Dynamics of polymer chains trapped in a slit

F. Brochard

To cite this version:

F. Brochard. Dynamics of polymer chains trapped in a slit. Journal de Physique, 1977, 38(10), pp.1285-1291. <10.1051/jphys:0197700380100128500>. <jpa-00208698>

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DYNAMICS OF POLYMER CHAINS TRAPPED IN A SLIT

F. BROCHARD

Laboratoire de Physique des Solides (*), Université Paris-Sud, Centre d’Orsay, 91405 Orsay, France

(Reçu le 3 mai 1977, accepté le 14 juin 1977)

Résumé. - On étudie les propriétés dynamiques des macromolécules en solution et confinées dansdes lamelles d’épaisseur d microscopique (20-200 Å). On utilise la technique des lois d’échelle pourinclure les effets de volume exclu et les interactions hydrodynamiques entre monomères. Pourcertaines valeurs de la concentration et de la masse moléculaire, on montre que les macromoléculesse comportent comme des chaines de Rouse à deux dimensions, l’unité étant le blob de taille d groupantun grand nombre de monomères. On distingue deux régimes « 2d » : 1) dans le régime dilué(chaînes séparées), on attend un coefficient de diffusion inversement proportionnel à la masse molé-culaire. Pour les modes internes, on obtient la loi de dispersion

039403C9q =T/~s d1/3 q10/3;2) dans le régime semi-dilué (caractérisé par une longueur de cohérence 03BE2 supérieure à d), on

a des modes de type gel pour $$ q03BE2 1 03C4-1q = Td1/3 q2/03BE4/32 et la structure des modes internes pour

q03BE2&#x3E;1.En augmentant la concentration, on passe continûment du comportement « 2d » au comportement

dynamique « 3d » des solutions massives.

Abstract. - We investigate the dynamics of polymer solutions confined in ultra-thin slits (20-200 Å),including both excluded volume effects and hydrodynamic interactions through a scaling analysis.1) In the dilute regime, for chain extension R larger than the slit thickness d, the overall translationaldiffusion coefficient Dt is predicted to scale like

$$

which is much larger than the value expected from the Debye-Bueche approximation

$$

(where RF2 is the chain size measured in the slit plane). For the internal modes, we find a structure ofthe Rouse type at wavelength larger than d. This is due to screening of hydrodynamic interactions, anintrinsic feature of slit systems. The eigenmode frequency scales like

$$ 039403C9q = Td1/3 q10/3/~s

2) In the « 2d » semi-dilute regime characterized by a monomer-monomer correlation length 03BE2 larger

than d, we find modes reminiscent of a gel for q03BE2 1 (with a relaxation rate $$ )

andinternal mode structure of a confined chain for q03BE &#x3E; 1. The self-diffusion coefficient of a chain is

predicted to scale like Ds ~ N-2 c-2 d-1/3. 3) At higher concentration, we reach the « 3d » semi-dilute regime (03BE d) and we recover the dynamics of the bulk polymer solution (Ds ~ N-2 c-1.75).

LE JOURNAL DE PHYSIQUE TOME 38, OCTOBRE 1977,

Classification

Physics Abstracts56.60 - 62.10 - 66.10

1. Introduction. - We consider flexible polymerchains trapped in ultra-thin slits. There are alreadymeasurements on partitioning and diffusion of macro-

(*) Laboratoire associe au C.N.R.S.

molecules in pores having the geometry of a slit [1].Such pores can be realized in various ways, forinstance in porous glass prepared by leaching or withcleavage fractures in mica. However such systemsgive little information on dynamical behaviour.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380100128500

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Lamellar phases of lipid-water systems ]2[ wouldseem at first sight a more promising system to test thecomplete dynamical behaviour of confined polymersolutions by using light scattering techniques. Weexpect that hydrophilic flexible polymers would betrapped in the water sheet of the lamellar phase, ofthickness varying from a few to 200 A.The statics of polymer chains confined in a tube and

in a slit has been discussed by a scaling approach [3].We have recently studied the dynamics of a singlechain trapped in a tube [8]. We investigate here thestatistical motions of polymer solutions confined in aslit using the dynamical scaling method [4] which isknown to give reasonably good results for bulk poly-mer solutions. Our assumptions follows reference [4] :we postulate that friction between monomer andsolvent is dominant; we ignore monomer/monomerfriction and also all internal viscosity effects. All ourdiscussion is restricted to chains dissolved in goodsolvents where monomers repel each other.

2. Summary of static behaviour : the concept of« blobs ». - The configuration of the chains under thesame conditions has been discussed by Daoud andDe Gennes [3]. They find five regimes depending uponthe monomer concentration c, the polymerizationindex N and the slit thickness d which are representedon the diagram of figure 1 .

2.1 SINGLE-CHAIN REGIMES. - In domain A, wehave a « 3d » conventional dilute solution of separate

coils of radius RF, (RF3 d), where RF3 is the Floryradius for bulk solution

In domain C (RF3 &#x3E; d), the coil becomes a flat

pancake of thickness d and radius RF2. We reacha « 2d » behaviour, characterized by a Flory radius RF2varying like N3/4. The chain can be represented as areal two-dimensional chain : the static units are notthe monomers of size a, but the blobs of size d contain-

ing gd monomers (Fig. 1) : in a scale smaller than d wehave a « 3d » behaviour and the blob size d is related

to gd by eq. (1)

on a larger scale, we have a o 2d » behaviour andthe « 2d » chain extension, according to Flory, is

This blob picture is very useful in understanding thedynamical behaviour. If we represent the chain by apancake with a uniform monomer concentrationc = N/ Rl d as in the Debye-Bueche approximation,we shall see that it would lead to too large frictioneffects. Because the monomer concentration is veryinhomogeneous, the solvent can flow easily throughthe chain.To describe the dynamical behaviour, we need to

know the elastic restoring force which opposes chain

FIG. 1. - The five regimes of flexible polymer chains trapped in aslit. We investigate the statistical motions of chains exhibiting a« 2d » behaviour : 1) in the dilute « 2d » regime where the isolatedchains can be represented by a 2-dimensional arrangements ofblobs of size d (the hatched zone corresponds to the Debye-Bueche

model where the concentration is supposed uniform); 2) in the« 2d » semi-dilute regime, where « 2d » chains overlap. The blobsare statically correlated (on a length’ 2) but hydrodynamicallyindependent because of slit wall proximity effects which screen thebackflows on a scale d.

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deformation. From the scaling analysis of reference [3],the elastic free energy associated with a chain extension

2.2 OVERLAPPING CHAINS. - If starting from

region C we increase the concentration, separatepancakes overlap and we reach the « 2d » semi-diluteregime represented by domain D. The coherencelength C2 which represents the distance between chaincrosslinks is larger than d. The chain can be seen as asuccession of non-interacting pancakes containing gmonomers. 03B62 is related to g through eq. (3)

with the relation g = cç 2 d, it leads to

At scales larger than ç 2, the chain is ideal and itsextension R is

For the dynamical behaviour, the SD solution behaveslike a gel at frequencies higher than the entanglementrelaxation rate. The gel modulus E is propostionalto the number of crosslinks :

If we go on increasing the concentration, (2 getssmaller and we reach the threshold line SN defined

by C2 = d. Above SN all correlation lengths are

smaller than d and local correlations are identical tothose of bulk solutions. The chain extension isR = (Nlg) 112 (3’ where C3 is the «3d» coherence

length defined by :

In domain E, R &#x3E; d and each chain occupies aregion with the shape of a pancake.

In domain B, R d and the chain occupies aspherical region.For both domains the gel modulus is

The aim of our investigation is to derive the dyna-mics of confined chains in the two-dimensionaldilute (o C ») and semi-dilute (o D ») regimes. Wefollow the analysis of reference [4] for bulk solutionswhich apply to domains A, B, E. We expect a smoothcrossover between the 2d-3d dynamical properties onthe threshold lines separating domains A, C and E, D.

3. The « Rouse » behaviour of confined chains. - Iffriction between monomers and solvent is dominant,the equation of motion for the nth monomer includinghydrodynamic interactions can be written as

where Bo is the mobility of a single monomer, Tnm theblackflow tensor and (p. the total force acting onmonomer m. (pm includes both external cpm ‘ andinternal (pint t contributions (cpm ‘ represents the localelastic force due to neighbouring units and excludedvolume effects between distant monomers).

In an infinite medium Tnm is long range and decreaseslike the inverse distance r = I r n - r m I

The essential point for the dynamics of confinedchains is that in a slit the range of the backflow kernelis d. The screening of the backflow is due to the

boundary conditions on the walls. At distance r muchlarger than d, the dominant factor in T(r) is exp - nr/d.This result can be understood qualitatively by treatingthe wall effect through a method of images ] 5[ as shownon figure 2. Thus the sum qJm in eq. (11) involves only afinite number of monomers which are within a dis-tance d from n.

If we are interested in space variations which areslow (qd 1) we may replace qJm by Tn in eq. (11) andwe arrive at an equation of the Rouse form

FIG. 2. - The method of images to treat boundary conditions atthe walls of the pore : at distances r &#x3E; d, the flow induced in a slit

by the point force T can be treated as the flow produced by (p and itsimages [5] (represented as dotted sources) in an infinite fluid. The

flow becomes very small at distances larger than d.

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where B is a renormalized monomer mobility :

We can estimate B by a scaling argument. Because therange of the kernel is just equal to the blob size d ofchain units in the 2d regime (domains C-D) wehave gd terms in the sum giving a contribution d -1.Thus we have

The mobility of one blob is

Successive blobs are independent dynamically andbehave like impenetrable spheres.

Remark. - We have taken into account only thegeometrical screening (d) due to boundary effects.There is also a backflow screening (K-1) due to

neighbouring monomers discussed in reference [4].As shown in reference [4], the screening length K -1is identical to the monomer-monomer correlation

length. It leads to K-1 = din domains C, D, K -1 == 03B63in domains B, E, and K - 1 = RF3 in domain A. Thescreening by surrounding monomers has therefore tobe taken into account only in the 3d regime where K-1becomes smaller than d.

Conclusion. - In the two-dimensional dilute andsemi-dilute regimes, the chains can be representedby « 2d » Rouse chains made of a succession of

impenetrable blobs of size d. The blobs are staticallycorrelated (excluded volume effects) but are hydro-dynamically uncorrelated.

4. « 2d » dilute regime. - 4.1 CHAIN MOBILITY.COMPARISON WITH DEBYE-BUECHE MODEL. - As

explained in part 3, the equation of motion forthe nth monomer of a single chain can be written as :

Summing over the index n, all the internal forces mustadd up to zero and we get an equation for the centreof mass

which defines the overall mobility of the chain

By Einstein’s relation, the corresponding diffusioncoefficient is

Comparison with the Debye-Bueche model. - Inthis paragraph, following the Debye-Bueche approxi-mation, we neglect the fluctuations of the monomerconcentration. We represent the chain by a pancakeof size RF2 and thickness d containing a concentration

of monomers (see hatched zone of Fig. 1).

As shown by Debye [6], the pancake as a porousmedium excludes the solvent flows except on a

thickness KDBI defined by KD = cBo/r¡; 1. Dimen-sionally Bo - 6 7rt7, a and KDBI is very small comparedto RF2 (KDS RF2 ^’ .Na/d &#x3E; 1). Thus the pancake movesas a rigid body in the slit.We calculate the friction force acting on the pancake

moving in the slit with the velocity U. A slit can beconsidered as a Hele Shaw cell [7] and the flow inducedat large distance by the motion of the pancake is

governed by the equation

plus the boundary conditions

In a porous medium there is no boundary conditionfor the tangential velocity Vo (vo falls to zero in adistance d for a rigid pancake or KDB in our case).Eqs. (20-21) imply V2 p = 0.The correct solution forp is

It leads to a viscous dissipation : .

The friction force acting on the pancake is

In the Debye-Bueche model, we expect a chain

mobility

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Comparing this result with eq. (18)

We find that the chain is much more mobile than

expected from the Debye-Bueche approximation.Conclusion. - A confined chain is much more

permeable than expected from a uniform concentra-tion picture. Because the concentration fluctuates

largely on a scale d (as represented by the blobs), thesolvent can easily flow through the chain. It leads to amuch larger chain mobility.

4. 2 INTERNAL MODE STRUCTURE. - 4. 2.1 Funda-mental mode qRF2 = 1. - The elastic restoring forcewhich opposes the chain extension (RF2 --+ RF2 + 6R)is, according to eq. (4)

If the time rate of change of R is bR, the velocity of allblobs is of order 03B4R and the friction force using theresult (17) is :

Equating these two forces leads to the relaxation rateof the first mode :

4.2.2 Internal modes qRF2 &#x3E; 1. - We derive the

dispersion law of internal modes by a scaling assump-tion, following reference [4]. We determine the struc-ture of internal modes by using the product of two-dimensionless functions f (qR2) g(qd)

a) The function f, which describes the crossoverbetween the diffusion mode of the whole confinedchain (qR2 1) and the internal modes must have thefollowing properties :- for qR2 1, we must recover f1Wq = D, q 2

Thus f (x --+ 0) _+ X 2. We verify that D, = R 2/0 1 isconsistent with result (14) ;- for qR2 = 1, f (1) = 1 ;- for qR2 &#x3E; 1, f(x) - xP. The exponent p is

derived from the condition that Ao-)q is independentof RF2 in the limit qRF2 &#x3E; 1. It leads to q = 10/3and to a characteristic frequency

b) The function 9 describes the crossover betweenthe two-dimensional (qd 1) and the three-dimen-sional (qd &#x3E; 1) regimes :

To satisfy the condition Acoq (qd &#x3E; 1) independentof d, we must have ’q = - 1/3. It leads to

in agreement with the Zimm eigenfrequency [4] in3 dimensions.

Remark. - A(o 17 is the characteristic frequencyassociated with the monomer concentration fluctua-tions measured by inelastic light scattering. We canderive from ð.Wq the structure of the N eigenmodes Tp(with p = N, ..., N/J, ..., ...1) which come into playin acoustic measurements. With a wave vector q,we explore a region containing n monomers such asq-1 - (n/gd)3/4 d in the limit qd 1. The corres-

pondence between p = N/n and q is q - p3/4. Weexpect then for the eigenmodes of a confined chain

In the limit p &#x3E; N/gd, the correspondence is q - p3/5and 1/Tp - p3X3/5 (= p1.8).

5. « 2d » semi-dilute regime. - 5 .1 GEL BEHAVIOUR(q’2 1). - At frequencies higher than the relaxa-tion time Tr for complete disentanglement of onechain, overlapping chains behave like a gel, of elasticmodulus ED defined by eq. (8). In the limit qC2 1,the polymer solution can be described by a continuumtheory and we specify by r the displacement of thegel and c(r) the polymer concentration.For longitudinal modes r = ro e"7x e-t/ , the elastic

restoring force per unit volume on the gel is

where ED = kB TldC’ (eq. (8)).We again use the blob concept to write the viscous

11

force. The blobs’ frictions (Bb- 1 = 6 nfls d ) are addi-tive and

Balancing these two forces, we are led to a relaxationtime

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On the threshold line SN, defined by (2 = ’3 = d,we have a smooth cross-over with the o 3d » gel ,relaxation time [4]

5.2 SINGLE CHAIN BEHAVIOUR (qC2 &#x3E; 1). - Forq(2 &#x3E; 1, we expect that the characteristic frequencyLBWq will become identical to the frequency measuredfor one isolated confined chain (eq. (31), (32), (33)).The cross-over between the gel behaviour (q(2 1)and the single chain behaviour (q03B62 &#x3E; 1) can bedescribed by a scaling law hypothesis :

where the characteristic frequency d separating thetwo regimes is given by (30)

and f(x) - xP, g(x) - xq for x &#x3E;&#x3E; 1.

p is determined by the condition that Ao-)q shouldbe independent of 03B62, which leads to p = 10/3 andthe result (32) ; q by the condition that Aa)q should beindependent of d, hence to q = - 1/3 and the resultis eq. (33).

5. 3 REPTATION TIME AND SELF-DIFFUSION COEFFI-CIENT. - We calculate the time Tr to disantengle onechain of total length L = (N/g) C2 following refe-rence [4]. The friction to pull one chain in the networkof the others is proportional to the blob number(N/gd). The diffusion coefficient involved in this processis then identical to the diffusion coefficient Dt of oneisolated confined chain (19) and Tr, defined by thediffusion law, is given by

The macroscopic self-diSusion coefficient D. of onelabelled chain is related to the chain extension R(c) by

i.e.

6. Concluding remarks. - Statistically a chainconfined in a slit behaves as a two-dimensional self-

avoiding walk if we take as units not the monomers(size a) but the blobs (size d). The main result of thedynamical analysis is that these « 2d » chains of blobs

behave as Rouse chains, i.e. the units are hydrodyna-mically uncorrelated. (In particular this leads to adiffusion coefficient for a single chain inversely pro-portional to the molecular weight.)

Experimentally the main directions of research

appear to be the following :

1) In porous media the cooperative diffusion coef-ficient of a semi-dilute solution should be measurablerather simply. A not too dilute polymer solutionpenetrates easily inside the porous medium (1). Ifthe porous sample is then placed in contact with asolution of slightly lower concentration some polymerwill diffuse back from the pores to the bulk solution :the time required for equilibrium will lead to a deter-mination of D.

2) We expect that the study of hydrophilic chainsdissolved in the lamellar phase of a lipid-water systemwill allow for more detailed investigations on thedynamics of confined chains. The use of light scat-tering [9] (2) should detect both cooperative gel modes(qÇ2 1) and single chain modes (qÇ2 &#x3E; 1) in the SDregime. Self-diffusion coefficients in the dilute regimeare also accessible (2).

The Rouse behaviour may also be tested more

directly in the dilute regime by a simple viscositymeasurement in shear flow. If the flow direction is

parallel to the sheets and the flow gradient is normalto them, the viscosity il is related to the monomerconcentration c in the hydrodynamically independentblobs picture by

where Tb = cd 3/gd is the volume fraction of the waterphase occupied by the blobs. If the chains had behavedas impenetrable pancakes, as in the Debye-Buecheapproximation, we would expect

where qJp = - RF2 d is the volume fraction occupiedN F2

by the chains, Thus simple viscosity measurements inpolymer-lipid-water mixtures may allow a test ofthe 2d Rouse behaviour of confined chains.

Acknowledgments. - The author has benefited frommany discussions with Professor P. G. de Gennes andwith F. Rondelez.

(1) Audebert, R., private communication.(2) Rondelez, F., private communication.

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References

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[2] GULIK-KRZYWICKI, T., TARDIEU, A. and LUZZATI, V., Mol.Cryst. Liq. Cryst. 8 (1969) 285.

DE GENNES, P. G., Solid State Phys. Suppl., to be published.[3] DAOUD, M., DE GENNES, P. G., J. Physique 38 (1977) 85.[4] DE GENNES, P. G., Macromolecules 9 (1976) 587.

[5] SONSHINE, R. M., BRENNER, H., Appl. Sci. Res. 16 (1966) 273.[6] DEBYE, P., BUECHE, A., J. Chem. Phys. 16 (1948) 573.[7] CHIA-SHUN YIH, Fluid Mechanics (Mc Graw Hill) 1969, p. 382.[8] BROCHARD, F., DE GENNES, P. G. (submitted for publication).[9] ADAM, M., DELSANTI, M., JANNINK, G., J. Physique Lett. 37

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