Chapter 7
Undulation-Enhanced Forces in Hexagonal Gels of Semiflexible Polyelectrolytes
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Theo Odijk Faculty of Chemical Engineering and Materials Science, Delft University of Technology, P.O. Box 5045, 2600 GA Delft, Netherlands
A theoretical analysis is given of undulation-enhanced forces in solutions of semiflexible polyelectrolytes. Undulation enhancement is fairly weak when the system is isotropic, nematic or cholesteric. Electrostatic repulsion can easily be enhanced by an order of magnitude in hexagonal gels. Van der Waals forces are perturbed mildly by configurational fluctuations although their enhancement does affect the free energy function near the secondary minimum. A stability analysis of enhanced electrostatic versus Van der Waals forces is presented in order to discuss experiments on gels of tobacco mosaic virus. Semiflexible polyelectrolytes are here defined as having a persistence segment of large aspect ratio i.e. the persistence length Lp is much greater than the effective diameter. A t high ionic strength, the latter reduces to a geometric diameter D ; at low salt, it is approximately proportional to the Debye screening length κ . Several important biopolymers are semiflexible polyelectrolytes exhibiting intriguing phase behavior. F o r instance, aqueous D N A forms a variety of lyotropic liquid crystals. Upon increasing the concentration, the following sequence of phases is seen (7): isotropic to cholesteric to two dimensional columnar hexagonal to three dimensional columnar hexagonal and finally to orthorhombic crystalline. There is a possibly optimistic tendency to believe that these phases ought to be understood in fairly simple statistical physical terms. It may be argued that the chemical detail of the polyion backbone is not critically important for long-range electrostatics swamps any other interactions, except possibly those o f dispersion type. Furthermore, our qualitative insight in the condensed matter physics of complex systems is increasing each decade, slowly yet steadily. The purpose of this paper is to show that the electrostatic interaction may be greatly enhanced by undulations or chain fluctuations in the hexagonal phase of positionally ordered polyions. In nonadhesive states, V a n der Waals forces are influenced to a much lesser degree at least within the Gaussian approximation adopted here. Nonetheless, undulation enhancement of dispersion forces cannot be neglected when trying to locate the secondary minimum for a gel stabilized by these interactions. Undulation enhancement is of minor import in solutions where positional order is weak or absent as I now show. 1
0097-6156/94/0548-0086$06.00/0 © 1994 American Chemical Society
Schmitz; Macro-ion Characterization ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
7. ODUK
Undulation-Enhanced Forces in Hexagonal Gels
87
Isotropic, Nematic and Cholesteric Phases Undulation enhancement of the electric potential exerted by a polyion was first considered by Odijk and Mandel (2) who developed a perturbation scheme in terms o f the large parameter Iyc. A similar procedure can be used to evaluate the average interaction between two nearby undulating polyions. However, I here present only a qualitative analysis for wormlike polyelectrolytes o f zero diameter, for simplicity. The case D > 0 leads to similar conclusions. A contour point of a wormlike polyion is chosen to be the origin Ο of our Cartesian coordinate system. The contour distance from Ο is specified by S j . The vector tangential to the chain at Ο is constrained to point in the ζ direction (see Figure 1). The first chain interacts with a second, one contour point o f which is fixed at (R,0,0) with the associated tangent vector pointing along the Z axis. The contour distance s is defined with respect to this contour point. The 2^ axis is skewed at an angle y with respect to the Z axis. I wish to investigate the average interaction between two nearby sections of the two chains (-2 > > d . A n excess of salt is present. b) L > > > d . This ensures that the hexagonal lattice exists. There is virtually no folding of the chains. Furthermore, a deflection segment of length X=L d > > d has a large aspect ratio so that the orientational fluctuations o f order d / λ are small. c) / c d R < < 1. This ensures that the asymptotic analysis of equations 9 and 10 is viable. d) i) d < < R ; ii) / /c d +3/cd+/cD > 1. The deflection segment has the electric properties of a charged rigid rod. f) (D + K ~ ) / K X > d / X . Electrostatic twist is negligible (3). The polyions are basically parallel. Orientation-translation coupling can be disregarded. Still, though very weak, undulations play an important role. g) R > D + 2 d + 2 / c " . Counterion fluctuations do not couple to undulations. h) X F / k T < 1 for κά > 1. The electrostatic interaction of an independently fluctuating deflection segment is small enough to circumvent a virial expansion (13). 1 / 3
2 / 3
p
3
2
1
2
2
2
1/2
,/2
,/2
1
el
Equations 12 and 13 have the following features. 1) F o r κά< < 1 , the osmotic pressure tends to a previous result (14) for an hexagonal array of rigid cylindrical polyions i.e. our reference configuration (6x)»&c»
e-
(14)
2) F o r κά< < 1, the first-order correction to equation 14 reads
where 7 r < < 7 T and 7r - exp(-/cR/4). This gradual decay agrees with a qualitative sketch given by Selinger and Bruinsma (75), provided we neglect their nematic interaction in accordance with condition f. A n enhanced decay length was first proposed by Podgornik and Parsegian (75) but for a Gaussian random coil enclosed in a tube. This calculation is valid when the step length is smaller than the tube diameter which is the reverse of the case focused on here O S ) 1
os0
osl
Schmitz; Macro-ion Characterization ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
92
MACRO-ION CHARACTERIZATION
( L > > R ) . Note that an enhanced decay length shows up within a perturbation that is weak. 3) The case o f experimental interest is κά = 0(\) for which no simple power or decay law can be derived. Recall that in view of restrictions c and d, we cannot take the limit o f dominating exponentials in equation 12 which would lead to d « 2R/K. 4) Similarly, we cannot address the approach to the uncharged hexagonal phase which is strongly fluctuating. In that case, d = 0 ( R - D ) and ir ~ k T L ( R - D ) " as discussed at length by Selinger and Bruinsma (75). 5) The list of restrictions (a-h) looks awesome. Still, as a rule, the hexagonal gels formed in osmotic stress experiments ( D N A (76), tobacco mosaic virus ( T M V , 14, 17), muscle filament (14,18) satisfy these requirements for the polyions are relatively thick, stiff and highly charged. F o r a detailed comparison of theory with experiment, the reader should consult ref. (70). A Lindemann melting rule has also been formulated with the help of equation (72) (T. Odijk, Europhysics Lett., in press). It appears to explain the stability of the hexagonal phase o f D N A (16). p
2
m
1/3
8/3
p
Van der Waals forces Stability theories of the D L V O type have been developed for hexagonal lattices of rigid polyelectrolytes by various authors (19-21). The work of Parsegian and Brenner (27) inspired Millmann et al (77) to reinvestigate the classic experiments of Bernai and Fankuchen (22) on gels of T M V . H o w do undulations influence the interpretation of these measurements? T o begin with, we must keep in mind that requirements a-h of the previous section are rather restrictive when attractive forces are taken into consideration. In particular, I disregard entirely configurations like those shown in Figure 3 in which chains adhere. If adhesion does occur, the Gaussian distribution evidently becomes a poor approximation. Given these limitations, the first problem to be faced is the possible effect of undulations on the V a n der Waals interaction itself. The bare V a n der Waals interaction per unit length between two parallel rigid cylinders is known in terms of a hypergeometric function (23) but physical insight is afforded by studying the expansions at small and large separations (24,25). F o r an hexagonal lattice we have A /kT =-
HD*
2(R-D)
l/2
D
3
8.2 (R-D) '
A /kT=w
9TTH
D
128D
R
(16)
R-D
