Conformation of confined macromolecular chains : crossover between slit and capillary L. Turban To cite this version: L. Turban. Conformation of confined macromolecular chains : crossover between slit and capillary. Journal de Physique, 1984, 45 (2), pp.347353. <10.1051/jphys:01984004502034700>. <jpa 00209762> HAL Id: jpa00209762 https://hal.archivesouvertes.fr/jpa00209762 Submitted on 1 Jan 1984 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The Chain J. Physique 45 (1984) 347353 FtvRmR 1984, 347 Classification Physics Abstracts 61.40 05.90 Conformation of confined macromolecular chains : crossover between slit and capillary L. Turban Laboratoire de Physique du Solide (*), ENSMIM, Parc de Saurupt, F54042 Nancy Cedex, France and Université de Nancy I, BP 239, F54506 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 semidiluée 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 semidilution pour une chaîne non confinée. 2014 Abstract conformation of macromolecular chains in dilute or semidilute 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 semidilution limit of unconfined chains. 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 [24]). In the present work, using the same methods, we study the crossover 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 semidilute solutions in a good solvent as a function of Di/D2, theslitlimitcorrespondingtodl/d2 + 0 and the capillary limit to Dl/D2 1. Polymer adsorption 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 problem is studied using the Flory theory, scaling arguments and the results are reinterpreted using the blob Fig 1. conformations in a finite slit. a) When R2 D2 the chain is pancakeshaped whereas, b) it is cigarshaped when R, > D2. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004502034700
Let The For 348 picture. The crossover between dilute (single chain) and semidilute (overlapping chains) behaviour is presented in 3. The local correlation crossover is studied in 4 and the chain conformation in semidilute solutions is discussed in the last section ( 5). 2. The single chain problem. 2.1 UNCONFINED ISOLATED CHAIN. chain with polymerization index N may be assumed to perform a walk on an hypercubic lattice with mesh size a as in the FloryHuggins 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 selfavoiding leading to a radius (1) : where vd is a critical exponent depending on the dimension 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 meanfield approximation (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 onedimensional ( 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 selfavoiding walk is related to the n 0 limit of the nvector model [11]. Equation 2.4 is exact in 2d [12] but disagrees with the texpansion result [13] (E 4 d) near d 4 : 2.2 ISOLATED CHAIN IN A FINITE SLIT (Fig. 1). 2.2.1 Flory approximation. R, be the length of the chain («cigar») elongated in the infinite direction (R1 > D2 > Dl) in such a way that the chain «sees» the walls in the other two directions; the 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 crossover 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 crossover 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. 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 twodimensional 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 onedimensional at large scale, the two length scales R2 and D2 must combine to give R1 N N when N + oo : means that nume (1) Here and in what follows a ~ sign rical coefficients are ignored
The The 349 The correct asymptotic behaviour is obtained when mi 1/3 and equation 2.7 is recovered 2.2. 3 Blob picture. preceding results are easily interpreted using the blob concept [1, 68]. 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, containing g monomers with : 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 pancakeshaped 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: The whole chain is built of Nlg blobs (Fig. 2) with mutual exclusion. As a result the blobs perform a 2d selfavoiding walk with step length D1 and the radius of gyration reads [1] : and : in agreement with equation 2. 7. 3. Crossover between dilute and semidilute solution. in agreement with equation 2.9. In the finite slit when R2 > D2 we are led to introduce, besides the spherical blobs with radius D, and g monomers, 2d «superblobs» with radius D2 containing gb blobs (ggb monomers). The 2d local correlations inside a superblob requires : It will be convenient to work with the following reduced units : The N/ggb superblobs perform a 1 d selfavoiding walk with step length D2 along the free direction and : 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 unconfined geometry : above which different spherical coils begin to overlap. When z 1 we are in the dilute regime, the semidilute regime is entered when z > 1. In a slit (D2 + oo) the critical overlap concentration is xdependent and reads [1] : Finally in a finite slit it depends both on x and y : Fig. 2. 2d pancake with radius R2 may be considered as a succession of blobs with radius D1 and local 3d correlations.
350 When Di D2, x y and the last equation gives back the capillary result [ 1 ] : These results may be also obtained via scaling [1] by requiring Nindependence for these local quantities and using the boundary conditions : When x 1 equation 3.6 reduces to the bulk result Equations 3. 6 and 3. 7 agree at the crossover between dilute cigars and pancakes (y 1). One may notice that the critical overlap concentration 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 Nindependent in cigars. In dilute solutions the chains are spherical when y x 1 a crossover occurs from spheres to pancakes at x 1 (y 1) and from pancakes to cigars at y 1 (x > 1) (Fig. 3). In this way, we get: in agreement with the previous results. A crossover between 2d and 3d local correlations occurs at a concentration C23 such that 3(C23) C;2(C23) Dl giving : 4. Correlation length and local correlations crossover in semidilute solutions. In dense 3d systems (semidilute solutions or melts) the chains are ideal at large scales according to the Flory theorem [5, 8]. This behaviour is a result of a competition between intra and interchain interactions. In 1 d systems selfavoiding 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 swollen [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 semidilute solution may be considered as a closepacked system of blobs. In a bulk solution the 3d correlation length Ç3 is the radius of a swollen sequence with internal concentration C containing [6, 8] or, in reduced units, in the plane The local correlations remain twodimensional as long as 2(C) D2 or, using equations 3. 7 and 4. 4, when : i.e. in the semidilute regime. 5. Chain conformations in the semidilute 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 : monomer units so that : and the N 1/2 ideal behaviour at large scale requires n3 1/8 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 : The chain may be considered as a succession of N/g uncorrelated blobs of size 3 containing g C3 monomers so that: so that : and the blob picture agrees with equation 5.2.
Chain 351 Fig. 3. conformations in the (x, z)plane for different 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 semidilute chains and the small letters to dilute chains (s sphere; p pancake; c cigar). The heavy line gives the limit of semidilution. The dashed line is the crossover line between 3d and 2d semidilute local behaviour. The prime on a letter indicates 2d local behaviour. Below the dotted line the chains are segregated
352 The internal filling factor 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 : which is a measure of the chains overlap, is smaller than one in the semidilute regime as required to get an ideal long range behaviour. In pancakes with 3d local correlations the components 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 : 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 semidilute 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 : 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 semidilute 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 semidilution so that : Rseg is independent of the local correlations because C C;nt NIR..G D, D2 for segregated macromolecules in contact. It may be verified that Rgeg R, at the semidilution limit C*(x, y). Semidilute spheres with 3d local correlations are observed as long as R(3d) D1. In reduced units the crossover occurs in the plane [1] : When x > y the crossover is towards 3d semidilute pancakes and a crossover between 3d semidilute pancakes and 3d semidilute 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 semidilute pancakes cannot be observed as expected. 2d semidilute cigars are obtained when R(2d) > D2 and C*(x, y) C C23. In reduced units the crossover plane between 2d semidilute cigars and 2d semidilute pancakes reads : and ideal behaviour requires n2 1/2 leading to : The conformational evolution of the chains in the sketched on figure 3 for different (x, z)plane is values of a Dl/D2 between the slit limit (a 0) and the capillary limit (a 1 ).
353 References [1] DAOUD, M., DE GENNES, P. G., J. Physique 38 (1977) 85. [2] BROCHARD, F., DE GENNES, P. G., J. Chem. Phys. 67 (1977) 52. [3] BROCHARD, F., J. Physique 38 (1977) 1285. [4] DAOUDI, S., BROCHARD, F., Macromolecules 11 (1978) 751. [5] FLORY, P., Principles of polymer chemistry (Cornell University Press, Ithaca, N.Y.) 1953. [6] DAOUD, M., COTTON, J. P., FARNOUX, B., JANNINK, G., SARMA, G., BENOIT, H., DUPLESSIX, R., PICOT, C., DE GENNES, P. G., Macromolecules 8 (1975) 804. [7] DE GENNES, P. G., Macromolecules 9 (1976) 587594. [8] DE GENNES, P. G., Scaling concepts in polymer physics (Cornell University Press, Ithaca, N.Y.) 1979. [9] DE GENNES, P. G., Phys. Lett. A 38 (1972) 339. [10] DES CLOIZEAUX, J., J. Physique 36 (1975) 281. [11] STANLEY, H. E., Phys. Rev. Lett. 20 (1968) 589. [12] NIENHUIS, B., Phys. Rev. Lett. 49 (1982) 1062. [13] WILSON, K. G., FISHER, M. E., Phys. Rev. Lett. 28 (1972) 240. [14] BROCHARD, F., DE GENNES, P. G., J. Physique Lett. 40 (1979) L399.