Conformation of confined macromolecular chains :
cross-over between slit and capillary
L. Turban
Laboratoire de Physique du Solide (*), ENSMIM, Parc de Saurupt, F-54042 Nancy Cedex, France
and Université de Nancy I, BP 239, F-54506 Vand0153uvre les Nancy, France
(Rep le 2 septembre 1983, accepti le 7 octobre 1983 )
Résumé. 2014 On examine la conformation de chaînes macromoléculaires en solution diluec ou semi-diluée en bon
solvant, confinées dans une fente d’épaisseur D1 et de largeur D2. L’évolution des différents régimes (sphère, gateau, cigare avec corrélations locales tri- ou bidimensionnelles) est étudiée en fonction du rapport D1/D2 dans le plan (x, z) où x = R3/D1, z = (C/C*)3/4, R3 étant le rayon de giration et C* la limite de semi-dilution pour une chaîne
non confinée.
Abstract 2014 The conformation of macromolecular chains in dilute or semi-dilute solution in a good solvent, confined into a finite slit with thickness D1 and width D2 is examined The conformational evolution (sphere, pancake, cigar with 3d or 2d local correlations) is studied as a function of D1/D2 in the (x, z)-plane where x = R3/D1,
z = (C/C*)3/4, R3 is the radius of gyration and C* the semi-dilution limit of unconfined chains.
Classification
Physics Abstracts
61.40 - 05.90
1. Introduction
The conformation and thermodynamics of long
flexible polymer chains in a good solvent confined into
a slit or a capillary have been studied some years ago
by Daoud and de Gennes [1] using scaling arguments and the blob concept (the dynamical aspects of the problem may be found in references [2-4]). In the
present work, using the same methods, we study the
cross-over between these two limits, the chains being trapped into a finite slit with thickness D 1, width DZ > D,, which is infinite in the third direction
(Fig. 1). The chain conformation is studied for dilute and semi-dilute solutions in a good solvent as a func- tion of Di/D2, theslitlimitcorrespondingtoDl/D2 -+ 0
and the capillary limit to Dl/D2 = 1. Polymer adsorp-
tion on the walls is supposed to be negligible as in
reference 1. In a good solvent the isolated chain is swollen and the radius of gyration is given, to a good approximation, by the Flory theory [5] which will be used throughout this work.
The outline is as follows : in § 2 the single chain pro- blem is studied using the Flory theory, scaling argu- ments and the results are reinterpreted using the blob
Fig 1. - Chain conformations in a finite slit. a) When R2 D2 the chain is pancake-shaped whereas, b) it is cigar-shaped when R, > D2.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004502034700
picture. The cross-over between dilute (single chain)
and semi-dilute (overlapping chains) behaviour is pre- sented in § 3. The local correlation cross-over is studied
in § 4 and the chain conformation in semi-dilute solutions is discussed in the last section (§ 5).
2. The single chain problem.
2.1 UNCONFINED ISOLATED CHAIN. - The chain with
polymerization index N may be assumed to perform
a walk on an hypercubic lattice with mesh size a as in the Flory-Huggins theory [5]. For an ideal chain the walk is random and the radius is
In the case of a real chain in a good solvent, the excluded
volume v is of order ad and the walk on the Flory- Huggins lattice is self-avoiding leading to a radius (1) :
where vd is a critical exponent depending on the dimen-
sion d of the system. In the Flory approximation RF
is obtained through the minimization of a trial free energy :
where the first term of entropic origin is the elastic energy for a streched ideal chain and the second gives
the interaction energy in the mean-field approxima-
tion (in units of kB T). Equation 2.3 overestimates the free energy of the real chain [7] but gives quite
accurate results for the radius of gyration :
When d = 1 the radius is proportional to N as it
should for a one-dimensional ( 1 d) chain with excluded volume and at the upper critical dimension d, = 4
above which the interactions become irrelevant the ideal chain behaviour is recovered in agreement with the magnetic analogy [6, 9,10] in which the self-avoid-
ing walk is related to the n = 0 limit of the n-vector model [11]. Equation 2.4 is exact in 2d [12] but disa-
grees with the t-expansion result [13] (E = 4 - d)
near d = 4 :
2.2 ISOLATED CHAIN IN A FINITE SLIT (Fig. 1).
2.2.1 Flory approximation. - Let R, be the length
of the chain (« cigar ») elongated in the infinite direc- tion (R1 > D2 > Dl) in such a way that the chain
« sees » the walls in the other two directions; the (1) Here and in what follows a ~ sign means that nume-
rical coefficients are ignored
Flory free energy in this situation (Fig 1 b) may be written as follows :
The length R1 of the cigar is given by the value of R for
which the free energy is minimum :
When R, D2 equation 2. 7 is no longer valid and we.
recover the infinite slit geometry for which one has to consider the following free energy (Fig.1 a) :
We get a flat « pancake » with thickness D, and radius
Equation 2.7 reduces to the capillary result [1] when Di = D2 = D and the cross-over between cigar and pancake occurs when Ri = D2. Solving equation 2. 7
for R1 with D2 = R1, equation 2.8 is recovered There is a cross-over between pancake and sphere when R2 = D 1 and solving equation 2.9 for R2 with D1 = RZ, we get the Flory radius for the unconfined coil :
the 3d version of equation 2.4.
2.2.2 Scaling argument. - For the infinite slit when
D 1 > R3, the chain is spherical with radius R3 ; when D 1 R3 the chain is confined with a pancake shape.
The system is two-dimensional for large N(R31D, > 1)
and according to equation 2.4 the radius R2 scales
with N as N 3/4 so that we may write [1] :
The N 314 asymptotic behaviour requires m2 = 1/4
in agreement with equation 2.9.
In a finite slit equation 2.11 remains valid as long
as R2 D2. When D2 is reduced below R2 we get a cigar of length Ri along the free direction. The system being one-dimensional at large scale, the two length
scales R2 and D2 must combine to give R1 N N when
N -+ oo :
The correct asymptotic behaviour is obtained when mi = 1/3 and equation 2.7 is recovered
2.2. 3 Blob picture. - The preceding results are easily interpreted using the blob concept [1, 6-8]. A
blob is a chain subunit within which the correlations remain of higher dimension when the chain is confined In the infinite slit the chain may be considered as a
succession of spherical blobs with radius D1, contain- ing g monomers with :
The whole chain is built of Nlg blobs (Fig. 2) with
mutual exclusion. As a result the blobs perform a 2d self-avoiding walk with step length D1 and the radius of gyration reads [1] :
in agreement with equation 2.9.
In the finite slit when R2 > D2 we are led to intro- duce, besides the spherical blobs with radius D, and g
monomers, 2d « superblobs » with radius D2 contain- ing gb blobs (ggb monomers). The 2d local correlations inside a superblob requires :
The N/ggb superblobs perform a 1 d self-avoiding walk
with step length D2 along the free direction and :
Fig. 2. - The 2d pancake with radius R2 may be considered
as a succession of blobs with radius D1 and local 3d correla- tions.
It is easy to check that equations 2.13, 2.15 and 2.16
give back equation 2. 7. The superblob picture breaks
down when gb - 1 i.e. near the capillary limit (D2 N D1). One may then consider pancake-shaped
blobs with radius D2, thickness Dl and the same type of correlations as in the pancake (Eq. 2.9). One
expects equation 2. 9 to remain valid in this limit since it crosses over smoothly to the 3d behaviour when
R2 = D 1. Then we get:
and :
in agreement with equation 2. 7.
3. Cross-over between dilute and semi-dilute solution.
It will be convenient to work with the following
reduced units :
It may be verified that :
so that with our conventions y x. In equation 3. 3
C is the monomer concentration in the solution and C* is the 3d critical overlap concentration in uncon- fined geometry :
above which different spherical coils begin to overlap.
When z 1 we are in the dilute regime, the semi-
dilute regime is entered when z > 1.
In a slit (D2 -+ oo) the critical overlap concentration is x-dependent and reads [1] :
Finally in a finite slit it depends both on x and y :
When Di = D2, x = y and the last equation gives
back the capillary result [ 1 ] :
When x = 1 equation 3.6 reduces to the bulk result
Equations 3. 6 and 3. 7 agree at the cross-over between dilute cigars and pancakes (y = 1).
One may notice that the critical overlap concentra-
tion is also the internal concentration C;nt inside the
isolated chain and according to equations 3. 5, 3. 6 and 3. 7, C;nt scales with the polymerization index as N -4ls
in spheres, N -112 in pancakes and is N-independent
in cigars.
In dilute solutions the chains are spherical when y x 1 a cross-over occurs from spheres to pan- cakes at x = 1 (y 1) and from pancakes to cigars at y = 1 (x > 1) (Fig. 3).
4. Correlation length and local correlations cross-over
in semi-dilute solutions.
In dense 3d systems (semi-dilute solutions or melts) the
chains are ideal at large scales according to the Flory
theorem [5, 8]. This behaviour is a result of a competi-
tion between intra- and interchain interactions. In 1 d systems self-avoiding walks are always fully extended
and the Flory theorem breaks down. 2d systems constitute a border case where the chains are slightly
swollen and strongly segregated [8].
One may introduce a correlation (or screening) length ç(C) within which the chains remain swol- len [6, 8], ideal behaviour being eventually observed
at larger scales. A chain may then be pictured as a
sequence of blobs of size ç( C) within which excluded volume effects are important and the semi-dilute solution may be considered as a close-packed system of blobs.
In a bulk solution the 3d correlation length Ç3 is the
radius of a swollen sequence with internal concentra- tion’ C containing [6, 8]
monomer units so that :
In a slit, when ’3 > Dl, one enters a regime with 2d
local correlations. Within a pancake with radius ’2
and thickness D 1, there are g, = CD 1 2 monomers
and according to equation 2. 9 :
so that :
These results may be also obtained via scaling [1] by requiring N-independence for these local quantities
and using the boundary conditions :
In this way, we get:
in agreement with the previous results.
A cross-over between 2d and 3d local correlations
occurs at a concentration C23 such that 3(C23) = C;2(C23) = Dl giving :
or, in reduced units, in the plane
The local correlations remain two-dimensional as long
as 2(C) D2 or, using equations 3. 7 and 4. 4, when :
i.e. in the semi-dilute regime.
5.’Chain conformations in the semi-dilute regime and
discussion.
The radius of gyration of unconfined chains with 3d local correlations is R(3d) = R3 when C = C* so that
we may write the scaling law :
and the N 1/2 ideal behaviour at large scale’requires
n3 = - 1/8 so that :
The chain may be considered as a succession of N/g
uncorrelated blobs of size ’3 containing g = C3
monomers so that:
and the blob picture agrees with equation 5.2.
Fig. 3. - Chain conformations in the (x, z)-plane for diffe-
rent values of a = D1/DZ : a) a = 0, b) a = 0.3, c) a . = 0.5, d) a = 0.7, e) a = 1. The capital letters correspond to
semi-dilute chains and the small letters to dilute chains
(s = sphere; p = pancake; c = cigar). The heavy line gives the limit of semi-dilution. The dashed line is the cross- over line between 3d and 2d semi-dilute local behaviour.
The prime on a letter indicates 2d local behaviour. Below the dotted line the chains are segregated
The internal filling factor
which is a measure of the chains overlap, is smaller than one in the semi-dilute regime as required to get
an ideal long range behaviour.
In pancakes with 3d local correlations the compo- nents of the blob random walk with step length Ç3 parallel to the walls are unaffected by the confinement.
It follows that the radius of gyration is still R(3d).
The internal filling factor is :
and the chains overlap since ’3 D, is required to
have 3d local correlations.
In cigars with 3d local correlations two regimes have
to be considered. First let us suppose that the chains
are ideal at large scale, then the length is R(3d) and
the internal filling factor reads :
This result is consistent with an ideal behaviour as
long as 4>c(3d) 1 i.e. when the chains overlap at high enough concentrations. 4>c(3d) > 1 means that
the chains segregate and are no longer ideal. The limit of segregation is given by :
In this regime one may assume that the chain length is
such that Pc(3d) = 1 giving :
The result of Brochard and de Gennes [14] for a melt is
recovered in the appropriate limit (Ca3 = 1 ; D1 = D2 = D).
Let us assume that semi-dilute pancakes with 2d
local correlations remain ideal at large scale; their
radius R(2d) must fit with the isolated chain value R2
at the limit of semi-dilution so that :
and ideal behaviour requires n2 = - 1/2 leading to :
The chain may be also considered as an ideal chain of
N jg blobs with radius Ç2’ thickness D, cointaining
g = C2 D1 monomers so that :
in agreement with equation 5.10. It is easy to check that R(2d) = R(3d) when C = C23. The pancake
internal filling factor reads :
This is a border case for which weak swelling and strong segregation may be expected
For semi-dilute cigars with 2d local correlations we
get :
and the assumed large scale ideal behaviour cannot be observed since R(2d) must be larger than D2 for cigars.
The 2d cigars are always segregated and their length
may be obtained by assuming complete segregation :
Rseg is independent of the local correlations because C = C;nt = NIR..G D, D2 for segregated macromo-
lecules in contact. It may be verified that Rgeg = R,
at the semi-dilution limit C*(x, y).
Semi-dilute spheres with 3d local correlations are
observed as long as R(3d) D1. In reduced units the
cross-over occurs in the plane [1] :
When x > y the cross-over is towards 3d semi-dilute
pancakes and a cross-over between 3d semi-dilute
pancakes and 3d semi-dilute cigars takes place when R(3d) = D2 i.e. in the plane :
When x = y (capillary) equations 5.15 and 5.16 coincide and 3d semi-dilute pancakes cannot be
observed as expected. 2d semi-dilute cigars are
obtained when R(2d) > D2 and C*(x, y) C C23.
In reduced units the cross-over plane between 2d semi-dilute cigars and 2d semi-dilute pancakes reads :
The conformational evolution of the chains in the
(x, z)-plane is sketched on figure 3 for different
values of a = Dl/D2 between the slit limit (a = 0)
and the capillary limit (a = 1 ).
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