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Reptation and tube renewal: Experimental and numerical simulation
Journal of Non-Newtonian Fluid Mechanics, 23 (1987) 215-228 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
215
REPTATION AND TUBE RENEWAL: EXPERIMENTAL AND NUMERICAL SIMULATION
G. MARIN,
J.P. MONTFORT
and Ph. MONGE
Universitt? de Pau et des Pays de I’Adour, L.ahorutoire de Physique des MutCriaux industriels, Institut Universituire de Recherche Scientifique, Avenue de I’Universitg, 64000 Pnu (Frunce) (Accepted
September
16, 1986)
Summary In the most recent theories, the diffusion of macromolecules in linear polymer melts is assumed to occur mainly through a reptation process. We have studied here both experimentally and numerically another possible diffusion mechanism called tube renewal. We find that, for a N chain isolated in a N,,,-matrix of the same polymer, the average tube renewal relaxation time rl( N, NnI) does not vary according to the reptation time rd( N,,,) of the matrix as given by the theories, but rather as rd( N,,,)/N,,. A numerical simulation has been used for testing the molecular models, and its results are in agreement with the above conclusion.
1. Introduction The molecular theories describe fairly well the viscoelastic properties of entangled polymer melts and concentrated solutions. However, some discrepancies remain and therefore improvements of the models may give a better description of the rheological behaviour, mainly in the terminal region of the relaxation spectrum, involving the whole motion of every macromolecule. The Doi-Edwards theory [l] describes the dynamics of entangled polymers by reptative motions of everyXlGin in a fixed tube constituted by the surrounding chains. The model predicts the zero shear viscosity no varying as the third power of the molecular weight M; experimentally, however, q, a M 34 . Doi [2] considered that the tube length could fluctuate and this assumption led to of q, very close to experimental values for samples not In the same time, various authors [3-61 suggested that could take place, like tube
216 renewal occurring by reptation of the chains constituting the tube, releasing the constraints acting on the chain imbedded in the tube. These theories forecast that tube renewal should be negligible for highly entangled monodisperse samples; it seems however that this mechanism should play an important role in polydisperse samples. This paper deals with (i) the experimental characterization of the tube renewal mechanism exhibited by the high molecular weight component in binary blends of polystyrene; and (ii) a numerical simulation of a one-dimensional diffusion of macromolecules involving reptation and constraint release. This analysis allowed us to test our interpretation of the molecular basis of tube renewal. 2. Reptation and tube renewal: theory The Doi-Edwards theory [l] represents a macromolecule of n monomers as embedded in a tube constituted by the surrounding chains. The diameter a of the tube is related to the average distance between entanglements; these entanglements are assumed to be localised every n, monomers. The real chain is modeled by a “primitive chain” of N = n/n, segments of length u. The N chain length is L = Na. The chains (real and primitive) are assumed ideal, so the mean square end-to-end distance R is equal to n’/*f or N’/*a and c1= n’,/‘l; I being the monomer length. The dynamics of such an N-chain in a melt is dominated by the mechanism of tube-disengagement or reptation. The characteristic relaxation time is
(1)
rd = L2/a2Dc, and the curvilinear
diffusion
coefficient
DC of the N-chain
along the tube is
0, = p&T/n,
(4
with pO the monomeric so n3
n3
M3
n,2
ne
Me
7da-a2a--12a-.
mobility
coefficient.
This time is related to the limiting parameters: steady state compliance .I,” through rd
=
(3) zero shear viscosity
qO and
?joJeO.
But, experimentally, q. scales as A43.4 and Js is roughly independent molecular weight so 7d varies like A43.4 and not according to relation 3.
(4) of
217 Doi [2] explained that discrepancy by considering tube length fluctuations. The length L of the primitive chain varies with time. He calculated an average of the chain contour length: L,, = Na and the fluctuation AL is (A L2) = id’. So, the largest relaxation time is: Td
=
7&(1 - N-1’q2
+Y[1_(~)1’2]
(9
For 2 < M/M, < 100, the above expression gives q a M3.4, in agreement with experimental data. In both cases (pure reptation and reptation plus tube length fluctuations) the tube is considered as fixed. The N-chain is assumed to exhibit only a sliding motion along the primitive path through the network of other entangled chains. However, the surrounding chains move also in their own tube, releasing some constraints along the N-chain. The diffusion of the surrounding chains by reptation allows the N-chain to leave its tube not only by the ends (reptation) but also sideways. This mechanism called tube renewal should be much slower than reptation for highly entangled nearly monodisperse samples. Besides, if the surrounding chains are shorter than the N-chain, tube renewal can be faster than reptation. Klein [3,4], Daoud and de Gennes [5] and Graessley [6] predicted that the characteristic time of tube renewal rl( N, N,,) of an N-chain diluted in a matrix of N,,,-chains scales as
with b = 3 (theory) or 3.4 (experimental) but the prefactors are slightly different for Klein and Graessley. Klein [4] considers that there is one efficient contact every g monomers, so n2 = n,/g constraints should be removed for every segment of the N-chain. For Graessley [6], every segment of the N-chain is related to z segments of the surrounding chains. A transverse motion of the segment is ‘possible when one of the z surrounding cells is free. So, relations 5 and 6 indicate that a crossover occurs ( rd( N) = 7,( N, IV,,,)) when
=$--(Klein) = L(g)’ .m2
(Graessley)
.
Recent diffusion measurements by Kramers et al. [7] suggest that the crossover occurs for a value close to 0.1 for expression 7. This corresponds to 2 < n2 -C 3 and 1 -Cz -C 2. Then for monodisperse samples (M = M,,) such that MB- 3M,, the tube renewal mechanism is negligible. But, for N-chains diluted in a matrix of shorter IV,,,-chains, the theories predict that tube renewal becomes dominant when Mhe2Mc2/M,: > 0.1.
218 3. Experimental (a) Samples We prepared binary blends of polystyrene whose molecular parameters are summarized in Table 1. Molecular weights of the two components were chosen in order to make tube renewal dominant or at least not negligible. The concentration $I = 0.02 g/g of the higher molecular weight N-component was below or close to the concentration &_ for overlapping of the N-chains, in order to get entanglement constraints (& - 2-3M,..M). We checked on the blend 2 700000-110000 that the variations of the relaxation time of the N-chains in the blend were of the same order of magnitude as the experimental uncertainty at concentrations below about 3-4 per cent. (b) Relaxation times We performed dynamic shear measurements to get the complex shear modulus G*(o) in the terminal and plateau regions of the relaxation spectrum. The experimental technique and the main results were reported in a previous paper [8]. We focus here in the different ways to determine an average relaxation time r(N, iv,,,) of the N-chain isolated in a IV,,,-matrix. Graessley [9] proposed to determine an average relaxation time equal to the reciprocal of the frequency w,, corresponding to the low frequency value G,;: is the weighting factor maximum of G”. The corresponding associated with the relaxation. For pure components, 0,’
- 0.45Tj0J>,
GZ - 0.28G;. The above determination was applied to polybutadiene blends for which the variations of G” exhibit two peaks. At low concentrations of the N-chains, however there is not a maximum of G” associated with the
TABLE 1 Molecular weights of polystyrene binary blends M
M”,
MM:/M;t
3MJM
390000 900000 1200000 2700000 3800000 2700000 2700000 2700000
110000 110000 110000 110000 110000 35000 2OOooo 390000
0.095 0.22 0.29 0.66 0.92 2 0.11 0.015
0.14 0.06 0.045 0.02 0.014 0.02 0.02 0.02
219
I
1
I
I
7-
/’ z3
____-----
___---
6-
6-
/
.-
.’
4-
3-
.t-
.:’
/’ .I
1’
H
___ .,.--*
/CC
.I
I/
‘I’ . T’
I’ If
l
M.2,00cl00 .
.
.**
.*
II
. L -5
I’
,I
.
2-
I
M,40000 I -3
I log
w
I I
-1
Fig. 1. Storage shear modulus G’(w) for a blend of polystyrene (M = 2700000; M,,, = 110000 and concentration +,,.,= 0.02 g/g) at 160°C. The dotted line represent the dynamic shear modulus of the pure components.
relaxation of the N-chains. Nevertheless, there is a low frequency reTaxation due to the presence of the N-chains, as shown in Figs. 1 and 2. This low frequency relaxation appears more distinctly on a Cole-Cole plot of complex viscosities (Fig. 3). We have proposed [8] to define the reciprocal of the frequency oM at the low frequency maximum of q”, as an average time of the N-chain relaxation in dilute blends (Fig. 4). The weighting factor of the low-frequency relaxation is Q, = qb - qS (Fig. 3) where TJ~ is the zero shear viscosity of the blend and qS is the zero shear
,r*_.,
c------------
.’
- - - *
-- +_., .y l/
.’ .J ./
,
./ ‘/’
./
.J
.3’
M,.110000
I
I -3
log w
Fig. 2. Loss shear modulus
G”(w)
-3
I
-’
,
I
I
for the same blend as in Fig. 1.
220
Fig. 3. Cole-Cole plot of complex viscosity of the same blend represents the complex viscosity of the pure matrix.
as in Fig. 1. The dotted
line
viscosity of the high frequency relaxation domain very close to that of the pure matrix (see the dotted line in Fig. 3). We suggested [lo] another definition of an average relaxation time for the N-chain. On Fig. 1, it is shown that, in the low frequency range, Gblend z++ GAatrix. So, we may consider the matrix as a pure viscous solvent of viscosity qs and express the complex shear modulus of the blend G,: as
G:(W) = G*(w)
+ jcqs.
(9)
G*(w) is the complex shear modulus of the N-chains We derive from the above relation:
J,ob= i qb
lim w-+0
” =L l7
Q
qb
in the blend.
2
lim
q
w-0
w
=
(10)
Jz.%, qb
where J,ob is the steady-state compliance measured for the blend and Jco is the steady-state compliance of the N-chains in the blend. Then, we define a
Fig. 4. Variations in Fig. 1.
of the imaginary
part of the complex
viscosity
$‘( w ) for the same blend as
221 relaxation time T( N, N,,) of the N-chains for pure components (relation 4) by
This definition
in the blend in the same way as
is very close to that the Watanabe
et al. [II]:
G,* is the complex shear modulus of the pure matrix and +> (- 1) is the concentration of the matrix in the blend. The relaxation time 7( N, N,,) calculated for the relaxation of the N chain diluted in the N,,,-matrix (relation 11) includes all the mechanisms: reptation plus tube length fluctuation and tube renewal. The first two are gathered in the time rd, and the tube renewal time rt is derived from 7 and 7,, assuming that reptation and tube renewal are uncorrelated. Then ,r,(N,
NJ1
=r(N,
NJ’
-Q(N)-‘.
(13-l)
On the other hand, experimental data have been extensively discussed in a previous paper [8]. The times frep and ? (relation 5, ref. 8) were the reciprocal of the frequencies w,, at the maximum of q”, respectively measured for the pure N-chains and for the N-component in the blend. So, relation 5 (ref. 8) can be rewritten as <(NY NJ’
= Q,(N,
N,,) - Q,(N).
(13-2)
(c) Results One can see in Figs. 5 and 6 that the two determinations of an average tube renewal time (relations 13-1 and 13-2) give very close values and, above all, the same molecular weight dependence. We have shown [12] that, for pure components: wM( N))’ - 0.7rd( N) and the results displayed here agree with a similar relation: 7,( N, N,,,) - 0.77,( N, N.,,). For polystyrene, at T’= 16O”‘C, the tube renewal time (relation 13-l) is given by r,( N, N,) = 2 x 10-21M’.9M;.3,
(s).
(14)
made by Watanabe et Our results seem to be confirmed by measurements al. [ll] on dilute polystyrene blends. A careful examination of their data in the range 23 000 < M, < 72400 and 472000 < M < 2 820000 gives the following result r1 = 5 x 10-%494~~6,
(s).
with values about twice ours, at the same temperature.
(15)
222
4
3
I= x
+ 2
. + I
,/,
,
5
6
7
log M
Fig. 5. Comparison of the two determinations of tube renewal time as a function of the molecular weight M of the long chains diluted in a same N,, matrix (M,, = 110000); (0) relation 13-1; (+) relation 13-2.
I
t/’
l
/
/
1
/ . +
/
I 6
I 5 log
M,
Fig. 6. Tube renewal time r,( N, N,,,) as a function of the molecular weight M,, of the matrix for a given N chain (M = 2700000). Points and crosses have the same meaning as in Fig. 5.
223 (d) Discussion For lightly entangled nearly monodisperse samples, tube renewal must affect the terminal relaxation time 7(N). So we recalculated T~(N) for the homopolymers, using relations 13 and 14 with N = N,,. We find [8]: Q(N)
= 3.3 x 10’7A43.3, (s).
(16)
confirming that tube renewal does not explain the difference between the theoretical exponent value of 3 and the experimental value of 3.4, for the molecular weight dependence of the reptation time. On the other hand, the experimental crossover between reptation and tube renewal occurring when Q(N) = T~(N, N,,) implies that: M’94~~9/M;~3
= 0.41.
(17)
For monodisperse samples, the crossover is for M/M, = 2.7 very close to the Kramers’ crossover occurring at M/M, = 3.2. We may then consider that for highly entangled monodisperse samples, tube renewal is negligible. Then the main meaning of relation 14 is that tube renewal theories are not confirmed as far as the N,,,-matrix dependence is concerned. We suggested [8] that multiple contacts between two N- and N,,-chains could explain the discrepancy. Recently, Klein [13] showed that the number of different N,-chains contributing to constraints on every blob of n,, monomers and and Klein N,,,-chains is roughly nk!‘. So, only n’,:” contacts are independent postulated that the relaxation time of every neighbouring constraints will be decreased by the same factor n,,‘I2 . This argument leads to a tube renewal time: (18)
Tt( NT N,, ) a N *% ( N,, )/NY’
which is not in agreement with experimental data (relations 14 and 15) when 7d( N,, ) a N,;34- We showed [14] however with the same argument that an N-chain is imbedded in a tube constituted by N’ = n/n’,:* different N,,,chains. So, the tube is equivalent to a Rouse N’-chain and there is one constraint released every time one N,,-chain has removed all its contacts with the N-chain. Then, the tube renewal time is given by r1 (N, N,,) a N’*rd( N,,,) a M*Mi.‘, in good agreement
with experimental
(19) data (relations
14 and 15).
4. Numerical simulation The experimental data shows that both reptation and tube renewal models give the general trends of the variations of the relaxation times as a function of the molecular weights. The exponents of the power law, however,
224 do not agree conpletely with the experimental data, indicating that these models may be oversimplified. In order to test the ideas we had about some modifications of the models to explain the experimental data, we have performed some numerical simulations. We report here the first results of this work: (a) a one-dimension simulation of pure reptation; (b) a one-dimension simulation of tube renewal; and (c) tube renewal with correlated constraints. (a) Simulation of reptation Here is a simply way to illstrate the reptation process (see Fig. 7): the tube (constituted of iV links) is symbolised by a set of N logical variables (YESNO(1) = TRUE or FALSE). Initially, the YESNO values of the tube are TRUE. Within this tube, a chain will diffuse by reptation: at regular time intervals, the chain will make a jump (either forward or backwards, randomly) of step unity. Each time the head (or the tail) of the chain go within the tube, the corresponding YESNO(1) value switches from TRUE to FALSE. The number of counts necessary to set all the YESNO values to FALSE defines the time for simple reptation Ts. Another assumption has to be made however: that is the diffusion coefficient of the chain along the tube is proportional to N-l, which means in our model that the time necessary to perform one jump is proportional to N. The relaxation time (or reptation time) of our chain will then be the average value of rd = T, x N
(20)
on a large number of simulations. The results of the simulation (log rd versus log N) gives an exponent of 2.84 (close to the theoretical exponent 3), for N varying from 5 to 200 links. (b) Simulation of tube renewal In the tube renewal process, the important function to evaluate is the average time for one constraint to disappear, the diffusion of the whole chain being a Rouse diffusion process. In order to focus on the constraint
Fig. 7. Pure reptation.
225 release process, we have performed a one-dimension simulation in the following way: We have a long N-chain (called the test chain) with N links (Fig. 8). Each link will have an entanglement point with a chain of the matrix containing N,, links. The N-chain is assumed, in that case, to be fixed. At each entanglement point, an N, chain will diffuse through a reptation process. The position on the matrix chain of the entanglement with the test chain is labeled by an integer number 1 < n < N,,,, at every entanglement point of the test chain. We will have then N such numbers, their initial values being defined randomly within the limits [l, N,,]. At regular time intervals, all the matrix chains will perform a random jump, either backwards or forward, so the concerned n value will be either decreased or increased. When at a given position, the value of n will be 0 or N,, + 1, we will say that the given constraint has been released. The number of counts necessary to free all the constraints, multiplied by N,,, (to take into account the one dimension diffusion coefficient), may define a tube renewal time. One has to stress however that this calculation can only describe the effect of the matrix N,,,-chains, the overall tube diffusing as a Rouse chain (i.e. the N length effect does not appear in that simulation). The calculations performed for N = 200 and N,,, varying from 5 to 200 links gives: 7 a N2.91, with an exponent close to the reptation exponent in agreement with tiein’s model 13741. (c) Tube renewal with correlated entanglements In the tube renewal model, all entanglements on the N-chain are assumed to be uncorrelated, i.e. each N,,,-chain is diffusing independently through a reptation process at a given entanglement point. The question we may ask is the following: is the lower experimental exponent for tube renewal (7, a N,$3 instead of Nn: or Nn:.4) due to correlated constraints? (i.e. a given N,,-chain could have several common entanglement points with the same N-chain). In order to try to anwer the question by a numerical simulation, we should first have an idea of the number of cross-overs of an N,, chain with the same
Fig. 8. Tube renewal.
226 N-chain: we have built randomly a three dimensional N-chain on a cubic grid. Then, at a given entanglement point of that chain, we have build another N, chain and counted the average number of crossovers <: the results is ; = A No.53 (21)
fir N = 2iO and N,, varying from 5 to 200 links (Fig. 9). The front factor A of eqn. (21) depends strongly on the topological model chosen here (i.e., on the flexibility of the chain), but the power law should not. This however has not been checked. We have then introduced the correlation assumption in the previous model (b), modifying it in the following way: a given N,,,-chain will have 6 common entanglements with the same N-chain. We will have p such ND,-chains entangled with the same N-chain: ,f.lnc;=pn,=N.
The < number was taken from the cross-over simulation. When a given N,,-chain (index i) makes a step forward, all the counters of the i-set (i-chain) will be increased together; when it makes a step backwards, all the counters will be decreased together. In that simulation, the tube renewal times are smaller than in the simulation without correlations (i.e. the chain is diffusing faster), but the exponent is still close to the reptation exponent (2.93). We have refined the simulation, introducing in the program the distribution of nCi given by the cross-over simulation, as
and we still obtained
an exponent
I
I
close to 3.
I
I
.
.
1.2 . /.
c” B
.
.
0.6
.
.
‘.OY 0.6 -,. b
: ’ 1.0
I “4 log N,
I 1.6
1 2.2
Fig. 9. Numerical simulation: number of common entanglements same N chain N = 200; N,,, = 5-200.
of an N,,, chain with the
227 If we assume now that all correlated constraints are released together when only one of them is released, that means that, for a given i-set (i.e. a given iN,,,-chain), all constraints will be removed together when one of them is removed. With this assumption, the exponent goes down to 1.89. As an example, we have reported on Fig. 10 the results of that simulation. If we take that picture, the tube renewal time should not depend on N,,, as the reptation time, but as the reptation time divided by N,,: 7
(NYNn,) a ‘d (Nn,)/N,n.
(24
This argument, as well as previous ideas of Klein [4,13], seems to show that the concept of entanglements as topological constraints localised every n, monomers is certainly too much simplistic and has to be revised: the average distance between entanglements is mainly a screening length, but this characteristic length would be “built” from true topological constraints whose number, distribution and correlation should be taken into account in the models. With regard to the simulations, an intermediate assumption could be made between the two last models: one may assume that to be correlated, the constraints should not only belong to the same N,,,-chain, but also be neighbouring constraints: this would be a numerical model of “loops” along the N-chain: this model should give an intermediate exponent (between 2 and 3), but should much depend on the topological model chosen, i.e. on the flexibility of the chains. 5. Conclusions The experimental part of this paper shows that pure reptation is not the only diffusion phenomenon occurring in high molecular weight polymer
Fig. 10. Numerical simulation: tube renewal with correlated constraints; constraints are removed when one them is released N = 200; N,, = 5-200.
all correlated
228 melts: the tube renewal effects have been characterized from the linear viscoelastic properties of long chains immersed in a matrix of shorter chains. The dependence of the tube renewal time on the length of the long chain agrees with the theoretical models (r a IV'). However, a smaller exponent is found ( r a N2.3 instead of IV:: theory) for the dependence on the matrix chain length. A simple numerical simulation of tube renewal has been made in order to understand this discrepancy. This simulation shows that a correlation of the constraints along the matrix chains (that can be seen as loops along these chains) may explain the lower exponent. Numerical simulation is found to be a helpful tool for testing and modifying the molecular models, and these calculations will be carried on in order to understand the diffusion of long chains in more complicated environments. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14
M. Doi and SF. Edwards, J. Chem. Sot. Farad. Trans., 2, 74 (1978) 1789. 1802, 1818. M. Doi, J. Polym. Sci., Phys. Ed., 21 (1983) 667. J. Klein, Macromol., 11 (1978) 852. J. Klein, A.S.C. Polym., Prepr. 22 (1981) 105. M. Daoud and P.G. de Gennes, J. Polym. Sci., Phys. Ed., 17 (1979) 1971. W.W. Graessley, Adv. Polym. Sci., 47 (1982) 67. P.F. Green, P.J. Mills, C.J. Palmstrom, J.W. Mayer and E. Kramer, J. Phys. Rev. Lett., 53 (1984) 2145. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 17 (1984) 1551. M.J. Struglinski and W.W. Graessley, Macromol., 18 (1985) 2630. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 19 (1986) 393. H. Watanabe, T. Sakamoto and T. Kotaka, Macromol., 18 (1985) 1436. J.P. Montfort, These d’Etat, Universite de Pau, 1984. J. Klein, Macromol., 19 (1986) 105. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 19 (1986) 1979.
215
REPTATION AND TUBE RENEWAL: EXPERIMENTAL AND NUMERICAL SIMULATION
G. MARIN,
J.P. MONTFORT
and Ph. MONGE
Universitt? de Pau et des Pays de I’Adour, L.ahorutoire de Physique des MutCriaux industriels, Institut Universituire de Recherche Scientifique, Avenue de I’Universitg, 64000 Pnu (Frunce) (Accepted
September
16, 1986)
Summary In the most recent theories, the diffusion of macromolecules in linear polymer melts is assumed to occur mainly through a reptation process. We have studied here both experimentally and numerically another possible diffusion mechanism called tube renewal. We find that, for a N chain isolated in a N,,,-matrix of the same polymer, the average tube renewal relaxation time rl( N, NnI) does not vary according to the reptation time rd( N,,,) of the matrix as given by the theories, but rather as rd( N,,,)/N,,. A numerical simulation has been used for testing the molecular models, and its results are in agreement with the above conclusion.
1. Introduction The molecular theories describe fairly well the viscoelastic properties of entangled polymer melts and concentrated solutions. However, some discrepancies remain and therefore improvements of the models may give a better description of the rheological behaviour, mainly in the terminal region of the relaxation spectrum, involving the whole motion of every macromolecule. The Doi-Edwards theory [l] describes the dynamics of entangled polymers by reptative motions of everyXlGin in a fixed tube constituted by the surrounding chains. The model predicts the zero shear viscosity no varying as the third power of the molecular weight M; experimentally, however, q, a M 34 . Doi [2] considered that the tube length could fluctuate and this assumption led to of q, very close to experimental values for samples not In the same time, various authors [3-61 suggested that could take place, like tube
216 renewal occurring by reptation of the chains constituting the tube, releasing the constraints acting on the chain imbedded in the tube. These theories forecast that tube renewal should be negligible for highly entangled monodisperse samples; it seems however that this mechanism should play an important role in polydisperse samples. This paper deals with (i) the experimental characterization of the tube renewal mechanism exhibited by the high molecular weight component in binary blends of polystyrene; and (ii) a numerical simulation of a one-dimensional diffusion of macromolecules involving reptation and constraint release. This analysis allowed us to test our interpretation of the molecular basis of tube renewal. 2. Reptation and tube renewal: theory The Doi-Edwards theory [l] represents a macromolecule of n monomers as embedded in a tube constituted by the surrounding chains. The diameter a of the tube is related to the average distance between entanglements; these entanglements are assumed to be localised every n, monomers. The real chain is modeled by a “primitive chain” of N = n/n, segments of length u. The N chain length is L = Na. The chains (real and primitive) are assumed ideal, so the mean square end-to-end distance R is equal to n’/*f or N’/*a and c1= n’,/‘l; I being the monomer length. The dynamics of such an N-chain in a melt is dominated by the mechanism of tube-disengagement or reptation. The characteristic relaxation time is
(1)
rd = L2/a2Dc, and the curvilinear
diffusion
coefficient
DC of the N-chain
along the tube is
0, = p&T/n,
(4
with pO the monomeric so n3
n3
M3
n,2
ne
Me
7da-a2a--12a-.
mobility
coefficient.
This time is related to the limiting parameters: steady state compliance .I,” through rd
=
(3) zero shear viscosity
qO and
?joJeO.
But, experimentally, q. scales as A43.4 and Js is roughly independent molecular weight so 7d varies like A43.4 and not according to relation 3.
(4) of
217 Doi [2] explained that discrepancy by considering tube length fluctuations. The length L of the primitive chain varies with time. He calculated an average of the chain contour length: L,, = Na and the fluctuation AL is (A L2) = id’. So, the largest relaxation time is: Td
=
7&(1 - N-1’q2
+Y[1_(~)1’2]
(9
For 2 < M/M, < 100, the above expression gives q a M3.4, in agreement with experimental data. In both cases (pure reptation and reptation plus tube length fluctuations) the tube is considered as fixed. The N-chain is assumed to exhibit only a sliding motion along the primitive path through the network of other entangled chains. However, the surrounding chains move also in their own tube, releasing some constraints along the N-chain. The diffusion of the surrounding chains by reptation allows the N-chain to leave its tube not only by the ends (reptation) but also sideways. This mechanism called tube renewal should be much slower than reptation for highly entangled nearly monodisperse samples. Besides, if the surrounding chains are shorter than the N-chain, tube renewal can be faster than reptation. Klein [3,4], Daoud and de Gennes [5] and Graessley [6] predicted that the characteristic time of tube renewal rl( N, N,,) of an N-chain diluted in a matrix of N,,,-chains scales as
with b = 3 (theory) or 3.4 (experimental) but the prefactors are slightly different for Klein and Graessley. Klein [4] considers that there is one efficient contact every g monomers, so n2 = n,/g constraints should be removed for every segment of the N-chain. For Graessley [6], every segment of the N-chain is related to z segments of the surrounding chains. A transverse motion of the segment is ‘possible when one of the z surrounding cells is free. So, relations 5 and 6 indicate that a crossover occurs ( rd( N) = 7,( N, IV,,,)) when
=$--(Klein) = L(g)’ .m2
(Graessley)
.
Recent diffusion measurements by Kramers et al. [7] suggest that the crossover occurs for a value close to 0.1 for expression 7. This corresponds to 2 < n2 -C 3 and 1 -Cz -C 2. Then for monodisperse samples (M = M,,) such that MB- 3M,, the tube renewal mechanism is negligible. But, for N-chains diluted in a matrix of shorter IV,,,-chains, the theories predict that tube renewal becomes dominant when Mhe2Mc2/M,: > 0.1.
218 3. Experimental (a) Samples We prepared binary blends of polystyrene whose molecular parameters are summarized in Table 1. Molecular weights of the two components were chosen in order to make tube renewal dominant or at least not negligible. The concentration $I = 0.02 g/g of the higher molecular weight N-component was below or close to the concentration &_ for overlapping of the N-chains, in order to get entanglement constraints (& - 2-3M,..M). We checked on the blend 2 700000-110000 that the variations of the relaxation time of the N-chains in the blend were of the same order of magnitude as the experimental uncertainty at concentrations below about 3-4 per cent. (b) Relaxation times We performed dynamic shear measurements to get the complex shear modulus G*(o) in the terminal and plateau regions of the relaxation spectrum. The experimental technique and the main results were reported in a previous paper [8]. We focus here in the different ways to determine an average relaxation time r(N, iv,,,) of the N-chain isolated in a IV,,,-matrix. Graessley [9] proposed to determine an average relaxation time equal to the reciprocal of the frequency w,, corresponding to the low frequency value G,;: is the weighting factor maximum of G”. The corresponding associated with the relaxation. For pure components, 0,’
- 0.45Tj0J>,
GZ - 0.28G;. The above determination was applied to polybutadiene blends for which the variations of G” exhibit two peaks. At low concentrations of the N-chains, however there is not a maximum of G” associated with the
TABLE 1 Molecular weights of polystyrene binary blends M
M”,
MM:/M;t
3MJM
390000 900000 1200000 2700000 3800000 2700000 2700000 2700000
110000 110000 110000 110000 110000 35000 2OOooo 390000
0.095 0.22 0.29 0.66 0.92 2 0.11 0.015
0.14 0.06 0.045 0.02 0.014 0.02 0.02 0.02
219
I
1
I
I
7-
/’ z3
____-----
___---
6-
6-
/
.-
.’
4-
3-
.t-
.:’
/’ .I
1’
H
___ .,.--*
/CC
.I
I/
‘I’ . T’
I’ If
l
M.2,00cl00 .
.
.**
.*
II
. L -5
I’
,I
.
2-
I
M,40000 I -3
I log
w
I I
-1
Fig. 1. Storage shear modulus G’(w) for a blend of polystyrene (M = 2700000; M,,, = 110000 and concentration +,,.,= 0.02 g/g) at 160°C. The dotted line represent the dynamic shear modulus of the pure components.
relaxation of the N-chains. Nevertheless, there is a low frequency reTaxation due to the presence of the N-chains, as shown in Figs. 1 and 2. This low frequency relaxation appears more distinctly on a Cole-Cole plot of complex viscosities (Fig. 3). We have proposed [8] to define the reciprocal of the frequency oM at the low frequency maximum of q”, as an average time of the N-chain relaxation in dilute blends (Fig. 4). The weighting factor of the low-frequency relaxation is Q, = qb - qS (Fig. 3) where TJ~ is the zero shear viscosity of the blend and qS is the zero shear
,r*_.,
c------------
.’
- - - *
-- +_., .y l/
.’ .J ./
,
./ ‘/’
./
.J
.3’
M,.110000
I
I -3
log w
Fig. 2. Loss shear modulus
G”(w)
-3
I
-’
,
I
I
for the same blend as in Fig. 1.
220
Fig. 3. Cole-Cole plot of complex viscosity of the same blend represents the complex viscosity of the pure matrix.
as in Fig. 1. The dotted
line
viscosity of the high frequency relaxation domain very close to that of the pure matrix (see the dotted line in Fig. 3). We suggested [lo] another definition of an average relaxation time for the N-chain. On Fig. 1, it is shown that, in the low frequency range, Gblend z++ GAatrix. So, we may consider the matrix as a pure viscous solvent of viscosity qs and express the complex shear modulus of the blend G,: as
G:(W) = G*(w)
+ jcqs.
(9)
G*(w) is the complex shear modulus of the N-chains We derive from the above relation:
J,ob= i qb
lim w-+0
” =L l7
Q
qb
in the blend.
2
lim
q
w-0
w
=
(10)
Jz.%, qb
where J,ob is the steady-state compliance measured for the blend and Jco is the steady-state compliance of the N-chains in the blend. Then, we define a
Fig. 4. Variations in Fig. 1.
of the imaginary
part of the complex
viscosity
$‘( w ) for the same blend as
221 relaxation time T( N, N,,) of the N-chains for pure components (relation 4) by
This definition
in the blend in the same way as
is very close to that the Watanabe
et al. [II]:
G,* is the complex shear modulus of the pure matrix and +> (- 1) is the concentration of the matrix in the blend. The relaxation time 7( N, N,,) calculated for the relaxation of the N chain diluted in the N,,,-matrix (relation 11) includes all the mechanisms: reptation plus tube length fluctuation and tube renewal. The first two are gathered in the time rd, and the tube renewal time rt is derived from 7 and 7,, assuming that reptation and tube renewal are uncorrelated. Then ,r,(N,
NJ1
=r(N,
NJ’
-Q(N)-‘.
(13-l)
On the other hand, experimental data have been extensively discussed in a previous paper [8]. The times frep and ? (relation 5, ref. 8) were the reciprocal of the frequencies w,, at the maximum of q”, respectively measured for the pure N-chains and for the N-component in the blend. So, relation 5 (ref. 8) can be rewritten as <(NY NJ’
= Q,(N,
N,,) - Q,(N).
(13-2)
(c) Results One can see in Figs. 5 and 6 that the two determinations of an average tube renewal time (relations 13-1 and 13-2) give very close values and, above all, the same molecular weight dependence. We have shown [12] that, for pure components: wM( N))’ - 0.7rd( N) and the results displayed here agree with a similar relation: 7,( N, N,,,) - 0.77,( N, N.,,). For polystyrene, at T’= 16O”‘C, the tube renewal time (relation 13-l) is given by r,( N, N,) = 2 x 10-21M’.9M;.3,
(s).
(14)
made by Watanabe et Our results seem to be confirmed by measurements al. [ll] on dilute polystyrene blends. A careful examination of their data in the range 23 000 < M, < 72400 and 472000 < M < 2 820000 gives the following result r1 = 5 x 10-%494~~6,
(s).
with values about twice ours, at the same temperature.
(15)
222
4
3
I= x
+ 2
. + I
,/,
,
5
6
7
log M
Fig. 5. Comparison of the two determinations of tube renewal time as a function of the molecular weight M of the long chains diluted in a same N,, matrix (M,, = 110000); (0) relation 13-1; (+) relation 13-2.
I
t/’
l
/
/
1
/ . +
/
I 6
I 5 log
M,
Fig. 6. Tube renewal time r,( N, N,,,) as a function of the molecular weight M,, of the matrix for a given N chain (M = 2700000). Points and crosses have the same meaning as in Fig. 5.
223 (d) Discussion For lightly entangled nearly monodisperse samples, tube renewal must affect the terminal relaxation time 7(N). So we recalculated T~(N) for the homopolymers, using relations 13 and 14 with N = N,,. We find [8]: Q(N)
= 3.3 x 10’7A43.3, (s).
(16)
confirming that tube renewal does not explain the difference between the theoretical exponent value of 3 and the experimental value of 3.4, for the molecular weight dependence of the reptation time. On the other hand, the experimental crossover between reptation and tube renewal occurring when Q(N) = T~(N, N,,) implies that: M’94~~9/M;~3
= 0.41.
(17)
For monodisperse samples, the crossover is for M/M, = 2.7 very close to the Kramers’ crossover occurring at M/M, = 3.2. We may then consider that for highly entangled monodisperse samples, tube renewal is negligible. Then the main meaning of relation 14 is that tube renewal theories are not confirmed as far as the N,,,-matrix dependence is concerned. We suggested [8] that multiple contacts between two N- and N,,-chains could explain the discrepancy. Recently, Klein [13] showed that the number of different N,-chains contributing to constraints on every blob of n,, monomers and and Klein N,,,-chains is roughly nk!‘. So, only n’,:” contacts are independent postulated that the relaxation time of every neighbouring constraints will be decreased by the same factor n,,‘I2 . This argument leads to a tube renewal time: (18)
Tt( NT N,, ) a N *% ( N,, )/NY’
which is not in agreement with experimental data (relations 14 and 15) when 7d( N,, ) a N,;34- We showed [14] however with the same argument that an N-chain is imbedded in a tube constituted by N’ = n/n’,:* different N,,,chains. So, the tube is equivalent to a Rouse N’-chain and there is one constraint released every time one N,,-chain has removed all its contacts with the N-chain. Then, the tube renewal time is given by r1 (N, N,,) a N’*rd( N,,,) a M*Mi.‘, in good agreement
with experimental
(19) data (relations
14 and 15).
4. Numerical simulation The experimental data shows that both reptation and tube renewal models give the general trends of the variations of the relaxation times as a function of the molecular weights. The exponents of the power law, however,
224 do not agree conpletely with the experimental data, indicating that these models may be oversimplified. In order to test the ideas we had about some modifications of the models to explain the experimental data, we have performed some numerical simulations. We report here the first results of this work: (a) a one-dimension simulation of pure reptation; (b) a one-dimension simulation of tube renewal; and (c) tube renewal with correlated constraints. (a) Simulation of reptation Here is a simply way to illstrate the reptation process (see Fig. 7): the tube (constituted of iV links) is symbolised by a set of N logical variables (YESNO(1) = TRUE or FALSE). Initially, the YESNO values of the tube are TRUE. Within this tube, a chain will diffuse by reptation: at regular time intervals, the chain will make a jump (either forward or backwards, randomly) of step unity. Each time the head (or the tail) of the chain go within the tube, the corresponding YESNO(1) value switches from TRUE to FALSE. The number of counts necessary to set all the YESNO values to FALSE defines the time for simple reptation Ts. Another assumption has to be made however: that is the diffusion coefficient of the chain along the tube is proportional to N-l, which means in our model that the time necessary to perform one jump is proportional to N. The relaxation time (or reptation time) of our chain will then be the average value of rd = T, x N
(20)
on a large number of simulations. The results of the simulation (log rd versus log N) gives an exponent of 2.84 (close to the theoretical exponent 3), for N varying from 5 to 200 links. (b) Simulation of tube renewal In the tube renewal process, the important function to evaluate is the average time for one constraint to disappear, the diffusion of the whole chain being a Rouse diffusion process. In order to focus on the constraint
Fig. 7. Pure reptation.
225 release process, we have performed a one-dimension simulation in the following way: We have a long N-chain (called the test chain) with N links (Fig. 8). Each link will have an entanglement point with a chain of the matrix containing N,, links. The N-chain is assumed, in that case, to be fixed. At each entanglement point, an N, chain will diffuse through a reptation process. The position on the matrix chain of the entanglement with the test chain is labeled by an integer number 1 < n < N,,,, at every entanglement point of the test chain. We will have then N such numbers, their initial values being defined randomly within the limits [l, N,,]. At regular time intervals, all the matrix chains will perform a random jump, either backwards or forward, so the concerned n value will be either decreased or increased. When at a given position, the value of n will be 0 or N,, + 1, we will say that the given constraint has been released. The number of counts necessary to free all the constraints, multiplied by N,,, (to take into account the one dimension diffusion coefficient), may define a tube renewal time. One has to stress however that this calculation can only describe the effect of the matrix N,,,-chains, the overall tube diffusing as a Rouse chain (i.e. the N length effect does not appear in that simulation). The calculations performed for N = 200 and N,,, varying from 5 to 200 links gives: 7 a N2.91, with an exponent close to the reptation exponent in agreement with tiein’s model 13741. (c) Tube renewal with correlated entanglements In the tube renewal model, all entanglements on the N-chain are assumed to be uncorrelated, i.e. each N,,,-chain is diffusing independently through a reptation process at a given entanglement point. The question we may ask is the following: is the lower experimental exponent for tube renewal (7, a N,$3 instead of Nn: or Nn:.4) due to correlated constraints? (i.e. a given N,,-chain could have several common entanglement points with the same N-chain). In order to try to anwer the question by a numerical simulation, we should first have an idea of the number of cross-overs of an N,, chain with the same
Fig. 8. Tube renewal.
226 N-chain: we have built randomly a three dimensional N-chain on a cubic grid. Then, at a given entanglement point of that chain, we have build another N, chain and counted the average number of crossovers <: the results is ; = A No.53 (21)
fir N = 2iO and N,, varying from 5 to 200 links (Fig. 9). The front factor A of eqn. (21) depends strongly on the topological model chosen here (i.e., on the flexibility of the chain), but the power law should not. This however has not been checked. We have then introduced the correlation assumption in the previous model (b), modifying it in the following way: a given N,,,-chain will have 6 common entanglements with the same N-chain. We will have p such ND,-chains entangled with the same N-chain: ,f.lnc;=pn,=N.
The < number was taken from the cross-over simulation. When a given N,,-chain (index i) makes a step forward, all the counters of the i-set (i-chain) will be increased together; when it makes a step backwards, all the counters will be decreased together. In that simulation, the tube renewal times are smaller than in the simulation without correlations (i.e. the chain is diffusing faster), but the exponent is still close to the reptation exponent (2.93). We have refined the simulation, introducing in the program the distribution of nCi given by the cross-over simulation, as
and we still obtained
an exponent
I
I
close to 3.
I
I
.
.
1.2 . /.
c” B
.
.
0.6
.
.
‘.OY 0.6 -,. b
: ’ 1.0
I “4 log N,
I 1.6
1 2.2
Fig. 9. Numerical simulation: number of common entanglements same N chain N = 200; N,,, = 5-200.
of an N,,, chain with the
227 If we assume now that all correlated constraints are released together when only one of them is released, that means that, for a given i-set (i.e. a given iN,,,-chain), all constraints will be removed together when one of them is removed. With this assumption, the exponent goes down to 1.89. As an example, we have reported on Fig. 10 the results of that simulation. If we take that picture, the tube renewal time should not depend on N,,, as the reptation time, but as the reptation time divided by N,,: 7
(NYNn,) a ‘d (Nn,)/N,n.
(24
This argument, as well as previous ideas of Klein [4,13], seems to show that the concept of entanglements as topological constraints localised every n, monomers is certainly too much simplistic and has to be revised: the average distance between entanglements is mainly a screening length, but this characteristic length would be “built” from true topological constraints whose number, distribution and correlation should be taken into account in the models. With regard to the simulations, an intermediate assumption could be made between the two last models: one may assume that to be correlated, the constraints should not only belong to the same N,,,-chain, but also be neighbouring constraints: this would be a numerical model of “loops” along the N-chain: this model should give an intermediate exponent (between 2 and 3), but should much depend on the topological model chosen, i.e. on the flexibility of the chains. 5. Conclusions The experimental part of this paper shows that pure reptation is not the only diffusion phenomenon occurring in high molecular weight polymer
Fig. 10. Numerical simulation: tube renewal with correlated constraints; constraints are removed when one them is released N = 200; N,, = 5-200.
all correlated
228 melts: the tube renewal effects have been characterized from the linear viscoelastic properties of long chains immersed in a matrix of shorter chains. The dependence of the tube renewal time on the length of the long chain agrees with the theoretical models (r a IV'). However, a smaller exponent is found ( r a N2.3 instead of IV:: theory) for the dependence on the matrix chain length. A simple numerical simulation of tube renewal has been made in order to understand this discrepancy. This simulation shows that a correlation of the constraints along the matrix chains (that can be seen as loops along these chains) may explain the lower exponent. Numerical simulation is found to be a helpful tool for testing and modifying the molecular models, and these calculations will be carried on in order to understand the diffusion of long chains in more complicated environments. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14
M. Doi and SF. Edwards, J. Chem. Sot. Farad. Trans., 2, 74 (1978) 1789. 1802, 1818. M. Doi, J. Polym. Sci., Phys. Ed., 21 (1983) 667. J. Klein, Macromol., 11 (1978) 852. J. Klein, A.S.C. Polym., Prepr. 22 (1981) 105. M. Daoud and P.G. de Gennes, J. Polym. Sci., Phys. Ed., 17 (1979) 1971. W.W. Graessley, Adv. Polym. Sci., 47 (1982) 67. P.F. Green, P.J. Mills, C.J. Palmstrom, J.W. Mayer and E. Kramer, J. Phys. Rev. Lett., 53 (1984) 2145. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 17 (1984) 1551. M.J. Struglinski and W.W. Graessley, Macromol., 18 (1985) 2630. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 19 (1986) 393. H. Watanabe, T. Sakamoto and T. Kotaka, Macromol., 18 (1985) 1436. J.P. Montfort, These d’Etat, Universite de Pau, 1984. J. Klein, Macromol., 19 (1986) 105. J.P. Montfort, G. Marin and Ph. Monge, Macromol., 19 (1986) 1979.