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Theory of the dynamics of tagged chains in interacting polymer liquids: general theory
T. Vilgis, U. Genz
To cite this version:
T. Vilgis, U. Genz. Theory of the dynamics of tagged chains in interacting polymer liquids: general theory. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1411-1425. �10.1051/jp1:1994196�.
�jpa-00247001�
Classification Physics Abstracts
05.20D 05.40 61.25H
Theory of the dynamics of tagged chains in interacting polymer liquids: general theory
T.A- Vilgis and U. Genz
Max-Planck-Institut für Polymerforschung, Postfach 3148, D-55021 Mainz, Germany (Received 24 February 1994, revised là May 1994, accepted 30 June 1994)
Abstract. The effective Rouse dynamics of a surgie polymer chain embedded in
an interact- ing polymer liquid, such as excluded volume melts, blends, or copolymer melts is studied. This
paper suggests an alternative formulation for the problem of the dynamics ofinteracting polymer liquids. This theory uses trie Martin-Siggia-Rose functional formahsm, instead of Mori-Zwanzig
projection operators. The theory is developed in different steps. The first approximation is an effective single chain approximation, that leads to a renormalized Rouse model. Taking into account ail interactions from other chains and performmg a preaveraging over the static and
dynamic structure factors leads to an effective friction function acting on the monomers of the tagged chain. The friction function is determined by ail interactions present in the system.
1. Introduction and motivation.
Static and dynamic properties of individual polymer chains in polymer melts, blends and block copolymers is in general an unsolved problem. The static problem seems to be quite simple
and has been studied previously for vanous situations: melts [1], polymer blends [2] and block
copolymers [3, 7-9]. The one component melt is relatively simple since the conformation of individual chains is ruled by excluded volume interactions and the chain connectivity only. In the case of polymer blends, the situation is more complicated. The underlying phase transi- tion of trie density fluctuations has a dramatic effect on the chain conformation of individual chains. Individual chains in a blend of two homopolyrners show a tendency to shrink when
approaching the phase transition and their radius of gyration reduces up to 15Sl. This has been demonstrated by numerical simulations [10]. In block copolymers, the situation is even
more comphcated. At the rnicrophase separation, the mean field collective structure factor derived first by Leibler [4] diverges at a finite wave vector k*, and not in the small k limit as in the case of a polymer blend. When fluctuations are taken into account it has been shown first
on a general basis by Brazovskii [5] that the divergence is removed and the two point density density correlation function remains finite. This theory has been apphed to block copolymers
[fil where it was shown exphcitly that the structure factor for block copolymer melts does not
diverge. The physical reason for this fundamental change between mean field theory and renormalized theory is that fluctuations drive the second order phase transition into a weakly
first order transition. Thus correlation lengths, structure factors, etc. remain finite.
The net effect on the static properties of a single chain is that individual blocks in the block
copolymer melt shrink, whereas, because of the strong repulsion between the two blocks, the different blocks are widely separated. This has until present times been found only in numerical
simulations [8] for high density (incompressible system) and for systems including vacancies [8, 9], 1e. compressible systems. In both cases the chain stretching effect has been seen. The
experimental situation is not clear at present. Whether this effect can be seen clearly in real
experiments remains to be demonstrated.
The calculation technique for the static single chain problem is very simple: on the basis of the Edwards Hamiltonian, the partition function for ail chains can be formulated. When
integrating over ail chains except chain 1, an effective single chain Hamiltonian is obtained.
It consists of two parts: the first contribution is due to the chain connectivity which is rep- resented by a Wiener term in the continuous description of the chain. The second term is an effective potent1al consisting of the excluded volume potential of chain 1 itself and the effect due to fluctuations and interactions with the surrounding medium. This effective Hamiltoman
depends on the positions R(s) of the segment s on chain 1 and can be expressed as
N ~ N N
H((R(s))) = ~~)/Î / ds
(~(~~~) + kBT /
ds / ds' U~R(R(s) Ris')) il-1)
2 s
N is the chain length and l~ is the mean squared distance between segments. The influence of the environment on the tagged chain is included in the effective potential UeR(R). Most remarkable is that in the case of polymer mixtures the spatial Fourier transform U~R(k) of this effective potential contains a singular part that changes sign before the phase separation occurs
[2]. This means that the effective potential acting on the chain becomes attractive before phase
separation, hence the radius of gyration of the chains shrinks. Such effects have been found numerically by Sanban and Binder [loi and very recently by light scattenng expenments il Ii.
It has also been shown that the effective potential calculated by reference [2] is responsible for
a weak locaJization of the chains when the size of the cntical droplets is of the order of the radius of gyration [12].
Motivated by the simple static case similar considerations should be possible for the dynamic behavior of tagged chains m interacting polymeric liquids. Unfortunately the dynamic problem
is more comphcated. Let us first summarize a few expenmental results. It is well known from neutron scattenng expenments [13] that in a dense polymer melt, interacting via the excluded volume, (below the cntical molecular weight from where reptation starts) the dynamics of an
individual chain can be described by the Rouse equation,
Ifôt~ ~~(~Î ôS~~) Ris, t) = fis, t) , (1.2)
where ( is the segmental friction coefficient and fis, t) is a random force with the correlation function
(f(s, t) fis', t'))
= 6kBT à(s s') à(t t') (1.3)
Equation (1.3) is the fluctuation dissipation theorem.
On the other hand, equation (1.2) is vahd exactly only for the dynamics of single, non- interacting random walk chains which are self-similar on ail scales. For this ideal system, the
Hamiltonian [14] is given by
N ~
~° ~ÎÎÎ/~~ ~ÎÎ~~
'
~~'~~
o
where N is the total counter length of the random walk. In the absence of interactions, the friction coefficient ( is the bare friction constant (o which is determined by the heat bath
govemed by the random force fis, t).
In the presence of interactions, it has been shown very often that the friction coefficient (o
is renormalized by the effect of interactions. The simplest model shows an additional term
[15~17]
t " to + j Idi (F(il F1°11 11.51
o
where Fit) is the force on a segment at time t due to excluded volume interactions. The above equation is not at ail as trivial as it looks. The first term represents the bare friction coefficient (o which stems from the heat bath. In Langevin dynamics this contribution can be thought to
have its physical reasons in local random forces. The second contribution, 1-e- the explicitly
written out force force correlation function is due to forces which cannot be assigned to the local fluctuations. Equation (1.5) can be intuitively understood in a coarse grained model.
The first term is due to stochastic processes within a lower cut off scale whereas the second term descnbes the stochastic forces at scales larger than the cut off. In polymer dynamics
such a widely used coarse grained model is the Rouse model [14] and the lower cut off length corresponds roughly to the (static) Kuhn segment length. This point will be discussed later in
more detail.
The aim of this paper is to find a field theoretical way to study the dynamics of interacting polymeric hquids. The dynarnics of a tagged chain in such a multicomponent interacting hquid
can be studied by various methods. The classical one is the Mori Zwanzig projection operator technique. This method was developed by Mon and Zwanzig [18]. Two types of variables are
distinguished: slow and fast variables. Trie fast variables are then projected out and bave an effect on transport quantities such as trie friction coefficient. This very successful standard technique provides a good basis for many problems in trie physics of liquids. Trie result is a
generalized Langevin equation (GLE). This method bas been used to study one component polymeric liquids, where trie chains are interacting by excluded volume forces by Schweizer [19- 22]. In these basic papers it was shown that excluded volume forces lead to several additional effects, including a possible explanation for the Rouse to reptation transition, that is well known
to occur for long chains [14]. In section 2, this method will be outlined and generalized to a
system containing several species, where in contrast to Schweizer we aim for the investigation of trie dynamics of tagged chains near (critical) fluctuations of a phase transition. Trie latter
generahzation is necessary if trie dynamical behavior near a theromdynamic phase transition
is studied.
One intuitively diflicult problem when applying Mon Zwanzig projection operator techniques
to treat a polymer melt is the following: a clear separation between time scales is not possible.
Ail time cales conceming the polymer molecules are essent1al since they contribute to the transport coefficients such as diffusion constants or viscosity. This does not at ail mean that the results derived by the Mon Zwanzig method are unphysical. In fact, the opposite is the
case as the Mon Zwanzig formalism denves a number of formally exact results, which are, however, sometimes diflicult to banale exphcitly without further assumptions. In section 3
therefore, another alternative method will be introduced which does not use this somewhat artificial separation of time scales by using a functional method. An exact dynamical generating
function is constructed such that correlation and response functions can be calculated simply by differentiation. This functional method based on the dynamical theory of Martin, Siggia
and Rose (MSR) [23] provides a framework which is analogous to trie statistical mechanics
developed for the static case [1-3, 7, 8]. Ànalogously to the Mori Zwanzig method the exact functionals cannot be used explicitly and several approximations have to be employed when
specific systems are discussed. One purpose of this paper is to present a dynamic functional for the behavior of the tagged chain analogously to the static case, where the effective potential
contains the information of the environment. However, for the simplest case, we expect a dynamic functional with a friction function that has most of the information of the environment,
i-e- the other chains in the system and their thermodynamic behavior. Another purpose of the present study is to provide an alternative formalism that uses functionals, for the study of the dynamics of tagged chains in interacting polymer fluids, which will be applied to more specific problems in subsequent investigations. Detailed applications of trie general result obtained in this paper to polymer blends and block copolymer melts are presented separateiy [25]. It
shouid be mentioned that the MSR technique has been appiied to the probiem of collective
dynamics of poiymers by Heifand et ai. [24]. These authors are interested in the dynamics of collective properties of concentrated poiymer solutions, whereas the present study is concemed
by a functional formulation of the tagged chain dynamics in interacting (criticai) polymer systems.
2. Discrete polymer mortels and projection operator techniques.
In a recent serres of publications [19-22], Schweizer addressed the problem of describing polymer dynamics in terms of microscopic chain properties. This avoids an a priori assumption on a
chain reptating in a tube. We present a generaiisation to a system containing severai types of poiymer species. Starting from the Liouvilie equation, Schweizer [19] empioyed Mon.Zwanzig
projection operator techniques [18] to derive an exact equation of motion for trie phase-space
distribution function fi((Ris), (Pis)) of a particuiar 'probe' chain under consideration.
This distribution function depends on the positions (Ris and the momenta (Pis) of ail Ni segments on this chain, which are indicated by trie indices s
= 1,.
,
Ni Trie mathematicai
procedure is niceiy presented in trie appendix of Ref. [19]. As a resuit of projection operator formalism, the time evoiution of fi is expressed in terms of a 'frequency matrix', which is local in time, and a memory contribution due to trie simuitaneous time deveiopment of the
surrounding 'matrix' polymers. The latter ieads to non-Markovian behaviour. In addition, a 'random force' term is present. From the equation for the time evolution of the distribution
fi ((Ris Ii) ), (Pis Ii) )), coupled equations for trie individual chain segments bave been denved.
~vithin a coarse-grained description for trie polymer chains, rapidly decaying contributions to the memory function are taken into account by a friction coefficient ii of a segment on the probe chain. In addition to the probe chain, the system contains several species a having va
polymers. Correlations of intermolecular forces F~, which act between segment s on the probe
chain and the matrix, decay on the time-scale of density fluctuations and have to be taken into account explicitly. For a Gaussian chain, the result can be expressed as
j~/
~~'~~i~'(~'~~) j'~l~~~'~ ~ ~l (2iÉ18 iÉl,S+1 iÉl,S-1) Ù~(~) (~'~)
~ 0
where ci
= 3kBT/1) is entropic spring constant in this bead spring model. The generalized
friction function (~,~> (t,t') is obtained as
i~,~,ji, i') = iiôjs si) à(i i')
+ £ (Fis expjoi£jt t')j Fis>
=
Îjs s') ôjt t') + Aj~,~>il, t') j2.2)
Fis gives the force on segment s resulting from intermolecular interactions with matrix poly-
mers. £ is the Liouville operator, and the projector Qi is orthogonal to the projection on the
probe polymer distribution function fi fi represents the random force and (...) refers to the
equilibrium average. The essential difference between equation (2,1) and the Rouse equation il.2) is the time dependence of the generalized friction function. If the second term in equation (2.2) vanishes, equation (2.1) reduces to the Rouse equation (1.2). The essent1al problem is
the evaluation of the generalized friction function, and a major difliculty is the treatment of the projected time evolution. Schweizer addressed this problem in references [19, 20]. Here,
we take a more conventional point of view and factorize trie four-point correlation function in equation (2.2) into a product of twc-point correlation functions while replacing trie pro-
jected time evolution operator by trie fuit Liouvilhan. This time of approximation is frequently employed when using a mode-coupling approximation [18]. It bas also been considered by
Schweizer [20, 22].
In a multicomponent melt, the force acting on polymer 1 due to the presence of the sur-
rounding matrix may be expressed in a Fourier presentation as v~
Fis Ii)
=
~ ~j ~j ~jV/~ exp [ik (Ris(t) Ris<Ii))] +
c-c-
,
(2.3)
2
, ~
where c-c- denotes the complex conjugate. In the case of excluded volume interaction, the
constants V?a are excluded volume parameters specifying the interaction between a segment
on chain and a segment of species a. The use of such simple (pseudo) potent1als is a simpli- fication. For microscopic descriptions, realistic potent1als, such as Lennard Jones potentials or
other models, should be employed. For the purpose of the present paper the pseudo potential approximation is suflicient. When inserting equation (2.3) into the memory function, the ad- ditional contribution to the friction function due to interactions with the polymer matrix can be written as
A(ss, ii,t')
=
~j ~j V/~V(~ (exp[ik RisÎPi exp[Qi £t] exp [iq Ris, P(
,
(2.4)
aT kq
~i>here pi is the Fourier transform of the partial density related to species a,
§ fiÎ~
PÎ " ~ ~ ~~P Î~~ iÉÎs(1)1 (~.5)
1 S
In equation (2.5), the summation is performed over ail segments belonging to species a. R$ ii)
gives trie position of segment s on chain at time t. Trie sum of four-point correlation functions in equation (2A) is factonzed in a product of two point correlation functions containing either
the matrix coordinates or the coordinates of the tagged chain. Simultaneously, the projected
time evolution operator is replaced by the fui] Liouville operator. This leads to
Î~Îss'Il>Î~) " ~ ~ Î~Î~Î~~~ (~XP Î~~ iÉls(~)Î ~XP Î~~' iÉls'(~')Î) (PÎ(Î)p~(Î~)) (2.6)
aT kq
Because the polymer system is isotropic, only terms q = -k contribute to the summation in
equation (2.6), which then simplifies to
Aiss>Ii,t')
=
~ fl V/~V/~ (exPlik lRis(t) Ris> (t'llll (P[(t)P[(t')1 (2.71
It can be seen that the dynamics of the tagged partiale is influenced by the dynamical properties of ail species. The above equation is closely related to the result given by Schweizer [19, 20], 1-e- the renormalized Rouse mortel. Trie difference to Schweizer's result is that here trie projected
propagators are replaced by trie unprojected ones. This possibility has already pointed out in reference [19].
À mode couphng approximation forms an alternative way to evaluate the additional, time-
dependent friction. It leads to more specific predictions on the dependence of the friction function on the segment indices s and s'. Whereas this additional step is very useful for the
explicit evaluation of the memory terms, it does not help to clarify the relation between trie
Martin-Siggia-Rose approach presented in trie following section.
3. Martin-Siggia-Rose formalism.
The aim of this section is the denvation of a generalized Langevin equation describing one tagged chain interacting with other chains. This linear Langevin equation will contain the influence of polymers interacting with polymer 1 via equilibrium properties of the surrounding polymers and will have the form of equation (2.1). So the procedure forms an alternative to trie Mon-Zwanzig projection operator formahsm. We start from u coupled, nonhnear, time local Langevin equations for u polymers including polymer 1. In a continuum description of trie chain, these equations describe trie time evolution of trie vectors R~(s,t) specifying trie
position of segment s on chain at time t,
l~ ~2
~~ ai ~~ ôs~ ~~~~'~~ ~~~~'~~ ~ ~~~'~~ ~ ~'' '~ ~~'~~
(~ is trie microscopic friction constant of a segment on chain1. In the overdamped case consid- ered here, the frictional force, which is trie first term on trie left nana side of equation (3.1), is
matched by ail other forces acting on segment s of chain 1. Three types of forces are present:
Very fast variables, which are not taken into account explicitly in this description, grue use
to random forces ((s,t), which are related to the microscopic friction coefficients (~ by the fluctuation dissipation theorem,
(i~(s, ii i~(s', il))
= 6k~T i~ô~~ à(s s'j à(i il)
,
(3.2)
where ô~j is the Kronecker symbol and à(z) is the Dirac à-function. The second term on the left hand side of equation (3.1) represents the entropic spring forces between subsequent
segments on chain 1. Àdditional excluded volume interactions between segments on the same chain and on different chains j #1 are included in F~(s, t). If this term vanishes, equation (3.1) reduces to the Rouse equation, equation (1.2). Because F~(s,t) depends on ail distances
(R~(s, t) Rj (s',t)) between segment s on chain to other segments s' on polymers j = 1,. u,
