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Concentrated polymer solutions in the presence of fixed obstacles
T.A. Vilgis
To cite this version:
T.A. Vilgis. Concentrated polymer solutions in the presence of fixed obstacles. Journal de Physique, 1989, 50 (21), pp.3243-3258. �10.1051/jphys:0198900500210324300�. �jpa-00211140�
Concentrated polymer solutions in the presence of fixed obstacles
T. A. Vilgis
Max-Planck-Institut für Polymerforschung, Postfach 3148, Ackermannweg 10, D-6500 Mainz,
F.R.G.
(Reçu le 24 mai 1989, accepté le 24 juillet 1989)
Résumé. 2014 On considère une solution concentrée et dense de polymères en présence d’obstacles fixes qui représentent le désordre après trempe. On montre que les fluctuations diminuent et que la longueur d’écran d’Edwards devient infinie pour une certaine concentration d’obstacles. On calcule la pression osmotique et la taille d’une chaîne dans ce système. On s’attend à une
transition de désordre, quand les chaînes s’effondrent et se localisent. Les résultats sont comparés
au cas du désordre après recuit.
Abstract. 2014 We consider a dense concentrated polymer solution in the presence of fixed obstacles, which represent the quenched disorder. It is shown that the fluctuations are reduced and that the effective Edwards screening length becomes infinite at a certain obstacle concentration. The osmotic pressure and the size of a chain are calculated for such a system. We expect a disorder transition, where the chains are collapsed and localised. The results are
compared with the case of annealed disorder (troubled solvent and troubled concentrated
solution).
Classification
Physics Abstracts
36.20 - 61.40 - 05.40
Introduction.
The shape and the behaviour of a polymer chain in the presence of fixed obstacles have found
some attention recently [1-3]. A localisation and collapse transition has been suggested, i.e.
the polymer becomes localised due to the disorder and the radius of gyration is smaller than that of the self-avoiding walk or even smaller than that of the random walk.
The results are very interesting, although the calculations suffer from crude simplifications
and approximations [2]. Exact calculations in terms of a renormalisation group theory are not simple, since the problem becomes very complex and suffers from non-stable fixpoints [4].
In this paper we present a more realistic calculation for dense systems, which, is solvable.
We consider a concentrated polymer solution in the presence of fixed obstacles. The concentrated regime emenables a mean field calculation which is exactly solvable in the Gaussian approximation for many chain systems [5]. The free energy and the osmotic pressure are calculated. We show that the free energy is reduced by the disorder, and that fluctuations are reduced dramatically by the presence of the quenched disorder. The effective
screening length, defined by the expressions of the osmotic pressure becomes infinite at a certain obstacle concentration.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500210324300
The effective Hamiltonian for such a concentrated polymer solution (or polymer melt) is
calculated. It exhibits a term which suggests strong influence of the disorder on the fluctuations.
We employ a perturbation calculation for the end-to-end distance of a typical chain in the
concentrated solution with the fixed obstacles. It is shown that the perturbation depends on
the chain length itself, which indicated a non-perturbative effect, i.e. the localisation transition.
We finally compare the results with those of the annealed disorder. In this case we also find
a reduction of the excluded volume parameter and thus a localisation at stronger disorder. It is shown that the first order perturbation term for the end-to-end distance is by a factor of
-,IN (N = chain length) bigger than in the case of annealed disorder.
Basic equations.
We consider Np polymers with interaction among themselves and with No quenched
obstacles. The polymers are described by random walks Rn (s), where s is the contour
variable, and n labels the different polymers. The polymers will experience an interaction
V(Rn(s) - Rm (s’ ) ) among themselves, and an interaction V(Rn(s) - ru, ) coming from the
fixed obstacles which are at fixed places ru. The Edwards Hamiltonian (Edwardsian) for the
system is :
where L is the contour length of the polymers and R’ = aR . The free energy is given by
as
where ( ) {ru} denotes the quenched average over the obstacle positions.
This expression for the free energy which uses the chain variables Rn (s) is not solvable in
general. In order to study the free energy of dense concentrated solution or melt we can make the following mean-field approximation for the functional Z({rU}) :
where
S°(k ) is the unperturbed structure factor. Np is the number of polymers, No is the number of
obstacles, and 1Ii is the volume of the system. Pk are the density variables for the polymers, Ck those of the obstacles.
The justification for such mean-field approximations can be found in reference [5], and they are valid whenever Np is a large number, i.e. the concentration fluctuations are small.
Thus we can write
with
We need the quenched average of Z((Ck)) over the {Ck}-variables, i.e.
and such quenched averages require in general replication, i.e.
but in the Gaussian approximation it turns out that the quenched average can be calculated
directly from equation (8). This will be shown first. Here we choose the replica formulation,
too and show that both methods will lead to the same free energy. The calculation without
using the replicas is much simpler, but we want to make the connection to reference [2],
where replica symmetry breaking in diluted systems has been discussed. We will see that this
problem does not arise in the concentrated solution case. The more general description is the replica formulation, which will lead to a problem with coupled replicas which is solvable in Gaussian approximation. Thus we formulate (Zn)
where 1 a n is the label for the replicas and and P (C k} ) is the distribution of the
quenched variables {C k}. For simplicity we assume a Gaussian distribution of the obstacle
density variables { C k } , i. e.
where X is a normalisation and Co is the mean density [2]. This assumption is certainly an oversimplification, but it will contain most of the physics as it is shown in reference [2].
Calculation of the free energy.
CALCULATION OF THE FREE ENERGY WITHOUT REPLICAS. - The free energy is given by equation (8) and (11), i.e.
Thus we calculate first
which is given by
We have to average log Z {Ck} over the Gaussian distribution of the variables Ck, which gives immediately the quenched average of the free energy
The sum over the k-values will be evaluated later. We now turn to the calculation with the
replica method.
CALCULATION OF THE FREE ENERGY WITH THE REPLICA METHOD. - Performing the quenched average with the replica method leads to
where the matrix Mk(af3) is identical to
with
and 8 af3 is the Kronecker symbol.
Wee see that we find a coupling between the replicas (a) and (/3) in (Z") which lead to
complications and nontrivial behaviour of the melt by the presence of fixed obstacles. A similar coupling between replicas has been found in reference [2] and replica symmetry
breaking has been discussed in the dilute system. In our case, i.e. the concentrated system we show that there is no replica symmetry breaking, since the free energy can be calculated
exactly in this Gaussian approximation.
The free energy which is experimentally relevant is given by :
The evaluation of (Z") requires the evaluation of the determinant of the matrix
where So is the unperturbed structure factor (see Eq. (4)). Note that this formula is in accordance with the random phase approximation result of de Gennes [6]. The matrix Mk/3 is a special case of circular matrix and has the form
with x = 1 + V - W and a = - W. The matrix is trivially circular and the determinant is
S (k)
given by [7]
and the quantity Zn> is the given by
To calculate the product we consider log Z> and replace the E by an integral, viz.
k
where f2 is thé volume. We find
Thus we find for the free energy by taking the derivative with respect to n in the limit n=0
which is identical to equation (15) if we replace the sum by the integral (see (24)). These integrals can be evaluated analytically if we use approximated forms for S°(k) (which are given by the Debye function) and short range potentials, i.e. the V and W s do not depend on
k. A good approximation for the structure factor is given by [5] :
where N is the number of Kuhn steps for one chain and C is the concentration of chains, i.e.
CN is the total number of segments per unit of volume. Rg2> is the mean square radius of
gyration of the Gaussian chain. It is given by (Ri) = Q2 N /6, where f is the Kuhn step length (see Ref. [5]). Equation (26) fits the Debye function very well and allows an analytical
evaluation of the integrals. The integrals are then of the type
in the limit of long chains, i.e. N --> 00, where factors of 1/N can be neglected.
V can be V or W.
The integrals are divergent but we use a cut off at short distances, i.e.
where A is a cutoff of the order of the inverse Kuhn length, i.e.
which is related to the shortest length scale in the problem. The free energy is then given by
or
This is of course the same formula as that we obtained by direct evaluation in the preceding
section. The first term represents the free energy of the melt or concentrated solution and if
we include the k = 0 term (see [5]) we find
Calculation of the free energy gives
In the limit xo - oo this reduces to
which shows an interesting cancellation of the last term at V = W, i.e. at a specific obstacle
concentration
or
if Vo is of the order of V. To discuss this further we calculate the osmotic pressure of the dense melt in the presence of fixed obstacles.
The osmotic pressure.
The free energy showed a cancellation of the disorder term with the excluded volume term.
Such a behaviour must be monitored in the osmotic pressure, which is a quantity that can be
measured. Therefore we consider the quantity (in analogy with Ref. [5])
where the first term in the exponential is the excluded volume interaction, i.e. the
k = 0 term (in the n-replica systems), the second term is the term of the mixed replicas, coming from the p (a) p ({3) contribution, and the denominator is the normalisation, which is chosen to be the completely free, non-interacting system, i. e. a system of chains without excluded volume interactions and without obstacles. This is a usual convention [5].
We call the last term arising from the fluctuations in equation (37) Kn and calculate
log Kn. Thus we have
and we need
The first term of equation (39) leads to the usual osmotic pressure of the concentrated solution and the last term comes from the obstacles. Thus we expect an effect of the disorder
on the osmotic pressure.
The osmotic pressure II is given by the usual thermodynamic relation
and we have for the free energy difference
The osmotic pressure is then [5]
where ncs is the contribution from the concentrated solution. It is given by [5]
where the screening length e is given by
The contribution of the concentrated solution is then the classical expression
as calculated first by Edwards [8]. The first term in (44) is van’t Hoff s law, the second term is the excluded volume effect and the third term is the contribution from the chain connectivity [5, 8]. Now we calculate the disorder part, i.e. 1Ii dW .
d03A9
Since W = C0 V 6 in a quenched number we assume that it does not depend on f2 explicitly,
i.e. aWla1Ii = 0. Therefore we are left with
since
The integral becomes then
and the osmotic pressure is
and the part coming from the disorder has the opposite sign as the term coming from the chain connectivity. We see that the fluctuation term becomes zero whenever
or the obstacle density Co becomes (if Vo is of the order of V)
i.e. an effective screening length
becomes infinite. Thus we are left with
It is important to notice that the presence of obstacles cancels the term of the chain
connectivity at a certain concentration.
The general limitation of this formula can be discussed within the framework of references [5, 8], i.e. it has to be
or
By rewriting equation (46) into the standard form (44) we can define a new effective screening length by
where e E is the classical Edwards screening length in concentrated solutions given by equation (43).
The effective single-chain Hamiltonian.
It is often important to discuss the effective Edwardsian Heff for one single chain in the presence of all the others and the obstacles. In the case of a pure concentrated solution or melt this leads to the concept of screening and a screened excluded volume potential
Most important is the result that the screened potential is = 0 at k = 0, and the chains behave
as Gaussian in the melt. The effective single chain Edwardsian is then
In the case where fixed obstacles are present we cannot expect such a simple form since we
have to carry out quenched averages. We start from the Edwardsian (1) which is in mean field
approximation given by the form of equation (3), i.e.
which contains all N chains and all No obstacles. The effective single chain Edwardsian is
given by the following procedure [8]. Label one arbitrary test chain in the concentrated solution. Consider the N-chain, No-obstacle partition function Z[N, No]
and rewrite H[N, No] as
where H[1 , No] is the Edwardsian of the test chain, H[N -1, No] is that of all the other
chains, and V [1, N - 1 ] is the interaction of the labeled test chain (with the label 1) with all
the other chains. This is identical to the excluded volume interaction V, of course. We write (57) as
To find the effective single chain Edwardsian, we have to integrate out the (N - 1) chains in
the partition function, i.e.
Performing this integration we are left with the effective one chain partition function and we
have with
where
and
Ho is the usual Edwardsian of a polymer chain in concentrated solution. We see that
equation (60) contains the disorder, which has to be averaged after any calculation of a
quantity using Heff.
It is interesting to discuss the replica form of Heff(1) for completeness. Therefore we take
the quenched average over the disorder, i.e. we consider
The average can be performed with the Gaussian distribution for the {Ck} equation (11)
which leads to
and we find (i) coupling of the replicas via the last term in the effective Hamiltonian and (ii) a divergence at Co - 1/V at k = 0 as before. Near this special concentration the replica mixing
term is most important. Consequences of such a behaviour will be discussed in a separate paper.
Perturbation theory of the radius of gyration for the molécule in the melt with obstacles.
The size of the chains in the presence of the obstacles is another question which has to be
mentioned in this paper. In the case of diluted systems an interesting localisation effect has been observed theoretically and numerically [1, 2].
In the case of dense systems we have to take the screening of the extended volume effect into account. An exact calculation is not possible and the use of the replicated effective single
chain Edwardsian is beyond the scope of this paper and will be published separately. Here we present a single first order perturbation analysis along the lines discussed in [5, 9] and based
on the form of the effective Edwardsian (60).
The general equation for the first order perturbation result for the end-to-end vector
(R2) is for long chains [5]
where VSO is the screened potential containing the disorder. It is (after Eq. (60))
Thus we have to average
where we have ignored linear terms in Ck, since they will be zero after averaging out the {Ck} over the Gaussian distribution of the {Ck}.
Now we take the average over the disorder and by (Ck C-k) {Ck} = C6 we find
We define a length connected to the strength of the disorder by
in analogy to the screening length of the polymer and rewrite the last part of equation (67)
The first part is the usual Gaussian end-to-end distance, the second part is the correlation in the concentrated case, and the last part comes from the disorder. This last part is divergent at
k = 0. This divergence comes from the fact that we have used infinitely long chain implicitly
and the singularity can be removed by a soft cutoff, i.e. if we replace
Alternatively a sharp cut-off at the lower integral boundary can be introduced, but the soft cut-off as given by equation (70) produces simpler mathematies: This integral can be
calculated and we find as final answer
The term arising from the fluctuations in the concentrated melt becomes renormalised by the
disorder. We see that the « disorder length » cr, and the solution screening length e are compared to the size of the chain f B/7V.
Here the perturbation is given by a big negative term f B//7 indicating a similar localisation effect as in the single chain case. This shows that the effect is not perturbative, and a more
detailed analysis of the higher order perturbation terms has to be done. Thus we expect the chains to be localised in the dense melt, but higher order terms of the perturbation theory
have to be discussed to justify this conclusion.
Nevertheless we can expect a « O-point» before the collapse or localisation transition at
where the square bracket term in equation (71) vanishes and we have the value for the end-to- end distance in the 0 solution, i.e. the ideal Gaussian limit. This happens to first order at
Comparison to the annealed problem - « troubled solvent ».
In this last section of the paper we study briefly and crudely the case of the dense concentrated solution by the presence of moving obstacles. Here the disorder is not quenched and the
annealed averages have to be taken. The corresponding problem in the dilute case has been studied extensively by Duplantier [10], where the term troubled solved has been introduced for solvent with impurities. The free energy of the annealed case is given by
The calculation is straightforward and the result is
This functional integral exists whenever V - V6 C6 > 0, i.e. for a larger density of obstacles
or alternatively for large Vo the functional integral diverges and the chains become collapsed.
A more detailed discussion has to include the three body repulsion term in order to justify
such results. This has been carried out in reference [10] for the dilute case. The free energy is
given by equation (73) and we find
Since only the excluded volume parameter becomes renormalised the corresponding end-to-
end distance of the chains in the annealed disordered environment is given by
where e,,, is the renormalised screening length in the annealed case it can be written as
If we compare equation (76) for the result for the end-to-end distance with the case of the
quenched disorder we see that the perturbation of the end-to-end vector is a factor of
B/Tv stronger.
The osmotic pressure is given by the same relation as in the case of a simple concentrated
solution, but with the renormalised excluded volume parameter, i.e.
This law is very different from that what we have found in the case of quenched disorder.
Here we find a renormalisation of the excluded volume term, i.e. it is reduced by the annealed
disorder, whereas in the case of quenched disorder this term was not affected by the obstacles.
The theory presented in this section is only valid for weak disorder, i.e. V > V20 C20, where
the osmotic pressure is positive. The case of strong disorder requires more sophisticated analysis, along the lines of reference [10].
Discussion.
We discussed a dense polymer melt (or strong polymer solution) in the presence of quenched
random obstacles. This case allowed the application of a mean field theory, i.e. the Gaussian
approximation of the free energy in density variables. In this case the quenched averages can be calculated exactly, and complications such as replica symmetry breaking as in the dilute solution [2] do not arise. We found that the free energy can be calculated in the Gaussian
approximation (analogously the Gaussian model in phase transitions). The osmotic pressure showed a cancellation of the fluctuation terms and the screening length becomes renormalised
by the disorder. The size of the chain is strongly influenced by the disorder, but a 8-point exists, where the ideal Gaussian limit of the radius of gyration is recovered. Such a discussion involved three length-scales (i) the distance of the obstacles or strength of disorder (- u) (ii)
the screening length of the excluded volume effect (- §) and the ideal chain dimension
(- ÎN 1/2). These length scales are the basis for an appropriate scaling theory for the dense systems which will be discussed in a separate paper.
The comparison of the results between the quenched and the annealed disorder showed that the first order term in the perturbation expansion of the end-to-end distance is of the factor Ùk bigger in the case of quenched disorder.
Acknowledgment.
The auther would like to thank Dr. Michael Brereton for a helpful discussion.
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