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The spinodal decomposition process of polymer films
B. Forrest, D. Heermann
To cite this version:
B. Forrest, D. Heermann. The spinodal decomposition process of polymer films. Journal de Physique II, EDP Sciences, 1991, 1 (8), pp.909-919. �10.1051/jp2:1991101�. �jpa-00247563�
Classification Physics Abstracts
64 60M 05 70
The spinodal decomposition process of polymer fihns
B. M Forrest and D. W. Heerlnann
Institut fur theoretische Physik and Interdisztphnhres Zentrum fur wissenschaftliches Rechnen, Umversitat Heidelberg, 6900 Heidelberg, Germany
(Received 25 February 1991, accepted l6May1991)
Abstract. We investigate the decomposition of a polymer blend film by Monte Carlo
simulations For this we have calculated the coexistence curve of polymers with length
N
=
10 monomers and present estimates of the exponents charactensing the growth of the typical
domains The scaling properties of the decomposition process is also investigated.
1. Introduction.
Two-dimensional systems are special. This is especially true for polymer systems. Due to the topological constraint, a single polymer chain can only interact with a limited number of other chains, unlike in higher dimensions, where chains can interpenetrate. Most of the contnbution to the energy comes from the intramolecular interaction and less from the intermolecular interaction This is also the basic argument why the spinodal decomposition process of a
polymer blend does not exhibit the initial exponential growth m the structure factor that is
observed for systems with fairly high molecular weight and for systems with long-range
interactions in three dimensions.
The study of the decomposition in two-dimensional systems should lead to more insight into the decoInposition process itself. Such a process for a polymer blend can be initiated by
changing the temperature rapidly Assume that the coemstence curve of the system looks like the one shown in figure I. A quench starts with the system prepared in a high temperature
state, where the distnbution of the polymer blend is homogeneous. There is no correlation among the chains. The second step consists of a rapid drop in the temperature such that the parameters concentration and temperature correspond to a state m the miscibility gap. This is
schematically shown in the figure I. The system, if quenched into the so-called spmodal
region, is then unstable against infinitesimal fluctuations and decomposes into regtons which
are nch m the polymer species A and those which are rich m B. Eventually the system has
decomposed and equilibrium is re-established
This bnngs up an 1mnlediate problem. where is the coexistence curve located? This is necessary to ensure that the quench is made into the spmodal regton and not into the metastable regton where states persist which are not the stable equihbnurn states. States in the metastable regton can have very long life-times and may not be clearly distinguishable
T i
i '
j
- -
-~-
-~-
SPI NODAL REGION
c
Fig I Genenc phase diagram for a polymer blend The outer curve shows the coexistence line, the
inner line indicates the gradual transition between the metastable and the spmodal region The arrow
shows how the system is brought from the stable state into the two-phase region.
from truly unstable states This inforlnation is also needed to estimate the supersaturation and establish how this vanishes.
In this paper we present data obtained from Monte Carlo simulations, both for the
coexistence curve of two-dimensional polymer blends and on the decomposition process. We focus on the time evolution of the structure factor, which measures how the charactenstic
domains evolve and coarsen and on the pair-distnbution function.
2. Notes on the shnulafion.
The program for the Monte Carlo simulation [1, 2, 3] of the polymer blend has been
previously described m [4]. It suffices here to say that we parallehzed the Monte Carlo process
by decomposing the underlying lattice into overlapping stnps. We can thus operate upon
some stnps simultaneously to obtain an almost linear seed-up This allows us to simulate large
lattices m a reasonable amount of computer and wall-clock time.
The simulation technique consisted m representing each polymer of N monomers by a self- avoiding walk of length N I on a two-dimensional square lattice The data presented below
were obtained from simulations on a 336x 336 lattice and for polymers of length
N
= 10 Since we were interested m simulating dense systems our system contained less than 5 fb empty lattice sites (vacancies) we employed the reptation method [5, 6] where the
vacancies on the lattice are regarded as the dynamic vanables A vacancy is picked out at
random and then one of its four neighbours is chosen with equal probability. If this
corresponds to an end monomer of one of the chains, the corresponding polymer is « pulled »
by the vacancy, which hops to the other end of the chain. The polymer has thus m effect slithered along its length, displacing itself by one lattice unit. The move is accepted or rejected according to the usual Metropolis cntenon. Note that we assign an interaction energy of
s or 0 between two neighbounng monomers on the lattice if they are of opposite or the same species respectively. Choosing e > 0 ensures that monomers of opposite species will tend to
repel each other. Of course, not every update of a vacancy will necessanly result m an actual update of a polymer. In fact, we have found that this occurs with a probability of roughly I/4,
i e., on average we need to update four vacancies to obtain one chain update.
Another variant of a simulation algonthm would be a generalized Kawasaki dynamics. In the Kawasaki dynamics for spin systems [7], neighbounng spins (or, equivalently, occupied
and unoccupied sites) are exchanged either if the energy is lowered or, m the case of an
increase m energy, if the Boltzmann factor allows it. Here we could search for an unlike
neighbounng chain along a test chain. Both chains would swap their identities and the
proposed exchange would be accepted or rejected on the basis of the associated energy change.
Other lattice algonthms [8] such as kink-jump dynamics can only be performed at much lower densities our system had a density of over 95 fb monomers Simulations employing such algonthrns have been performed, for example, by Sanban et al [9]
The algorithm was tested and run on the Superduster parallel computer of the
Interdisziphnares Zentrum fbr wissenschaftltches Rechnen at the University of Heidelberg
The machine has currently 128 T-800 transputers. The parallel program [10] was wntten m
OCCAM.
3. Estimates of the coexistence curve.
Predicting or computing the coexistence line between the gas and the liquid phase is a formidable task even for very simple systems. Analytic methods are not able to take into account the full interaction among the chains and one has to resort to approximative methods,
as for exaInple the Florylnean field theory [11] Silnulationlnethods [1, 2, 3] on the other hand allow one to calculate the coexistence of species m a systematic way.
For lattice polymers a grand-canonical enseInble method was developed [9, 12] which allows one to obtain the equilibnum concentration of polymer A and of B as a function of
temperature and of the chain length The idea is the same as the one which is employed m the simulation of simple lattice systems like the Ising model Instead of holding the concentration of the species fixed one allows a transformation from one kind into the other. A chain which is
designated as A can be transformed into one labeled B Such a transformation is connected with an energy change. If the overall energy is lowered by the change we accept the move, while if it is raised we accept such a change only with a Boltzmann weight corresponding to the specified temperature The method begins with a concentration of I, i e., all of the chains
are of one species (type A, say) The system is then allowed to equilibrate at a specified temperature, after which the corresponding equilibration concentration is determined. By repeating this process at vanous temperatures we can obtain an estimate for the coexistence
curve
The resulting phase diagram is shown m figure 2. This plot shows clear finite-size effects m the location of the coexistence line This is of course especially true close to the transition
temperature Note that the transition temperature is fairly close to the one for the two-
dimensional Ising model ~
= l/2 In (1 + /) [13] This
is because the chain length kB Tc
which we have chosen is fairly small, so that the shift m temperature, if we consider the
equivalent neighbour model [14], is small compared to the mean-field cntical temperature.
In the temperature region where we have performed our quench to, we do not see large finite size effects m the location of the coexistence curve. There may however be finite-size effects because eventually m the decomposition process there will be modes comparable with
the system size. Our simulations cannot be extended up to times where such modes become very important. We are limited to the investigation of the initial stages of the decomposition
process. Nonetheless from figure 2 we can be sure that our quench to T
= I (temperature will
always be given in units of e/kB) corresponds to a state well inside the two-phase region (deep quench)
c a.
o
«
#
c
w u
g O
. o
uench
o
Fig.
L~ = (circles ) and L~ = 336~ squares).
Thetemperature is given m umts of the coupling
The line shows w the ystem
is brought from the stable state into the two-phase regionThe cross marks the parameter
combination for the uench
4. The sbucture factor.
The coarsening of the system from an initial homogeneous state can be seen most prominently
in figure 3. We have shown in this figure snap-shots of the system. The first picture shows the initial state where the system was prepared at T= oJ. The second picture is after 20000 Monte Carlo steps and the last one at 10~. Only one type of chain is shown. Note that we are
measunng time in units of Monte Carlo sweeps after one such sweep all the chains in the system will have been updated on average once.
Initially there are no domains m the system which would give rise to correlations and structure. As time evolves and the system is brought into the two-phase region more and more structure begins to appear in the form of domains. These domains show a very homogeneous
concentration profile. There are only exceptional domains present where an A polymer is
inside of a domain of B polymers and vice versa.
The coarsening of the domains can be quantified by considenng the structure factor which
is the Founer transform of the pair correlation function G (r, t)
=
( £ j(c(r~, t) co)(c(r~
+ r, t ) co)] (I)
S(k, t)
= £ G (r, t exp (tk r (2)
Here co = 0.5 is the average concentration. Usually one considers the sphencally averaged
structure function so that S only depends on the absolute value of the wave vector
k : S(k, t).
In figure 4 the evolution of the structure factor is displayed as a function of the wave vector k and of time t. We notice that the structure function is peaked and that the peak shifts to
smaller k values with increasing time, indicating the coarsening. The data presented were averaged over 20 statistically independent runs. Each run, performed on eight processors,
'
I
~ ~(f
m
jL-,..a
Fig 3. Evolution of the system after a quench into the two-phase region. Shown are snap-shots at three different times. The picture shows the tnitial state and the second at 20000 and the last at 10~ Monte Carlo steps Shown is only one type of polymer
£~ JOfiq$
f ~
*
#
,
~
~
l~j~
a
e
'
~
J#
i
.
~
@«
2fW
«--
4l~
Fig 3 (continued)
120
ioo -- isi
- 12t
- 8t
Bo -- 6t
- 31
~ ~- l
6Q W
0l
4o
2o
o
o
k
Fig. 4 Shown is the structure factor at different times (15t corresponds to 10~ Monte Carlo steps)
consisted of an initial equilibration phase at T
= oJ followed by the quench to T
=
I (In all 9 600 transputer hours were used, which is equivalent to roughly 500 hours on an IBM-3090
mainframe
We have mentioned in the introduction that two-dimensional polymer systeIns are special.
In two diInensions we do not a priori expect to see potentially Inean-field behaviour in the evolution of the peak of the structure factor. In the Inean field theory of spinodal decoInposition [15] it is predicted that during the initial stages of the decoInposition the peak
should not shift with tiIne and that S(k, t) should grow exponentially m time for all wave vectors smaller than a critical wave vector k~. According to the cntenon which was given by Binder [16] we should expect to see such a behaviour only in dimensions higher than two.
Indeed such a behaviour has been observed m expenments [17] as well as m simulations [18].
In our simulations we neither see that the peak is stationary nor do we observe the
exponential behaviour. This is consistent with results for various other systems, among them the Ising system [19], 4~ theories [20], and numencal integration methods of vanous
Hamiltomans [21]. However, we cannot rule out the possibility that the absence of an early-
time regime with a time-independent peak may be due to the shortness of the chains and not due to the dimensionahty
In figure 5 we show the peak position as a function of time The peak position is an
indication of the typical domain size which is proportional to kit Even for the very early
times we do not see a time independence Rather, the evolution tends to a power law
behaviour
kmax CC t ~. (3)
Such a power law behaviour would indicate a scaling behaviour of the structure factor at intermediate or later times. This is further exemplified m the typical domain size which we get from the pair distribution function G(r, t), where r denotes the distance from any fixed
2
2
# ~
E m C
2
24
2
0 2 3
In t
Fig 5. Position of the peak m the structure factor as a function of time A power law behavJour indicates a possible scaling of the structure factor
reference (cf. Fig 6). This function gives the correlation or probability of finding like atoms m
the neighbourhood of a monomer of a chain. Initially the probability of finding a like chain in the iInmediate neighbourhood is roughly one half as the chains are randoInly mixed, but as the system evolves toward equilibnum the probability is rising.
We can get another estimate of the typical domain size from the first zero in
G(r, t). Tltis is shown in figure 7 where we have plotted the position of the first zero m
G as function of time. Again the quantity is not constant in time. Rather, it indicates a power law behaviour once more. The exponent should be the same as the power n which we have
introduced for the typical wave vector
R cc tn. (4)
We expect such a behaviour only if the first two moments of the structure factor
« «
ki( t)
= kS(k, t dk/ S(k, t) dk (5)
o o
« «
k~(t)
=
k~S(k, t) dk/ S(k, t dk (6)
o o
(7) obey the necessary condition
1,e., the ratio must be time-independent. From figure 8 we see that this ratio becomes
independent only very late m the decomposition process. This is consistent with the
03
a ~
0.2 j " ~
.
" ~~
m y . 1m9
_ , m f " ~*~~
2 0,1
~ m f ° ~*~~
- g
© m o g
m . g
°
, o g
, m o j
a a m m ~ m a m, a m m m m j j j
, , f. I. j .
a , i 1
-0.1
0 5 0 5 20
r
Fig 6 Pair distnbution function at dJfferent times fouowing a quench (t =15 represents 10~ MCS)
C
21
20
2 3
In t
Fig 7. Position of the zero m the pair distnbution function as a function of time
1.3
f
# ~
~ .
i-io
s
observation that only for large t do we see a power law behaviour of k~~. We can estimate the exponent n as
n~~~ = 0.198 (9)
n~R~ = 0.21 (10)
(11)
In the case where the necessary condition for the scaling of the structure factor (time independence of the ratio of the moments) is fulfilled we expect the scaling to be
s(k, t
=
kij(t) i(k/k~~(t)) (12)
The scaling of the structure factor is shown m figure 9. Only those data scale which come from the later stages of the decomposition process. Initially the ratio of the moments is not time independent and the scaling does not work We have only touched the potential scaling
region. However, for the times for which we were able to do the simulation we have just about reached a scaling regtme.
5
m
a ~lS
4 °#f
. ~12
f~om f~
m ~9
~°
~ o id
= 3 @$° "~
. %3
~ ~ " °°
a ~l
, m , a
fl
°
~
2 o' " .~
m m
o
aa a a am
a ~
a '
I I
~o q©
" $~m°o~.
~
~ l 2 ~ ~
k/k~ (t)
Fig 9 Shown is the scaling of the structure factor This plots shows that the scaling only works in the later stages of the decomposition
5. Conclusion.
In this paper we have demonstrated that it is possible to obtain a very good estimate for the
coexistence curve for polymer blends. We have used these estimates to perform a deep
quench into the two-phase region. Our results indicate a possible power law behaviour of the
typical domain size which forms after a quench even for films We have also shown that the structure factor scales in the later stages for such polymer films
Acknowledgement.
We would like to thank the members of the Interdisziphnares Zentrurn fbr wissenschaftliches
Rechnen for their support dunng this work Funding from the Bundesministenum fur
Forschung und Technologie (BMFT) under project number 0326657D and the BAYER AG under project number 03M40284 is gratefully acknowledged.
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