E. During the “Rouse” stage this modulus effectively slightly decreases from E, to i = FE0 giving a possibility of some reduction of the residual fluctuation (which remains after the cooperative stage). The amplitude of the last, reptational stage, A
rept
=
i?l(i + K)
2:
s”,
(4.6)
is smaller than A,, but higher than AR. The dependence essentially determined by two large parameters u and r: u = q2R212,
r=c,rx,
of grept on t is
(4.7)
where 77s G/(1 + s”)
(4.8)
(note that r + u due to inequalities
(2.4)). If r -G u’, then
(P(t/Tl)
t-et”,
7
&&
= i (2ulf)
exp(-m’tlt*)
(4.9) ,
tat*,
where 71 =
(Uq2t*
.
(4.10)
Thus, grept- 1 for t S T~; in the region or =St =St” the function decreases according to the power law grept 0~t-“2, and the exponential decrease is predicted at t b t*. If I’ % u”, the function g ,,,t(t) is characterized by the following asymptotics:
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
(1 - 2u*/r) exp(-t/r*) f3rep
+ 2(~*/r)(p(tlr~)
,
277
t+t*,
=
(4.11) i (2ulT) exp(-n2tlt*)
,
tBt*,
where r2
2t*
=
/r
7,
)
=
t*
/4u2
(4.12)
.
Thus, in this regime an almost purely exponential relaxation with the characteristic time r2 is changed (at t P T*) by a long-lived “tail” with a small relative amplitude of order u*/lY Note that the longest relaxation time is always of order t* m N3. Nevertheless, the relaxation may be crudely characterized by a shorter time r,,rt corresponding to a decrease of grept from 1 to, say, l/e. Eqs. (4.9)-(4.12) suggest that (Lgg4t*,
bsq-‘s5*,
t*/r
,$*sq-kR,
7rept ‘-
(4.13) )
where (*
=
r-l’*R
2:
(bR)“2
= aN;‘4N”4
(4.14)
(it is taken into account that $4 1 and r = r”v,NIN, 2: N/N,). Consequently, for q 9 1 /t* the time crept varies according to an anomalous law: r,,rt m q4; whereas for longer waves, l/t* B q B 1 lR, the time tends to a constant value 7
rept = 2t* /r - (N2/D,,)(N,IN)
- W;‘N2 )
(4.15)
which is comparable with the maximum “Rouse” time. The characteristic scale ,$* dividing these two regimes is of order of the geometrical mean of tube diameter and of macromolecular size. Let us consider the physical meaning of the obtained results (for the reptational stage). Note that during the first two stages of relaxation the initial concentration fluctuation 6c(r, Oj would decrease by a factor of u/u”= (1 + s”) /g. Therefore, the molecular field h(r) would also decrease (see eq. (2.9)). This can be alternatively represented as an effective decrease of the parameter of volume interactions, u, by the same factor u/u”, i.e. as a renormalization u + u”. This renormalization effectively accounts for the coupling between a deformation of the entanglement network and the reptational motion of polymer chains. Thus we can consider the reptational stage assuming
A. N. Semenov
278
I Polymer density fluctuations in concentrated solutions
that the network is undeformable; the omitted effects would be taken into account at the very end by the simple substitution u + u”in all final equations. Let h(r, 0) be the molecular field corresponding to the initial concentration fluctuation 6c(r, 0) via eq. (2.9). Consider a macromolecule with end links r0 and rN (see fig. 2). Under the influence of the molecular field the macromolecule would creep along the tube (created by other polymer chains) toward that end, where the potential h is lower, since in this case the macromolecular potential energy would decrease. The velocity of the motion, As/At, must satisfy the following relation: (As/At) = D,,[h(r,,
t) - h(r,,
(4.16)
t)]/T
(compare with eqs. (3.5)). Thus, if h(r,) < h(rN), then the macromolecular links would “flow” into vicinity of the point r0 and raise the local concentration there. The corresponding rate of the concentration change is
[(h(rNJ t>) - h(r,,
h(r,, t>, where 2c,lN taken second h(r,, Using
is the mean concentration into account that the mean end, (h(r,, small almost uncorrelated. h(r,, t) (4.17) and we get
t) =
Fig.
(4.17)
2. The links
0) exp(-2c,uD,,tlN)
move the tube be accumulated
ends. In eq. (4.17) of the molecular field (for qR % 1) the quantities
.
towards
the end
rO if point r,,.
< h(r,).
Thus,
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
279
After the substitution u 4 u”we obtain the relaxation time r = N/2c,v”D,, which coincides with r,,rt = r2 (see eq. (4.15)). So far we neglect random motions of a polymer chain along the tube axis. The characteristic value of the random displacement, As during the relaxation time r is (see eq. (3.5)) As - (D,, ~)l’~ - (N/c,@)“* .
(4.19)
This displacement can be neglected only if the spatial size of the polymer chain segment consisting of As links is smaller than the characteristic scale q-l, i.e. (4.20)
(As)“~u 4 1 /q . Substituting 1 /q
eq. (4.19) into (4.20) we get
P a(Nlc,v”)“4
= [*
.
(4.21)
Note that this inequality coincides with the condition of applicability of eqs. (4.11), (4.12) (r P u’). Ob viously eq. (4.18) is in accordance with eqs. (4.11), (4.12) apart from small corrections of order of u*/r. The considered relaxation process is characterized by an important feature: the initial inhomogeneity Sc(r, 0) relaxes primarily due to a redistribution of the end polymer links or of the links which are close to the ends. As a result the fluctuation of the total concentration decreases considerably during the time of order crept = N/2c,v”D,, . Nevertheless the system remains in an essentially nonequilibrium state since the spatial distributions of the end and the middle polymer links appear to be considerably different. This difference relaxes much more slowly, during the time of order t* % T,,+ giving rise to the long-lived “tail” which was mentioned above (see eq. (4.11)). Let us pass to a more short-wave region, q-l -+ a(Nl~,u”)“~ 2: t*. Here we meet an unusual situation: random shifts of a macromolecule along its tube slow down the relaxation rather than accelerate it. In fact, these random movements tend to smooth out the difference between the distributions of end links and of middle links, which has appeared as a result of the “rapid” relaxation of the total concentration fluctuation (during the time T,,~~). Consequently, these movements hinder the relaxation. Obviously, this effect becomes more pronounced as the scale of inhomogeneity q-’ is decreased (in comparison with the scale of random motions -a(As)‘12 - a(D,, ~)l’~). As a result we have an anomalous dependence of the characteristic time r,,+ on the wavelength 2rrlq: rEeptincreases as q-l is decreased. One may perform a more
A. N. Semenov
280
/ Polymer density fluctuations in concentrated solutions
detailed qualitative analysis and obtain the law: r,,rt CCq4 for q-l G &*, in accordance with eq. (4.13). The above consideration suggests the following interpretation of the “dynamical” length 5 * : This length is of order of the lower limit of the (average) displacement of a macromolecular end during the characteristic relaxation time crept. Some comments about the relaxation of composition in polymer mixtures which was considered in refs. [6-81. Let us consider a symmetrical (NA = NB = N; uA = a, = a; cA = cg = c,/2) and incompressible (c, + cg = const) mixture of homopolymers A and B. Obviously, the cooperative stage of the relaxation which is connected with a “longitudinal” deformation of the entanglement network, is forbidden for such a system (since a longitudinal deformation contradicts the incompressibility)#‘. On the other hand, the reptational stage of the relaxation must be qualitatively the same for a mixture and for a solution#‘. Thus, the results obtained in this section for the reptational stage can be compared with the corresponding results of refs. [7,8]. The relaxation time rs of fluctuations with wave-vector q (see eq. (4.1) of ref. [7]) in our designations and in the region limited by inequalities (2.4) can be represented as rs = const(l\rlc,uD,,)l[l
- exp(-q2Na2)]
,
(4.22)
where const is some numerical factor. This result is in accordance with our eq. (4.13) which is valid in the region q s 1 /t*; nevertheless in a more short-wave region b =Sq-l < [* a coincidence is lacking. This discrepancy is connected with the fact that the relaxation of a (harmonical) inhomogeneity was assumed in ref. [7] to be purely exponential (see section 1) whereas actually the relaxation is essentially nonexponential (see eqs. (4.9)). It is for this reason that the dynamical length [* had not been obtained in ref. [7]. Note also that the method used in ref. [7] cannot “catch” the long-lived tails with characteristic time of order t*. Using eqs. (4.9), (4.11) we obtain the behavior of the concentration fluctuation at the longest times t %-t*: the amplitude 6c decreases as 6c
K
exp(-yt)
,
tst*,
(4.23)
*’ Even if a longitudinal deformation would be allowed, it could not result in a change of the composition since the A and B polymer chains form a common entanglement network. H The only difference of minor importance is that one should assume that r = c,uN rather than r = c,u”N in eqs. (4.9)-(4.12).
A. N. Semenov
I Polymer
density fluctuations
in concentrated
solutions
281
where b+q-‘+R,
y = r2/t* - W,N,IN3 , i.e. the relaxation The relaxation was investigated regime and under y - c,uD,,lN
(4.24)
rate is much slower here than at shorter (of composition) in the mixtures on the theoretically in ref. [8]. The result (for the conditions (2.4)) can be represented = c,uW,,N,lN2
times. longest time-scales the entanglement as
bGq-‘
,
(4.25)
in contradiction with eq. (4.24). I comment that the result of ref. [8] (eq. (4.25)) is incorrect due to two independent reasons. Firstly, a one to one correspondence (at a fixed q) was assumed [8] between the current of polymer links, J, and the gradient of the chemical potential, Vp (a relation of type (1.2)). However, such a relation is lacking even in the region t P t* (J depends not only on Vp but also on conformational distribution of macromolecules). Secondly, an incorrect expression for the coherent structure factor of a single reptating chain, K,(q) was used in ref. [8] (in the region t S t*, qR 4 1): K,(
4) m exp(-q*D,t)
D, - R2/t* ,
,
instead of the correct one [lo]: K,(q) a exp(-n2t/4t*)
.
5. The results for the long-wave
fluctuations,
qR G 1
Let us consider a relaxation of fluctuations (or, equivalently, correlation function) for q-l P R. Let us start with the region R G q-’ <
= X, ,
the dynamical
(5.1)
where the relaxation is characterized by three stages as before (see eq. (4.1)). The characteristics of the first, cooperative stage remain the same (see eqs. (4.2), (4.3)). The amplitude and the relaxation law for the second, “Rouse” stage are A,=-
Eli E,+K
E -y-2:--_ E+K
’
&x(t) = l-4%)
2
(5.2)
282
A. N. Semenov
where
s”= F76= AS,
teristic
time
rR = r,,N2/4
of the “Rouse”
time for an isolated The amplitude sponding
I Polymer density Jfuctuations in concentrated solutions
Rouse
(see eqs.
relaxation
A rept = EI(K
stage#“,
,
+ (ru))“‘+(rutlt*)
exp(-m2tlt*) r is defined
Thus,
the charac-
with the maximum + E)
-
s”;
the corre-
law is
exp(-rut/t*)
where
(3.16)). coincides
coil.
of the reptational
relaxational
(3.18),
roughly
by eq. (4.7)
+ $
exp(-20.2tlt*),
tG t* ,
(5.3a)
t%- t* ,
(5.3b)
and
a
n-‘-f(epzx -
q(x) =
epx)( 1 -
z)-~z~‘~dz
(5.4)
.
0
The second term in eq. (5.3a) is small since TU %-1. Thus, reptational stage) is initially governed by the exponential (5.3a)) with characteristic time 7
rep
t*lTu.
=
the relaxation (at the law (the first term in
(5.5)
The exponential relaxation is changed by a long-lived “tail” (the second term in (5.3a)), the “tail” being “cut off” exponentially at t b t*. Note that eq. (5.5) is in accordance with the result of Pincus, eq. (4.22). The time r,,rt can be represented as crept = 1/D,,,,q2, where D
rest
=
TR212t*
is much smaller
(5.6)
QCN-'
than the cooperative
diffusion
constant
D, = Tc,u~, m No but is
much larger than the self-diffusion constant, D, = R2/2t* m Ne2. The physical meaning of these results can be clarified as follows. The (reptational) relaxation in this regime is governed by the translational diffusion of macromolecules (since the scale l/q is larger than R). If the volume interactions were absent, the rate of the relaxation would be controlled by the self-diffusion constant D,. The volume interactions increase the rate of the process by a factor of order Fi,,lFi, - couN, where Fin, and Fid are the characteristic values of interactional and ideal-gas contributions to the free energy of the system. Thus we obtain the effective diffusion constant Drept c,uN. D,. *lo This stage
was considered
previously
[21].
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
283
Let us consider now a more long-wave region, e q-l e X2 = N”‘X,
X, = N,“‘Nu
.
(5.7)
Note that the characteristic scale A, is of order of the (total) length of the tube axis (which can be represented as N/N, steps of the length b = UN:“). The cooperative and “Rouse” stages proceed in this regime as in the previous case provided that q-l G Nu. If on the contrary q -I 9 Nu, then the time of the cooperative stage, T,, appears to be longer than the “Rouse” time TV. Thus, the “Rouse” stage superposes the cooperative stage so that the former becomes almost indistinguishable against a background of the latter:
The characteristics A
rept
=
of the reptational
EI(K + i) = 5,
stage are
‘!Lept(4 = /-4tlt*) f
(5.9)
The dependence grept( t ) can be qualitatively understood in the following way: a complete relaxation cannot be achieved at the cooperative stage because of the elasticity of the entanglement network; in time the effective elastic modulus decreases proportionally to p(tlt*) making possible more and more complete relaxation. Let us compare the obtained results with the results of ref. [14]. Note that a (rather specific) case was considered in this reference: it was assumed that elastic modulus of the entanglement network, E,, exceeds the bulk modulus K (E,IK B 1, i.e. cOuNe 4 1). On the other hand, eqs. (5.8), (5.9) were obtained in the opposite limiting case cOuN, % 1. Nevertheless the general approach [14] can be easily applied to the case cOuN, % 1 so that the following result may be obtained (in our designations):
S,(q)=u?
4 -l
(
&
=sxx,.
0
exP(- Qq2t) + $$-
exp(-v2t/4t*) 0
, 1
(5.10)
Eq. (5.10) is in qualitative agreement with our results for the long-wave region (5.7), see eqs. (5.8), (5.9), (4.2). On the other hand, for shorter waves, the results of ref. [14] become incorrect: in this region 4 -’ G X, = NuIN:“,
284
A N. Semenov
! Polymer density fluctuations in concentrated solutions
direct reptational motions of polymer chains (which were not taken into account in ref. [14]) play an important role for the relaxation process#“. In the most long-wave region, q-l %- X,, the cooperative relaxation time becomes longer than the reptation time: rc = l/ D,q2 P t*. Thus, the topological constraints (entanglements) do not affect appreciably the relaxation process. Consequently the relaxation must be exponential: S,(q)-u-’
exp(- D,q2t) ,
q-l % X2 = N3’2N,1’2u
.
(5.11)
6. Conclusions and comparison with experiments The main results of previous sections can be summarized as follows: (1) A relaxation of concentration fluctuations in a semidilute or concentrated polymer solution proceeds in three stages with considerably different times rc, provided that the scale of inhomogeneity ‘R and ‘rept h=q-‘
Fig. 3. A schematic (double logarithmic) plot of cooperative (T,), “Rouse” (TV) and reptational microscopic (T,.,,) relaxation times vs. the scale of inhomogeneity X E 4-l; 7Ois the characteristic time.
#I’ Note that this type of motion implies the directed tube axis under influence of the molecular field.
reptation
of a macromolecule
along
the
A. N. Semenov
I Polymer density fluctuations in concentrated solutions
285
according to the law rC = X “/DC. The time TV0: X 4 at X c R, and it tends to a constant value rR ccN2 in the region R 6 11 s Nu. The third time 7,ept is of order of the longest reptation time, T,,, 0: N3, if X - b (b is the tube diameter); in the region b =SX s (Rb)“’ the time 7,,pt decreases (as X increases) according to an anomalous law: T,,~~0: x -4; then (in the region (Rb)“2 s 11 s R) it tends to a plateau value 7,,pt 0: N2; in the region R 5 X i N,1’2Na it increases, 7rept o( NA ‘9 and finally it again tends to another constant of order of T,,,~~CCN3 for the most long waves, X b N,“2Na. (2) As a rule the amplitude of the first, cooperative stage, A,, is the largest, and the amplitude of the second, “Rouse” stage is the smallest. (The case of a theta solution near the dilute-semidilute transition region is an exception: here A, may be smaller than A,_.) (3) The relaxation on the reptational stage is not single exponential: it is characterized by a spectrum of times between T,,,,~ (see eq. (4.13)) and 7maxCCN3. A long-lived “tail” of the relaxation usually appears as a consequence, the “tail” being cut off at t - T,,,~~. (4) The rate of the (reptational) relaxation is determined primarily by the time 7maXand by the dimensionless parameter r (see eq. (4.7)) which can be represented as r = (QJE + Lri,JK)-l, where II,,, = c,TIN is the ideal-gas part of the osmotic pressure, K is the bulk osmotic modulus and ,?? is the elastic modulus of the entanglement network. Some comments concerning the experimental results obtained in refs. [l-5] using the methods of dynamical scattering of light and neutrons for qR b 1. Semidilute solutions of polystyrene in good [l, 21 and theta [3-51 solvents were studied. In all cases a number of relaxation modes were revealed, the fastest mode (f) being always of cooperative nature: 7;’ = D,q2, where the diffusion constant D, almost does not depend on the molecular weight of the polymer, M. A second, slow mode (s) with the inverse time T,’ = D,q2 was also revealed [ 11, the corresponding diffusion constant being weakly M-dependent. Another law for the slow mode, T,’ 0~q”Mp3.’ was also reported [3]. Three modes: fast, intermediate (with the inverse time 7,’ which almost does not depend on q) and slow (Q-,’ = A + Bq2, where A and B are constants) were observed in other experiments [2]. A somewhat different behavior of the inverse times was reported in refs. [4,5]: T,’ 2: D,q2, where D, weakly depends on M for qR31 and D,,,ccM-“, n = 0.5 to 0.6, for qR < 1; 7,’ CCq”Mey, y = 3 to 3.8. Thus, the experimental data are not in a complete agreement with each other. Doubtless that there are a lot of relaxation modes, and this observation is obviously in agreement with the theoretical predictions of the paper. Moreover, the fastest mode can be identified with the cooperative stage which has been considered above. The relative amplitude of this stage is (see eq. (4.2)) A, = K/(E, + K). Taking into account that (for a good semidilute
A.N.
286
Semenov
I Polymer
0
density fluctuations
I
I
I
I
2
4
6
8
in concentrated
Ib
solutions
c(% 1
Fig. 4. The dependence of the relative amplitude of the fastest mode, 2, - A,/(A, + A,) on the concentration c (in weight %) of PS solution. The solid line corresponds to the theory (eq. (6.1), K = 0.95), the circles to experimental data [l].
polymer solution) the osmotic modulus K CCc2.25 [6] and the elastic modulus of the entanglement network, E, 0: c2 [22], we get A, = ~c’.~?(l
+ ~.(7’.~~),
K =
const .
(6.1)
The theoretical dependence (6.1) is compared with experimental data (concerning the amplitude of the fastest mode) [l] in fig. 4. A more complete verification of the theory requires experimental data for the largest possible N/N,, i.e. for concentrated solutions of ultra-high molecular weight polymers.
Acknowledgment
I am grateful to I.Ya. Erukhimovitch
for valuable discussions.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12]
W. Brown, Macromolecules 18 (1985) 1713. P. Stepanek, J. Jakes, C. Konak, R. Johnsen and W. Brown, Polym. Bull. 18 (1987) 175. M. Adam and M. Delsanti, Macromolecules 18 (1985) 1760. W. Brown, Macromolecules 19 (1986) 387, 3006. W. Brown, R. Johnsen, P. Stepanek and J. Jakes, Macromolecules 21 (1988) 2859. P.G. de Gennes, J. Chem. Phys. 72 (1980) 4756. P. Pincus, J. Chem. Phys. 75 (1981) 1996. K. Binder, J. Chem. Phys. 79 (1983) 6327. P.G. de Gennes, J. Chem. Phys. 55 (1971) 572. M. Doi and S.F. Edwards, J. Chem. Sot. Faraday Trans. II 74 (1978) 1789, 1802,.1818. I.Ya. Erukhimovitch and A.N. Semenov, Sov. Phys. JETP 63 (1986) 149. W.W. Graessley, Adv. Polym. Sci. 16 (1974) 1.
A. N. Semenov
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I Polymer density fluctuations in concentrated solutions
287
P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1979). F. Brochard, J. Phys. (Paris) 44 (1983) 39. S.F. Edwards, Proc. Phys. Sot. London 88 (1966) 265. L.D. Landau and E.M. Lifshitz, Statistical Physics, part 1 (Pergamon, Oxford, 1980). A.N. Semenov, Physica A, submitted. P.G. de Gennes, J. Phys. (Paris) 42 (1981) 735. M. Rubinstein, E. Helfand and D.S. Pearson, Macromolecules 20 (1987) 822. J. des Cloizeaux, Europhys. Lett. 5 (1988) 437. A.N. Semenov, Sov. Phys. JETP 63 (1986) 717. M. Adam and M. Delsanti, J. Phys. (Paris) 43 (1982) 549.