The diffusion of flexible macromolecules in a limited volume

2608

T . P . ZHMAKINAet aL REFERENCES

1. Adv. Polymer Sci., Crazing in Polymers. B., Springer Press, 52/53: 983. 2. A. L. BOLYNSKII and N. F. BAKEYEV, Vysokomol. soyed. A17: 1610, 1975 (Translated in Polymer Sci. U.S.S.R. 17: 7, 1855, 1975) 3. R. P. KAMBOUR, J. Polymer Sci. Macromolec. Rev. 7: 1, 1973 4. A. V. YEFIMOV, V. P. LAPSHIN, V. I. FARTURNIN, P. V. KOZLOV and N. F. BAKEYEV, Vysokomol. soyed. A25: 588, 1983 (Translated in Polymer Sci. U.S.S.R. 25: 3, 692, 1983) 5. A. V. YEFIMOV, V. M. BULAYEV, A. N. OZERIN, A. V. REBROV, Yu. K. GODOVSKII and N. F. BAKEYEV, Vysokomol. soyed. A28:1986 (Translated in Polymer Sci. U.S.S.R. 28: No. 8, 1986) 6. Yu. K. GODOVSKII, Teplofizicheskiye metody issledovanya polimerov (Thermophysical Methods of Polymer Study). p. 280, Khimiya, Moscow, 1976 7. V. L GERASIMOV and D. Ya. TSVANKIN, Pribory i tekhnika eksperimenta (Experimental Apparatus and Technique). 204, 1968 8. O. KRATKY, Y. PILZ and P. I. SCHMIDT, J. Colloid Sci. 21: 24, 1966 9. E. PAREDES and E. W. FISCHER, Makromolek. Chem. 180: 2707, 1979 10. S. V. BUENRLL, D. CLAYTON and M. KEAST, J. Mater. Sci. 8: 513, 1973

Polymer Science U.S.S.R. Vol. 28, No. 11, pp. 2608-2616, 1986 Printod in Poland

0032-3950/86 $10.00+ .00 © 1987 Pergamon Journals Ltd.

THE DIFFUSION OF FLEXIBLE MACROMOLECULES IN A LIMITED VOLUME* T. P. ZI-IMAKINA, L. Z. VILENCHIK a n d B. G. B~LEN'KIt Polymers Institute, U.S.S.R. Academy of Sciences (Received I April 1985) The translational diffusion of flexible macromolecules in a cylindrical channel, with a radius exceeding by twice or more that of the mean square gyration radius of the macromolecule, has been studied. The essential difference in diffusional mobility in the limited volume of an impermeable rigid sphere and of a flexible macromolecule was demonstrated. For the latter, this mobility is determined not only by the ratio of the mean dimensions of its corresponding limited volume, but essentially depends on the internal macromolecular structure and on the value of the hydrodynamic interaction of its segments. A STUDy o f the differential m o b i l i t y o f flexible m a o r o m o l e c u l e s in l i m i t e d v o l u m e s is o f s u b s t a n t i a l value in u n d e r s t a n d i n g m a s s t r a n s f e r p r o c e s s e s in p o r o u s m e d i a a n d finds p r a c t i c a l a p p l i c a t i o n in such fields a s m e m b r a n e filtration, high velocity, high efficiency exclusion c h r o m a t o g r a p h y etc. * Vysokomol. soyed. A28: No. 11, 2348-2354, 1986.

Diffusion of flexible macromolcculesin limited volume

2609

As shown in numerous experimental papers, the diffusion coefficient of flexible macromolecules in a limited volume, Ds appears to be lower than that in free solution, Do, since "diffusional retardation" DJDo depends on the ratio of dimensions of the macromolecule and the pore. The existing theoretical models [1, 2] consider the diffusion of an impermeable sphere in a cylindrical pore and satisfactorily explain the experimental data in the diffusion of coiled macromolecules. In the case of flexible macromolecules there is a series of experimental facts, which are unexplained by the notion of a macromolecule as an impermeable sphere when simulating the hydrodynamic properties of macromolecules in limited volumes [4]. Retardation of diffusion appears to be less than predicted by theory [5-8], the diffusional mobilities of flexible and compact macromolecules in the same, porous medium are essentially different [5-8], retardation of diffusion in porous media of similar structure but of different pore size, is not a synonymous function of the ratio of sizes of macromolecule and pore [8]. From the above, it is evidently necessary to construct a diffusional mobility theory for flexible macromolecules in a limited volume, taking account of their microscopical structure. Besides this, in relation to the undisturbed macromolecular dimensions, e.g. its mean square radius of energy ($2) ~ and pore radius, ro, it is necessary to differentiate the following cases (S2)½>>ro,

(S2)~,,,ro,

($2)~<~ro/2

The first case was examined by Brochard and De Gennes [9]. The second, is probably the most complex to study; it may be studied using simulation of the polymer chain motion with the EVM. This case was considered in reference [10], without taking account of the hydrodynamic interaction of segments. The present work is devoted to the study of the third case. Translational diffusion of flexible macromolecules in a cylindrical pore. To describe the diffusional mobility of a macromolecule in a pore, the Kirkwood-Riseman theory Ill] was used, in the framework of which the hydrodynamic properties of flexible macromolecules are studied from the viewpoint of segment interaction. We shall represent the macromolecule as a series of N spherical beads, combined together by insubstantial bonds. Each bead describes a hydrodynamic property of a Kutm segment. The essential approximations to this theory are the previously neutralized tensor for hydrodynamic interactions (the Oseen tensor) for conformations and also neglecting the micro-Brownian motion of the segments. It is known, however, that in the case of stationary flow, the Kirkwood-Riseman theory gives results, practically coinciding with the conclusions of more rigid theories and with results of modelling the hydrodynamic properties of the macromolecule on the EVM without preliminary neutralization o f the conformations [4, 12, 13]. We shall examine translational motion of an undeformed macromolecule with velocity U in a still liquid in respect of the axis of an infinitely long cylindrical pore o f radius, ro (Fig. 1). We shall consider a solvent with a compact medium in relation to the macromolecule and to each of the segments. As also in the Kirkwood-Riseman

2610

T.P. Zm~,rdNA

et

al.

theory, we will describe each segment by a bead of radius a, the translational frictional coefficient of which we will define from Stokes' law, ~=6m/a. We note that a unique, precise solution of the problem of many molecules was given by Stimson and Jeffrey [14] for slow motion of 2 spheres in parallel to the line of their centres in an inorganic liquid. They used a bipolar coordinate system, a unique system, permitting simultaneous satisfaction of the boundary conditions in 2 spheres, located one above the other. For a large number of molecules it is impossible in general to seek a coordinate system having a similar property [14]. We attempted therefore to find a regular scheme of sequential interation, with which the marginal problem may be solved to a desired approximation, by considering the boundary conditions of only one of the molecules. Such a scheme may be constructed based on reflection methods, first applied by Smohchowsky to a system of N spheres, sinking in a boundless liquid

[14].

1

"

t

t

i+l

FIG. 1. Schematic representation of a flexible macromoleeule in a cylindrical pore of radius ro; be-separation between 2 neighbouring chain segments; Rt~-separation between the i-th and j-th segments; a-bead radius, simulating a Kulm segment; bt-separation of i-th segment from pore axis. We will start from the Stokes equation [14] for the stationary tension of an incompressible liquid r/oV2V-VP=0, VV=0 (1) with the limiting conditions %°=0, (2) where V is the velocity of liquid motion, P - t h e pressure, ~7-dynamic viscosity. The reflection method is based on the linearity of eqn. (1) and (2), which allows one to represent the velocities of liquid motion in the i-th segment, U , in the form of a superimposition of velocities of a nondisturbing current (in our case this velocity is zero) and disturbance velocities, caused by motion of the segments and by the presence of the cylindrical walls 13, -- u} + .... (3)

2611

Diffusion of flexible macromolecules in limited volume

where U~t) corresponds to the case of a clear solution and U~z) and U~3) are related to the presence of the boundaries. In its turn, each of the terms in the right hand side of eqn. (3), represents a linear superimposition of disturbance velocities of each segment. We note that in the Kirkwood-Riseman theory, by virtue of the smallness of the spherical radius a compared with the separation between the segments, the heterogeneity of velocities with an order of separation of a was neglected, which allows one to describe the action of the segments on the current by precise forces.

3

_/djJ o /

/

i " ~

. / / / ~

/

11 I-

/1,


f

o"i

0

I

I

gO

100

I

N 150

FIQ. 2. Dependence of equivalent hydrodynamic radius of macromolecule in clear solution on number of segments in the chain, for a/bo = 0-07 (I) and 0.39 (2). 1", 2 ' - r e s u l t s obtained from appro-

(/(8)

ximate Kirkwood-Riseman.formula [11]. fo = N~ 1 +-~-x , where x=

~ / . The points (6it) t/o be] show results of numerical calculationof a freelyjointed chain by the Monte Carlo method, without preliminary neutralizationof the Oseen tensor and taking into account the correct bead volume [15]. We will calculate the force acting from the sides of the macromolecule as a whole N

on the liquid in the presence of cylindrical boundaries: F = ~, Fl i=l

We shall consider the macromolecule as a gradually migrating particle; we shall assume Ul = U. The force, acting from the sides of the i-th segment, using eqn. (3), is written in the form

Fi=F~l)+F~a)+ ...

(4)

As also in the Kirkwood-Riseman theory, we will describe the hydrodynamic interaction of the segments with one another by using the neutralized Oseen tensor [11] ^

1 /1\^

where Rig is the separation between the i-th and j-th segments and I is the unitary N tensor. The obhqueness of the neutralized Oseen tensor permits substantial simplification of the problem and consideration of only the axial velocity components of the force.

2612

T.P.

ZHMAKINA et

al.

The value of Fi(i) is determined by considering the macromolecule in an inorganic liquid [11 ]. ( N / \1 .) F~a ) = ¢ U - - X ( - - ) F j (6) 6m/o j = 1 \ R u / i#j

By defining the column-vector V(1) by the equality F~I)=V~I)¢u,

(7)

We obtain a system of N algebraic equations A~u(1)=E,

(8)

Where the matrix A is give~ by the relation

Aij=6U+(1--~U) ( ( L ~

Ont]Ok~Rij]

(9)

Here J~j is the Kroneker symbol and E is the unitary vector. The F~(1) forces determined by solving eqn. (8) allow one to find the disturbance velocity at an arbitrary point r above the macromolecule (first reflection). N

U(ri)= E rrl F(1) j=!

It is obvious that such a velocity field does not satisfy the conditions necessary to introduce the velocity field U (2) (second reflection [14]), determined by the limiting conditions on the channel walls.

v % = , o = - v(1)l,=,o

(10)

An additional force z~(~3), acting from the sides of the macromolecule on the liquid in presence of the cylindrical wails, is determined by the interaction of all its segments with the U (2) field (third reflection). Calculation df this force was carried out similarly to that of ~1). The difference is that in neglecting the disturbing action of the segments on the flow close to the i-th segment being considered, the relative rate will not be the same for all the i's, as in the case of an inorganic liquid. W e note that we must first take account of the screening of the i-th segment by all the others. Thus we obtain

F(a)=f(flilaF} 1)+ ~ g(fli, ~j, z,j, ro

j = 1

p,

wq)~-~-l - 7 - - - - ~ \ - i f - / " 1 , o O n t o j = t \JxO/

(11) ro

This relation is justified for afro~ 1 [14]. The first two items in the right hand side coincide with velocity U (2) with a precision up to a constant factor (coefficient of friction), The

Diffusion of flexible macromolcculesin limited volume

2613

first takes account of the disturbance "reflected" from the walls, due to motion of the /-tit segment, the second-the motion of the remaining segments. The function f(fli) is tabulated in references [14, 15]. The function g depends on the separation of segments i and j from the cylindrical axes and their mutual disposition (a cylindrical coordinate system is used in equation (11)) and may be calculated by the means described in reference [14]. However, we will use, in place of an exact g function, its evaluation in the form, f(fl~)go(Rij), where go(Rij) is determined from the known solutions of the motion of 2 similar spheres in a cylindrical channel (15). For ~u(3), we obtain a system of N linear equations A~(Z)=B,

(12)

where N

a

a

Bi=f(fl,) ~- ~}')+ ~f(fli)go(Rij) re

-(')

(13)

j~i

Thus the problem consists in the solution of 2 systems of linear algebraic equations with the same matrix coefficients, with unknown but different right hand sides [16]. In the first case (eqn. (8)), the right hand side corresponds to the by-passing of the macromolecule by a homogeneous solvent flow, in the second (eqn. (12)) to an inhomogeneous flow, arising by reflection of disturbing actions of the macromolecule, during its motion in the pore. Obviously, the frictional coefficient of the macromolecule in a cylindrical pore will be N

f~ = ~ (~}1)+ ~}a)) ~,

(14)

i=l

and the coefficient of translational diffusion:

ns=kT/L Modelling the process and discussion of the results. In the calculations, we were limited to the case of linear macromolecules for which according to Peterling and Ptitsyn=Eisner [12]: N l+e

2 =

Ii-jl' +ebo~, /6-/

*-

( 6 + 5 e + e 2)

1+.

(RO>-t=~J-~/(li-jl'-T-bo), where g is the excluded volume parameter. We shall consider some calculated results. To test the model, the coefficients of translational friction of a macromolecule without bulk interactions (g=O) in a clear solution and N~< 150 were calculated, for a series of a/be values. Figure 2 shows the fo relations obtained for equivalent hydrodynamic radius Rh=6m/o be' where fo is the

2614

T. P. ~ A

et al.

friction coefficient of the macromolecules for a/bo=O.07 and 0.39, to the number of segments in the chain. The dashed lines show the results obtained by the approximate formula of Kirkwood and Riseman [12]. The points are the results of calculating Rh by the freely-jointed Monte Carlo chain method, without preliminary neutralization of the Oseen tensor and by taking into account the particular bead volume [13]. The higher Rh values, obtained in reference [13] with a/bo=0"39,are explained by the effect of the particular bead volume, which was not considered in our calculations.

1.0

\

\

0.8- \

t~6

"\

x.

\

Fro. 3. Dependence of ratio of coefficient of diffusion in the pore to its value in clear solution on ratio of mean square inertial radius of a macromolecule to pore radius (explained in text); bo/ro---O'l (1) and 0.2 (2).

Figure 3 represents the dependence of the DJDo ratio on (S2)~./ro for a/bo=0.39 and other a/bo values, 0.1 (curve 1) and 0.2 (curve 2) corresponding to 2 different pore radii, with a fixed length of linkage b0. The continuous lines correspond to a solvent of good thermodynamic quality (8=0.2), the broken lines to a 0-solvent (8=0). It is seen that the presence of the walls affects the diffusional mobility of the macromolecule in the 0-solvent more strongly than in the good one. The dotted and dashed line shows the corresponding dependence for axially symmetric motion of a rigid globule in a cylinder [14]. Comparison of these solutions confirms that the walls affect the diffusioual mobility of the globule to a larger degree than that of the flexible macromolecules. This is related to the fact that the presence of the walls causes weakening of hydrodynamic interaction of segments with one another. Moreover, the appreciable difference between a flexible macromolecule and a sphere consists in that with the former, D,/Do depends not only on the ratio of macromolecular to pore size (S2)~t/ro,but also on pore sizes (see Fig. 3, curves 1 and 3). The fact is there are 2 parameters with linear dimensions (spherical pore radii) involved in diffusion of a sphere in a pore. It is obvious that the dimensionless ratio, D,JDo, may depend only on the ratio of the radius of the sphere to that of the

Diffusion of flexiblemacromolcculosin limited volume

2615

pore. In the case of a flexible macromolecule there.are 4 such parameters (pore radius to, the mean square inertial radius of the macromolecule, (SZ) ~, bead radius, a, aud segment length, bo). It is clear that they can give 3 independent, dimensionless ratios, on which Ds/Do will depend. It follows from this that even within the limits of one homologous series (constants a and b), it is impossible to obtain a single dependence: it will be differ~ob°)for this case. ent for pores of different size. We may write DA'D0 = J"/'(S2)~;, [ ~ We will examine the relation of D~/Do to bo/ro for a constant (S2)~/ro ratio. It follows from calculations that bo/ro grows with increase in Ds/Do (Fig. 3). In other words, with a constant (S2)½/ro ratio, D~IDois less for larger diameter pores. This apparently unexpected result has a simple physical meaning. As pore size increases, constancy of
2616

T. P. Z~WA~NA et al.

falls a n d this causes a w e a k e r r e t a r d a t i o n o f a flexible m a c r o m o l e c u l e , a n d e x p l a i n s w h y the e x p e r i m e n t a l l y o b s e r v e r d differences o c c u r in diffusional m o b i l i t y o f flexible and compact macromolecules. Translated by C. W. CAPI,

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. I0.

J. L. ANDERSON and J. A. QUINN, Biophys. J. 14: 130, 1974 H. BRENNER and L. J. GAYDOS, J. Colloid. Sci. 58: 312, 1977 J. L. ANDERSON, J, Theor. Biol. 90: 405, 1981 H. YAMAKAWA, Modern Theory of Polymer Solutions, N. Y. Haper Row, 119 pp., 197I C.K. COLTON, C. N. SATTF_,RFIELD and C. -J. LAI, A.I.Ch.E.J. 21: 289, 1975 J. KLEIN and M. GRUENNEBERG, Macromolecules 14: 1411, 1981 J. H. WONG and J. A. QUINN, J. Colloid Sci. 5: 169, 1976 W. M. DEEN, M. P. BOHRER and N. B. EPSTEIN, A .L Ch.E. J. 27: 952, 1981 F. BROCHARD and P. G. DEGENNES, J. Chem. Phys. 67: 52, 1977 A. M. SKVORTSOV, V. N. GRIDNEV and T. M. BURSHTEIN, Vysokomol. soyed., 21: 219, 1979 (Not translated in Polymer Sci. U.S.S.R.) 11. J. G. KIRKWOOD and J. J. RISEMAN, J. Chem. Phys. 16: 565, 1948 12. V.N. TSVETKOV, V. Ye. ESKIN and S. Ya. FRENKEL', Struktura makromolekul v rastvorakh (The Structure of Macromolecules in Solution). 719 pp., ~lauka, Moscow, 1964 13. J. G. DE LA TORRE, A. JIMENEZ and J. J. FRIERE, Macromolecules, 15: 148, 1982 14. J. KI-IANNEL' and G. BRENNER, Gidrodinamika pri malykh chislakh Reinordsa (Thermodynamics with Small Reynold's Numbers), 630 pp., Mir, Moscow, 1976 15. T. GREENSTEIN and J. HAPPEL, Appl. Sci. Res. 22: 345, 1970 16. T. P. ZHMAIONA, Avtoref. dis na soiskaniye uch. st. kaud. fiz. mat. nauk, (Discn on thesis, canditate in physical materials science). 21 pp., IXC AI~ISSSR, Leningrad, 1984