J. Phys. Chem. 1993,97, 3924-3926
3924
Self-Consistent-FieldTheory for Confined Polyelectrolyte Chains
S.J.
Miklavic
Departments of Physical Chemistry and Food Technology, Chemical Center, P.O. Box 124, Lund, Sweden Received: February 24, I992
In a recent article appearing in this journal, Podgornik' continued his calculations on the properties of a system of polyelectrolytes confined between charged walls. This appears as third in a series of articles on the same theme2.' and follows recent simulation and mean-field studies on the subject conducted in this d e ~ a r t m e n t j - While ~ Podgornik's results (the pressure and overall monomeric density distribution between the two walls) behave, in the main, as anticipated, there remain some confusing implications of his model which we aim to address. However, the problem is not necessarily with Podgornik's general approach or with its execution but with a more fundamental issue upon which he has based his calculations. In fact, it is the question of the proper theoretical treatment of polymers as continuous "strings" near impenetrable boundaries which should be addressed. For this reason, it is probably more pertinent to formulate the discussion in a broader sense, encompassing and analyzing earlier works which have led to what I see as a difficulty with the formalism as applied by Podgornik and others tovarious problems. In a study aimed at determining the proper accounting procedure of polymer configurations near a wall, Di Marzio8 had considered alternative means of obtaining the correct polymer statistical weights. By way of a simple, one-dimensional, lattice random walk he has shown that onecan satisfy the basic statistical mechanical postulate of equal a priori probability by imposing on the walk an absorbing "boundary" condition at the lattice site nearest, but beyond, the actual wall location (see part a of the Figure 1). This method has long since been acknowledged as fundamentally sound, in direct contrast to an earlier belief that one should apply a reflecting boundary condition a t the wall itself (see Di Marzios for appropriate references). In the sequel to his discussion he argued that, in taking the continuum limit of this discrete random walk (i.e., a vanishing lattice spacing together with an increasing number of steps in the walk), in which one obtains the classical diffusion equation representation for the configuration probability, the absorbing site adjacent but exterior to the polymer accessible region "converges" to the real wall. While the solution of thisdiffusion equation, G(z,t), the probability that the diffusive "particle" (or polymer configuration) starting anywhere ends at position z in a "time" (i.e., continuous polymer index), t , indeed complies with the condition G(z=O-,t) = 0 where z = 0- denotes the wall position approached from the left (Le,, z=O - e as e 0), the continuum limit, he argued, justified the apparently trivial step of replacing eq 1 with
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G(z=O,t) = 0 (2) However, it should be realized that taking the continuum limit of discrete systems is generally a nontrivial exercise and that there is, in this case, a subtle but important difference between these two conditions! Nevertheless, the classical diffusion equation together with eq 2 has thus far been employed in situations pertaining to polymer-induced colloidal stabilizationlo-I I and polymer adsorption,l2-I4 as well as to the study of the interaction of charged colloidal particles mediated by a polyelectrolyte sol~tion.l-~*~~-~~ 0022-3654/93/2091-3924f04.0010
Edwards and Freed]I have since considered the configurational problem of polymers which are not confined to a lattice but are allowed to sample all available free space subject to potential bonding constraints between monomers. In the continuum (polymer) limit the chain of connected point particles becomes a continuous polymer "path" and the resulting configuration probability, G(r,r';L), can be represented as an integral over all paths of length L between fixed end points, rand r'. This integral has been shown to be reducible to a diffusion equation. To treat the wall constraint these authors, for sake of argument, replaced the hard wall with a more smoothly varying 'field" potential which, more or less, confined the polymer to the region of interest (termed the polymer accessible volume, V). They then turned up the strength and sharpness of this confining potential until it resembled that of a hard wall (part b of the figure). The result of this procedure was simply the statement that their G 'vanished a t and outside the walls". Freed]* later advanced an even simpler terminological determination of the boundary condition based on the assumption of continuity of the probability function. As this distribution function, G(r,r';L), can be viewed as the Green's function for the diffusion equation, being zero outside the field of interest and continuous and differentiable everywhere inside V (except at r = r'), its continuity was said to extend across the boundary, S, of V, 'thus" being zero a t the boundary. To best exhibit the physical consequences of a purely absorbing boundary condition, one need only focus attention on the relatively simple problem of an ideal chain confined between two impenetrable walls, infinite in lateral extent. In this case, from translational invariance, the probability function depends only on the normal coordinates z and z'. In accordance with the absorbing boundary conditions, the planes coincident with the two walls, at z = f a , are nodal planes, that is, at which the solution to the diffusion equation vanishes. In fact, the most general solution is an infinite sum over all eigenfuncti~ns~ each of which has a node a t thelocations of the walls (each eigenfunction of increasing order having an increasing number of nodal points between the planes). With the probability distributiondetermined one can calculate the monomer (density) distribution between the walls which, in the case of discrete monomers on chains starting and ending anywhere, is written as N l N C(Z)= ( C ~ ( Z - Z , ) ) = - - C G ( z , n ) G(z,N-n) n= I
Zn=l
(3)
It is obvious from this equation that because G vanishes a t the walls then so too does the monomer density, c. Consider the limit of a vanishing surface separation in which the polymer (of point particles) essentially sample a two-dimensional space: where are the monomers to sit if not on the walls? This feature is exhibited in the above-mentioned applied works. In particular, the seriousness of this unphysical density profile (being zero a t the walls) cannot be ignored in those cases where the walls really do adsorb monomers (that is, when there is an attractive monomerwall potential)12-14or when the walls are oppositely charged to the monomeric units of a polyelectrolyte which has been studied by Podgornik and o t h e r ~ . ' - ~ J s In - ~ ~these cases, it would be expected for energetic reasons that monomers would be driven to the walls resulting in a nonnegligible density there! The zero density result also has grave thermodynamic implications. We have shown recently6 that (for this ideal case of a noninteractive system) an exact statistical mechanical analysis predicts that the pressure in this system is directly proportional to themonomericdensityat contact with the walls: P = kT(c(*u). Although this derivation was based on a discrete treatment of monomers, it is trivial to show that an analogous result is valid on the continuum level. A zero contact density thus implies a zero pressure, a result which would then seem to be applicable 0 1993 American Chemical Society
Comments
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3925
\ Polymer Inaccessible Region
Polymer Accesible Region
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(a)
(b)
Figure 1. Schematic diagrams of cases referred to in the text. Part a is an illustration of Di Marzio’s one-dimensional random walk problem, shown
in the ( H I ) plane. The real ‘wall” is on the zeroth site while the absorbing boundary is placed on the (-1) site. In the continuum limit, the absorbing boundary approaches the real wall from the left. Part b is this author’s schematic interpretation of the limiting procedure imagined by Edwards and Freed. A repulsive potential, replacing the real wall (at z = O), keeps the polymer to the right-hand side. It is progressively increased in magnitude and steepness in their conceptualization.
over all surface separations, if the boundary condition were correct-a seemingly impossible conclusion. It is this manner of inconsistency apparent in Podgornik’s work (and others) which is disturbing. Given that the zero density is an unphysical conclusion what then is the source of the problem? One explanation to resolve this difficulty directly pertaining to the use of the conditions (1) and (2) will be advanced here.lg The idea can be expressed in very simple terms. In fact, in one way or another, the arguments employed by Di Marzio,8Edwards and Freed,’* and Freed18 arriving a t the absorbing boundary condition all suffer from a common principle flaw which can be summarized in the following way. All these authors take the knowledge of what the function G should behave like external to the system of interest (that is, zero) and take the outer limit to the wall, adopting the erroneous premise of continuity that it applies a t the wall itself and thus determines the properties of the system in the interior. In short, the error refers to the formal replacement of eq 1 with eq 2. In this system, the assumption of continuity of the function across a discontinuity (in reality an infinite potential barrier) cannot becorrect! In thegeneral theory of boundary value problems, there is said to be an equivalence between the values of a function at a boundary and values internal to the boundary; there is no mandatory stipulation of continuity across a boundary. The suggestion thus is that in order to obtain the true boundary condition for this problem one should consider a limiting process applied from the interior, V. With this philosophy the limit taken from theoutside (eq 1) is an appropriate boundary condition for the exterior problem. Take as an example the discrete lattice-based work of Di Marzio.8 From his discrete statistical mechanical demonstration, wecan safely assume that he is correct in proposing the absorbing condition a t the site adjacent and external to the wall (at I = 0). What then happens at the wall itself? One should perhaps analyze the effect the externally applied absorbing boundary condition has on the values of the configuration probability a t the wall site and on those interior sites nearest the wall and then take the limit
focusing on these values. Perhaps it should not be surprising that a more general boundary condition would ensue, for example, the mixed boundary condition, dG/&(O,f) - aG(O,t) = 0 (4) for some (constant) value of a which should be determinable from the properties of the polymer. This equation suggests that, throughout the limit process, while the value of the function decreases (as it should because the wall does exclude configurations) its rate of decrease also decreases proportionally. This would result in a small but finite probability (hence a discontinuity in value across the wall), intermediate between the results given by the more severe conditions of perfect absorbing and perfect reflecting walls, which would not necessarily conflict with Di Marzio’s original analysis. The use of eq 4 as a means of physically reconciling the use of the absorbing boundary condition has been a point of considerablediscussion in theliterat~re.*&*~ Theidea is to assume the existence of a phenomenological, short-ranged polymer-wall potential which leads “naturally” to the use of (4) with a now a property of the interaction potential: a < 0, an attractive wall; a > 0, a repulsive wall. The solution to the governing equation with (4) as boundary condition has been given p r e v i ~ u s l y *(see ~.~~ also the reply to this comment). Starting from the assumption of a repulsive potential, the limit a m, i.e., an infinitely repulsive wall, results in a vanishing G at the wall. While this has been considered adequate justification for the absorbing condition, it should be noted that the limit a +-a, Le., an infinitely attractive wall, leads in eq 4 to an identical analytical condition on G a t the wall. It seems then that, with this existing ambiguity, the G = 0 condition cannot be absolutely guarenteed on this phenomenological basis. It is more than likely that an even more general condition will apply
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F(G(O,t),dG(O,r)/dz, dG(O,t)/dt) = 0 (5) which is a reasonable eventuality for if one fixes one end of the
3926
The Journal of Physical Chemistry, Vol. 97, No. I S , 1993
polymer in space and then the condition to be satisfied by the remaining end should depend on the polymer index, t . As there is a formal analogy between polymer configurations and particle diffusion in the limit of long polymers, it might be insightful to recall that the problem of diffusing particles is in reality described by an integro-differential equation which, in principle, includes all the necessary information of the system of interest including the trueboundary condition. While thediffusion equation is but an approximation to the true solution, it can represent a quite accurate description if used together with a more correct boundary condition derived from the full integral formulation. Even in the simplest polymer problems the detailed structure of the latter varies from case to case. Thus, it is not unlikely that the actual boundary condition will depend on the specific system and that there is no universal condition. Such is the manner of result suggested by the work of Ploehn, Russel, and HaP4who have derived a complex boundary condition specific to the adsorption problem. To get this result they have taken into consideration the discontinuity introduced by an adsorbing hard wall and examined the physics of what occurs within, what can be called, one "mean-free path" of the wall. It is of interest to note that their result, obtained from the integral form in the limit of zero wall potential, in our notation is, aG(0 t ) x ac(0,t) G(0,t) - A = - at 2 aZ which is one form of eq 5 (here Xis their segment length). While the validity of eq 6 is in need of strict independent verification (possibly by simulation), its appearance demonstrates that an alternative to the pure adsorbing condition is obtained when one abandons the notion of continuity, or conversely, when one retains in the deduction process the physics of what occurs inside and near the boundary. To summarize, it has been argued that the absorbing boundary condition traditionally employed is considered to be in error. This has led to some thermodynamic inconsistency in the light of our recent analysk6 It was pointed out by one referee that the use of eq 2 does not introduce serious qualitative anomalies into thermodynamic quantities (such as the pressure) as derived from the system free energy, a fact which was acknowedged in regard to Podgornik's work earlier in this paper. However, the incon-
Comments sistency explicit in the contact formula, which arises from the assumptions in the model, means that one is automatically at a disadvantage when making comparisons with more exact treatments (such as simulation for which it is more difficult to obtain the free energy). While it is true that the explanation put forward here does not represent a quantitative resolution in the sense that it provides a definite practical alternative, it is hoped that a deeper and more careful analysisof the problem will have been stimulated.
References and Notes (I) (2) (3) (4) 2461. (5)
Podgornik, R. J . Phys. Chem. 1992, 96, 884. Podgornik, R. Chem. Phys. Letr. 1990, 174, 191. Podgornik, R. J . Phys. Chem. 1991, 95, 5249. Akesson, T.; Woodward, C. E.; Jiisson, Bo. J . Chem. Phys. 1989,9/,
Miklavic, S. J.; Woodward, C. E.; Jonsson, Bo; Akesson, T. Macromolecules 1990, 23, 4149. (6) Miklavic, S. J.; Woodward, C. E. J . Chem. Phys. 1990, 93, 1369. (7) Granfeldt, M. K.; Jdnsson, Bo; Woodward, C. E. J . Phys. Chem. 1991, 95,4819.
( 8 ) Di Marzio, E. A. J . Chem. Phys. 1965, 42, 2101. (9) Morse, P. M.; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, 1953; Chapters 6 and 7. (IO) Edwards, S. F. Proc. Phys. SOC.1967, 92.9. ( 1 1 ) Edwards, S. F.; Freed, K. F. J . Phys. A . 1969, 2. 145. (12) Dolan, A. K.; Edwards, S. F. Proc. R. Soc. London 1974,337, 509. (13) Dolan, A. K.; Edwards, S. F. Proc. R. Soc. London 1975,343,421. (14) Jones, I. S.; Richmond, P. J . Chem. Soc. Faraday Trans. 2 1977,73, 1062. (15) Muthukumar, M.; Ho, J-S.Macromolecules 1989, 22, 965. (16) Muthukumar, M. J. Chem. Phys. 1987,86, 7230. (17) van Opheusden, J. H. J. J . Phys. A . Math. Gen. 1988, 21, 2739. (18) Freed, K. F. Adu. Chem. Phys. 1972, 22, I . (19) An alternative view suggested independently by D. Chandler and C. E. Woodward holds that the difficulty presented here has its origin in eq 3.
Starting with a finite N polymer, the probability of finding a monomer on the The density, eq 3. wall is finite and a decreasing function of N, say F( I/"). is an increasing sum of N contributions of such quantities. It could be that in the limit N -, there would be a cancellation offactors which leaves the wall density finite. The error would thus lie with the premature interchange of limit processes. It may be that the way to overcome this difficulty would be to essentially renormalize the distribution functions which may then lead back to the choice of boundary condition (?). (20) de Gennes, P-G. Rep. Prog. Phys. 1969, 32, 187. (21) Eisenriegler, E.; Kremer, K.; Binder, K. J . Chem. Phys. 1982, 77,
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6296. (22) Freed, K. F. J . Chem. Phys. 1983, 79, 3121. (23) Nemirovsky, A. M.; Freed, K. F. J . Chem. Phys. 1985, 83, 4166. (24) Ploehn, H . J.; Russel, W. B.; Hall, C. K. Macromolecules 1988,21, 1075.
