J. Phys. Chem. 1984, 88, 6661-6667 AI = (h2/2rmlkT)1/2
with h, k , and ml being Plank’s constant, Boltzmann’s constant, and the mass of a molecule of species 1, respectively, and Iluo is an integral
6661
Aql loses the contribution from kinetic energy (first term on right-hand side of eq 41) and measures the difference AZlno of interactions of a molecule of species 1 with the two pure solvent environments. The activity coefficient is represented as follows n
R T In y1 = In this integral Nu is the number of solvent molecules in the system of total volume V, ulu(r)is the potential energy of a molecule of species 1 and a solvent molecule separated by distance r in the absence of any other molecules (Le., a vacuum potential energy), and gluois a “weakened” radial distribution function. We refer elsewhere for the meaning of the weakening, or coupling, parameter X.30~31The important point for us here is the correspondence between the superscript on g and the limiting operation of eq 39: glue, hence Zloo, reflects the mean potential energy of interaction of a molecule of species 1 with a pure solvent environment (one mole of species 1 added to an infinite amount of pure solvent). Note that when two different solvents are compared,
E l l j + ( I I u- ZlUo)
j- 1
(44)
The integrals I look just like Ila0in eq 43, but with j replacing (r (and the obvious meanings for Nj and u,,) and without a superscript on g,,, the “weakened” radial distribution function for solute molecules of species 1 and species j (where j can be 1 or some other solute species). The first sum in eq 44, then, represents the effect of solute-solute interactions in the solution at the specified concentrations. These interactions are mediated by, and so y 1 depends on, the solvent species, which appears implicitly in g l j . The second term in eq 44 is the effect of the change in solute-solvent interaction due to the nonzero concentrations of all solute species.
A Lattice Theory of Polyelectrolyte Adsorption H. A. Van der Schee and J. Lyklema* Laboratory for Physical and Colloid Chemistry, Agricultural University, De Dreijen 6, 6703 BC Wageningen, The Netherlands (Received: May 29, 1984)
The lattice theories for polymer adsorption of Roe and of Scheutjens and Fleer have been extended to include an electrical term. Equilibrium segment profiles and adsorbed amounts are computed as a function of chain length, solvent quality, electrolyte concentration, chain charge, surface charge, and adsorption energy. The most conspicuous difference with the adsorption of uncharged polymers is that, due to the repulsion between chain segments, loop and tail formation is suppressed, so that flat adsorption ensues with no influence of molecular weight. Addition of electrolytes screens this repulsion but very high ionic strengths are required to approach the behavior of uncharged polymers. Agreement with experiments is satisfactory.
Introduction The adsorption of polyelectrolytes is a matter of considerable practical and theoretical interest. The phenomenon plays a role in fields as disparate as water purification (adsorption of polyelectrolytes can lead to flocculation of particles, rendering them more easily filterable), food technology (e.g., as emulsifiers in food emulsions), pharmacy (also as emulsifiers), medical science (the polyelectrolyte heparine is an anticoagulant), soil structure, paint manufacture, etc. For a good understanding it is usually mandatory to have an insight into the conformation of the polyelectrolyte at the phase boundaries involved. From the theoretical side, very few attempts have been made to describe polyelectrolyte adsorption. The relatively most elaborate picture has been put forward by H e s ~ e l i n k - ~Although this theory is able to explain a number of influences on the adsorption at least semiquantitatively, it has a number of limitations, the most serious one being that a step function is assumed for the polymer segment distribution in the adsorbate whereas this distribution is just one of the principal parameters sought. As in Hesselink‘s model, Silberberg’s4model does not contain an ab initio derivation of the segment profile.
That theories for the conformation of adsorbed polyelectrolytes have not yet been developed is not surprising, since the underlying theory for the adsorption of uncharged macromolecules has only recently become available, mainly through the contributions of Scheutjens and Fleer.5~6Essentially, this theory is a lattice theory in which lateral interaction is accounted for in the mean field approximation. A characteristic feature is that individual conformations can be discriminated so that the contributions of trains, loops, and tails can be evaluated. The earlier Roe’ theory is similar to SF theory in that it is capable of describing the segment profile. However, it does not distinguish between individual conformations. Here we present extensions to the SF and Roe theories for the case where the adsorbed macromolecules are charged. Hence this is the first ab initio theory for the conformation of adsorbed polyelectrolytes. The most detailed picture is of course obtained with the SF theory. However, the Roe theory is relatively more satisfactory for polyelectrolytes than for uncharged polymers because with polyelectrolytes the formation of tails is largely suppressed, virtually ignoring that tail formation is the main actual shortcoming of Roe theory as compared with the S F approach. As the Roe procedure offers fewer computational problems, a large part of our results will be based on this theory.
(1) Hesselink, F. Th. In “Adsorption from Solution at the Solid-Liquid Interface”; Rochester, C. H., Parfitt, G. D., Eds.; Academic Press: London, 1983. (2) Hesselink, F. Th. J. Electroanal. Chem. Interfacial Electrochem. 1972, 37, 317. (3) Hesselink, F. Th. J. Colloid Interface Sci. 1977, 60, 448. (4) Silberberg, A. In “Ions in Macromolecular and Biological Systems”; Everett, D. H., Vincent, B., Eds.; Scientechnica: Bristol, England, 1978; p 1.
General Theoretical Principle The liquid phase adjacent to the adsorbent is mimicked by a lattice of coordination number z. The lattice layers are numbered ( 5 ) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979,817, 1619. ( 6 ) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (7) Roe, R.-J. J. Chem. Phys. 1974, 60, 4192.
0022-3654/84/2088-666l $ O 1.50/0 0 1984 American Chemical Societv
Van der Schee and Lyklema
6662 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 i = 1, 2, ..., M , 1 representing the surface layer and the layers beyond M constituting the bulk. Each lattice site can be occupied by either a polymer segment or solvent molecule. In order to better imitate real molecules, variations are possible in which polymer segments occupy more than one site or where the small ions have a finite volume. Backfolding is allowed, which is a restriction of the theory, especially in the case of polyelectrolytes, but this limitation is partly offset by the fact that it is also accepted in the bulk to which all interfacial quantities are normalized. At present, we discuss homopolymers only. In SF theory, the conformation of a molecule is taken as fully defined by the number of segments the molecule has in any layer i. A molecule is counted as adsorbed if it has at least one segment in i = 1. The probability P, of finding a certain conformation c is given by
P, = w,
i = 1 2 3 4
M
n p;' i1
with w,
=
xoqx1F-1-q
(2)
In these equations, p I is the weighting factor for individual segments in layer i (referred to p* = p M + 1 = p M + 2 = ... l), also called free segment probabilities. They represent the probability of finding a segment in i if this segment were not covalently bound in the polymer chain. ri is the number of segments that conformation c has in i. The factor w, accounts for the lattice-determined number of steps that can be made along the chain from a certain site to the next: Xo is the fraction of the number of possible steps remaining in i, XI is the same for a step from i to (i + 1) or ( i - l), e.g., for a hexagonal lattice Xo = 0.50 and A, = 0.25. r is the number of segments per chain and q the number of steps in i, all of this applying for conformation c. The degeneracy of conformation c is related to w, through
a, = Lz'lw,
(3)
where L is the number of sites per layer. The free segment probabilitiesp , depend on the available volume in i, given by 4°i/$o.(the superscript 0 refers to the solvent, the subscript * to the bulk, Le., 4°* = 4°M+l= = ...) and on the interaction energy with its neighbors, written according to Flory-Huggins as
-2x((4O,) - 4*) where
(4OA =
W01-+1X04O1 + X14O1+1
(4)
Here, x is the Flory-Huggins interaction parameter. For segments in the first layer, an adsorption energy parameter xs must be introduced. This quantity has been defined by Silberberg.* Finally, for charged macromolecules an electrical energy term for monovalent ions amounting to -me($i - $.)/kT = -abl -ye) must be added, where a is the degree of charging and the average potential at i. Consequently
PI =
4°1(40M)-1 exP[-2X((4°1)
- 4O*)1 exP(~1,Xs)exP[-a(Yl
- Y*)l (5)
It is inherent in this picture that for the average potential at i is chosen instead of the potential of mean force. Although for dissolved polyelectrolytes the difference is not too great because of compensation of corrections? it is difficult to assess the quality of the approximation for adsorbed polyelectrolytes. A rigorous analysis also has to involve chain rigidity which is beyond the present approach. In principle, the segment density profile (&) (Le., q5i as a function of i) can be obtained from (1) and ( 5 ) if $i(C$,) is available. (8) Silberberg, A. J . Chem. Phys. 1968, 48, 2835. (9) Fixman, M. J . Chem. Phys. 1979, 70, 4995.
E
bulk
4 4 01
4
Figure 1. Pictorial representation of an adsorbate and the potential distribution. The top shows two molecules, A is adsorbed, B is not. For A r = 32 and q = 14; for B r = 27 and q = 14. Bottom diagram: potential profile for a positive surface potential #,, and a positive adsor-
bate. Th? bulk potentials have maxima value is J/..
J/. at the plates and their average
Iteration is necessary as the free segment probabilities themselves depend on this profile. The equilibrium distribution is obtained by maximizing the canonical partition function with respect to the numbers of conformation and chains are generated by a matrix procedure, as described by Scheutjens and Fleer.s,6 If the Roe theory is used, a somewhat different path is followed. In this case the grand canonical partition function is written (eq 5 of ref 7) and maximized with respect to {C$J. If the adsorbate is charged, the electrical free energy of each profile has to be computed and included in the maximization.
Incorporation of Electrical Terms Most of the computations have been done on the basis of a model that is consistent with the mean field approach for the underlying picture of uncharged polymers, Le., the polyelectrolyte charges are thought to be concentrated in the centers of the lattice layers and smeared out. The charge on each layer per unit area is ui = f m e 4 , / a o
(6)
where a. is the area of the unit cell. Counterions and co-ions are considered point charges and distributed between these plate charges according to Poisson-Boltzmann (PB) statistics. As a rule, the full PB equation is used, Le., the Gouy-Chapman (GC) approximation is analyzed. As this equation cannot be solved analytically in our case, some computations have also been done using the Debye-Hiickel (DH) approximation. It is along the lines of the model to treat the bulk solution in equilibrium with the adsorbate also as a repeating array of plate charges; here the D H approximation is used throughout. In order to verify the quality of some of the above model assumptions, some computations have also been done with polyelectrolyte charges not confined to lattice layers but smeared out over the cells and another one with small ions of finite sizes. Figure 1 visualizes the model. Let us first consider the bulk situation. Following Moller et a1.,I0 we introduce averaged (10) Moller, W. J. H. M.; van Os, G. A. J.; Overbeek, J. Th. G. Trans. Faraday SOC.1961, 57, 325.
Lattice Theory of Polyelectrolyte Adsorption
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6663
stoichiometric concentrations fi+ and E- of the small ions per unit cell of volume aorO,where ro is the plate distance. Because of electroneutrality fi+
- It-
f a&/aoro = 0
a.e
(7)
The sign in front of the term containing 4. is the sign of the polyelectrolyte. If it is positive, fi+ may be identified as the stoichiometric salt concentration in the bulk polyelectrolyte; if the polyelectrolyte is negative, the same applies to fi-. The actual concentrations n,(x) at any position between two plates is given by n*(x) = fi* exp[YJ(x) - ?+I
i, so that it covers the entire bulk phase. At the crests
(8)
where p+ = e$+/kT, etc., $(*) being those values of $(x) where na(x) = fi+. For the region between the two plates the Poisson equation reads
In the adsorbate layer, potentials can become so high that application of the D H approximation is no longer justified. Also, the additivity and the symmetry with respect to the mid-distances r0/2 are lost. Because of the high y values, (9) must now be solved numerically without approximation. There are two boundary conditions. (1) At the adsorbent surface (dy/dx),=, = -uoe/ttokT, where uo is the surface charge density. (2) At the solution side, i = M , potential and field strength contain contributions from both the surface adsorbate charge and from the repeating bulk plane charges. The potential y Mdue to the bulk charges differs from y* in (18) since in (15 ) only the sum for x > xM is needed. It reads
+
yM(bulk side) = where tois the permittivity of free space and e the relative dielectric constant. As in the bulk the potentials are usually small, (9) will be analyzed in the D H approximation. Defining the reciprocal Debye length aslo
= (fi+
K'
+ fi-)e2/ttokT +
(11)
&a&/aoro- fi+jj+ - fi-jj-
e(ii+
+ ri-)/kT
Levine and Neale" pointed out that for the space between two layers (1 1) is valid to lower potentials than for a semiinfinite space near one plate, where fi+ = E-. The choice of b is equivalent to choosing a value for the average potential in the bulk. As our analysis contains only the difference in potential between bulk and adsorbate, this choice is arbitrary and for sake of convenience we set b = 0. This quantity does, however, play a role in describing the Donnan potential between bulk polyelectrolyte and dialyzate where the equilibrium electrolyte concentration equals 2fi+fi-/(fi+ fi-). Setting b = 0 implies that
+
1
+ jj+)
= fi-(l - j j - )
p+ = j j -
I jj*
(13) (14)
(dashed horizontal line to the right of Figure 1, bottom) In the DH approximation, the contributions to y(x) at any position x in the bulk due to all plane charges uI are additive. For one such layer, say layer i positioned at x = xi, the solution of (11) is y(i) =
- bulk side)]
K ~ M
(20) - K ( $ ~
- 2qM(bulk side)).
with
fi+(
[d~(x)/dxl,i,, = -
so that the total field strength at the solution side equals
d2y(x)/dx2 = K ' ~ ( x ) b]
(19)
and its contributions to the field strength is [dy(x)/dx],=,, = xyM(bulkside). The contribution of the charges on surface and adsorbate are given by
(10)
the PB equation becomes
b=
ale 2tto~kT
-[exp(Kro) - 1]-'
ule 2t€oK k T
-exp[rK(x - Xi)]
where the - and + sign apply to the semiinfinite spaces x > xi and x < xi, respectively. The total potential at x due to an infinite array of parallel charges ur separated ro from each other is
Subject to the two boundary conditions, (9) is solved by a sixth-order Runge-Kutta integration. For details of the mathematical procedure and iteration involved, see the Appendix. The obtained potentials are substituted in ( 5 ) . If the analysis is based on the Roe theory, the electrical free energy Pel of any preset segment profile is needed. Generally, F,, follows from integration of
where u[ is the charge on plate i during the isothermal-reversible charging process. Different paths are open to carry out this integration, e.g., depending on whether or not the small ions are concomitantly charged and whether or not the bulk polyelectrolyte is simultaneously charged. The equivalence of the former of these alternatives has been verified by Casimir in the absence of polyelectrolyte;12the equivalence in the presence of polyelectrolyte has now also been e~tab1ished.l~In view of the demands of the numerical procedures we found it most convenient to charge the adsorbed macromolecule with the bulk polyelectrolyte already having the final charge, the free energy of which acts as the reference. Computational details are collected in the Appendix.
Solutions for Low Potentials Especially if xs is not very high, the amounts of polyelectrolyte adsorbed remain relatively low and so do the potentials in the adsorbate. Under that condition the Debye-Hiickel (DH) approximation is also applicable for the adsorbent and analytical solutions can be given for the free energy. At any x in the adsorbate the potentials are additive contributions of all plate charges. Instead of (16) we have now for x between layers i and (i 1)
+
Y(X) = u0e
(16) ale exp[-K(x - xi)] + exp(-xro) exp[x(x - xi)] 2tco~kT 1 - exp(-nro)
=-
exp(-xx) (22)
The last term is the contribution of the surface charge go on the (17)
This equation applies to the region between xi and xi + r,, for any (1 1) Levine, S.; Neale, G. J . Colloid Interface Sci. 1974, 49, 330.
CEOKkT
(12) Casimir, H. B. G. In "Theory of the Stability of Lyophobic Colloids"; Verweij, E. J. W., Overbeek, J. Th. G., Eds.; Elsevier: Amsterdam, 1948; p 63. (13) van der Schee, H. A. Thesis, Agricultural University of Wageningen, Netherlands, 1984.
Van der Schee and Lyklema
6664 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
-
0-
;-.c.’
rl
-81, -10
,
10
5
,
, 20
1s
i
Figure 2. Segment concentrationprofile for an uncharged (a = 0) and fully charged (a = 1) polyelectrolyte: x = 0.5, x s = 2, r = 2000, uo = 0, 6. = lo4, a. = 1 nm2, ro = 1 nm. Hexagonal lattice. The concentration (M) of 1:l electrolyte is indicated. Roe’s theory.
adsorbent. Unlike in (16) all plane charges are now different, but the distribution can be computed if (q+) is known, using ( 6 ) . By the same token, for the potentials at the planes -
1
I
I
I
5
10
15
Figure 3. Effect of chain length on the segment concentration profile: x = 0.5, x s = 4, a. = 1 nm2, ro = 1 nm,uo = -e/lOao, 1:l electrolyte,
0.01 M. SF theory. OF,
h l
The ( i in the last term derives from the fact that the first layer is at r0/2 from the surface. In the SF picture these potentials are substituted in (5). If the Roe theory is used, the free energy is needed. Extending (21) to a multilayer system having in each layer an excess of charge over that on bulk layers amounting to (ai - a,) dX, if X is the charging parameter, gives dFeI = C+j(X)(aj i
US)
dX
I
20
i
f
(24)
a=l
which in view of the proportionality of to give FeI = XC$jCuj i
with X can be integrated
- a*)
(25)
Results Figure 2 is a typical example of a segment density profile. In order to exaggerate the qualitative features the plot is made semilogarithmic. The most conspicuous conclusion is that at low salt content, where charge effects of the polyelectrolyte count most heavily, a region of negative adsorption occurs. Significant adsorption takes place only in the layer(s) adjacent to the surface. In other words, polyelectrolytes adsorb in flat layers because the mutual repulsion between the charges on the chain inhibits loop formation. If electrolyte is added, the profile progressively resembles that of an uncharged polymer. For a = 0, the part at low i is linear, Le., the $i(i)distribution is exponential, but at large i @i decays more slowly to its bulk value 4.. Considering the experimentally established fact that hydrophobic colloids are coagulated by not more than 0.2 M (1:l) electrolytes, the very high salt concentrations that are needed to screen the polyelectrolyte charges completely are unexpected. However, this feature seems well-established; it is also observed if the theory is modified such as to let each small ion occupy a
J LA I
I
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6665
Lattice Theory of Polyelectrolyte Adsorption 0
h
-2
:1,
-
r.10
O01. 1
Q.
lo-'
0
5~10-~
Figure 7. Theoretical adsorption isotherms for polyelectrolytes as a function of the number of segments r per chain: x = 0.5, xs = 4, a = 1, u0 = -e/lOao, a. = 1 nm2, ro = 1 nm, M 1:l electrolyte. The amount adsorbed 0 is expressed in equivalent monolayers.
c
-8
I
1
I
I
5
10
15
20
Figure 5. Comparison between profiles in the Gouy-Chapman and Debye-Hiickel approximation. Poor solvent (x = 2) x, = 5 a = l, r = 100, otherwise as in Figure 3.
-3t I rl
,
Figure 8. First layer volume fraction as a function of xo,cff.Drawn curves: polyelectrolyte-0.05 M 1: 1 electrolyte. Dashed curves: polyelectrolyte4.1 M electrolyte. x = 0.5, a = 1, 6,= The chain length r is indicated. For comparison, the corresponding curve for the unch,arged polymer is also given (dotted curve).
tails only
-5
I 5
10
15
2o
I
25
Figure 6. Segment concentration profile for 4. = x = 0.5, xs = 4, r = 1000, a = 1, uo = -e/lOao, a. = 1 nm2,ro = 1 nm. No electrolyte
added. SF theory. more extended decay for CY = 0 in the S F case. However, for polyelectrolytes, loops and tails are suppressed and then the adsorbed amounts are almost indistinguishable. The difference in depth of the profile minimum does not measurably contribute. Figure 5 illustrates the effect of invoking the DH approximation: it overestimates the potentials and therefore predicts too deep minima and too low adsorptions in the first two layers. An interesting feature is observed at high 4. (Figure 6 ) . In this case the contribution of the tails to the profile has a maximum beyond the generally observed minimum. The high potentials at a short distance from the surface tend to expel tail segments from that region, but further away, where potentials are low, the chain can "curl up". This phenomenon can have some relevance for colloid stability, notwithstanding the fact that there are only few of them (about one per ten chains) so that on a weight basis they contribute by less than 0.3% and hence are easily overlooked. Typical adsorption isotherms are presented in Figure 7. They are of the high affinity type if r is not too low and, in agreement with the conclusions arrived at above, the plateau value is then insensitive to r. As is the case with the adsorption of the uncharged polymers, a certain critical value of the adsorption energy parameter xs is needed to let the polyelectrolyte adsorb at all, but once this critical value has been surpassed, the adsorption rises sharply with x,.
e,,:
I
CM-
-- ------__
x*=4
c*.10-'
0 . 7 ~ 0.60.5-
---->,=4
--------
.--2s=lo-'
--_----.
a. ( c i a , )
Figure 9. Influence of the surface charge: (-) Gouy-Chapman; (---)
Debye-Hiickel. xs and salt concentrationsc, are indicated. = :loo, ro = 1 nm, a. = 1 nm2.
x = 0.5, r
Figure 8 illustrates this by representing the volume fraction #J1 in the first layer as a function of the effective adsorption energy parameter, defined through (26) Xs,eff = Xs - CYY 1 Under any condition dl is less than for uncharged polymers. The rea.son is that the buildup of #J1 is entropically favored by the formation of loops and tails, but with polyelectrolytes this process is suppressed.
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
6666 1 .o
c 0
01
I
-3
I
-2
J 0
I
-1
log ( c , l M )
Figure 10. Adsorption of oligo- and polylysine on silver iodide as a function of chain length and concentration of HN03. Drawn curves: theory for x = 0.6, xs = 4.86, I#W = lo4, 01 = constant = 0.78, uo = 0.22e/a0,ro = 0.55, a. = 0.3 nm2.
The influence of the surface charge is demonstrated in Figure 9. The decrease of Oexc with uo is linear and steeper in more dilute electrolyte. The DH approximation overestimates the generated potentials and therefore predicts too low adsorptions. However, if the adsorbed amounts are low (this is the case for xs = 1) the difference between the GC and D H approximation vanishes. Increase of c, also makes the DH approximation relatively better because the potentials are then kept low.
Comparison with Experiment and Further Discussion Although the present paper is primarily concerned with theory, a brief comparison of some characteristic results shows that our conclusions do agree well with experimental trends. A series of experiments with oligo- and polylysine on silver iodide and polystyrene latices have been carried out, partly in cooperation with Dr. B. C. Bonekamp. At low pH, where the oligo- or polypeptide is fully charged, it was found that the adsorption rises with r if r is low but becomes independent of r at higher r, in agreement with Figures 3 and 5 . However, at high pH the adsorbate becomes uncharged and then 19 continues to increase with r, because then loops and tails can develop. A similar trend has been observed with the effect of electrolytes: the higher c,,,,, the higher the adsorbed amount and the more sensitive it is to r. In agreement with theoretical prediction (Figure 2) electrolytes continue to have an effect even at concentrations exceeding 0.5 M, a feature that was also observed with the adsorption of poly(styrenesu1fonate) on ~i1ica.l~More examples of the polylysine work have been reported e l s e ~ h e r e . ’ ~ . ~ ~ Several results of other authors may be quoted that at least semiquantitatively support our picture. For instance, Horn found flat adsorption for polyethyleneimines on polystyrene latex at low pH, where the adsorbate is charged16and Williams et al. reported the same for carboxymethyl cellulose on barium sulfate;” these authors found thicker adsorbed layers at elevated ionic strengths. Flat adsorption for charged polyelectrolytes has also been reported by Mabire et al.ls and by PavloviE and Miller.Ig All told, the predicted behavior is well-documented. Also the consequences for colloid stability are entirely in line with the present picture. A very typical demonstration is that at low pH, AgI sols undergo charge reversal due to the adsorption (14) Marra, J.; van der Schee, H. A.; Fleer, G. J.; Lyklema, J. In “Adsorption from Solution”; Ottewill, R. H., Rochester, C. H., Smith, A. L., Eds.; Academic Press: London, 1983; p 245. (15) Bonekamp, B. C.; van der Schee, H. A.; Lyklema, J. Croat. Chem. Acta 1983, 56, 695. (16) Horn, D. In “Polymeric Amines and Ammonium Salts”;Goethals, E. J., Ed.; Pergamon Press: Oxford, 1980; p 333. (17) Williams, P. A,; Harrop, R.; Phillips, G. D.; Robb, I. D.; Pass, G. In “The Effect of Polymers on Dispersion Properties”; Tadros, Th. F., Ed.; Academic Press: London, 1982; p 361. (18) Mabire, F.; Audebert, R.; Quivoron, C. J. Colloid Interface Sci. 1984, 97, 120. (19) PavloviE, 0.;Miller, I. R. J . Polym. Sei., Part C 1971, 34, 181.
Van der Schee and Lyklema of polylysine, but the critical coagulation concentration remains very similar to that in the absence of polyelectrolyte, whereas at high pH no charge reversal should occur although the critical coagulation concentration rises sharply. In line with our picture, charged polylysine adsorbs in flat layers and no steric stabilization ensues, whereas uncharged polylysine does stabilize AgI sols sterically because now loops and tails can develop. If we accept that there is a gratifying qualitative agreement between theory and experiment, the question rises as to how good the quantitative accordance is. This can only be judged on the basis of the question how realistic the parameters are that fit experiment to theory. Before discussing this, it should be realized that for the time being there are no satisfactory experimental data available that may be considered discriminative. First, because of analytical and surface area problems, experimental values of 0 are seldom better than a few tens of percents, all polyelectrolytes studied so far are heterodisperse copolyelectrolytes and most adsorbents are heterogeneous. Direct data on the distribution (&) are all but absent. From the theoretical side, apart from the inherent limitations of lattice theories, a number of other approximations have been made, of which it is difficult to assess the quality, e.g., the substitution of the average potential for the potential of mean force. Of others, such as the constancy of a and t and the zero volume of the small ions and the assignment of the charges to plates, rather than assuming a certain spatial distribution, it was verified that they had a relatively minor influence. Against this background it is too early to seek full quantitative agreement. However, as a general trendl3-I5 all data and trends are explainable with our theory, using reasonable parameter values. By way of illustration Figure 10 is included. The values for a. and ro correspond well with the volume of the lysyl group; further that x > 0.5 agrees with the fact that at high pH, where polylysine is uncharged, phase separation takes place. In conclusion, our theory predicts a behavior of polyelectrolytes at interfaces that is a t least in qualitative and semiquantitative agreement with the facts. Further elaboration of this theory and designing better experiments are therefore warranted.
Acknowledgment. The authors have benefited greatly from the help of Mr. J. M. H. M. Scheutjens with physical and mathematical problems. Appendix Calculation of the Potential Profile. The potential profile is calculated by integration of the Poisson-Boltzmann equation (9) between the two boundary conditions at the surface and at layer M . At each of the M planes through the centers of the lattice layers there is a discontinuity in the field strength amounting to
Because of these discontinuitiesthe integration has to be performed from layer to layer, with the end values of y and dyldx of the previous layer as starting values. To this end value of dyldx the discontinuity according to eq A1 is added. Within each layer the integration was performed using a sixth-order Runge-Kutta method.20p21To this end eq 9 is written as a system of two coupled first-order differential equations
-dY1 = - dY dx dx
(A3
with yl defined as y and y 2 as dyldx. The integration starts at y Mand dy,/dx, which are interrelated by eq 20. The value dy,/dx must obey the boundary condition (20) Zonneveld, J. A. “Automatic Numerical Integration”: Mathematisch Centrum: Amsterdam, 1964; Mathematical Centre Tracts 8. (21) Gear, C. W. “Numerical Initial Value Problems in Ordinary Differential Equations”; Prentice-Hall: Englewood Cliffs, NJ, 1971. (22) Stigter, D. J. Colloid Interface Sci. 1975, 53, 296.
J. Phys. Chem. 1984,88, 6667-6670 at the surface. To achieve this the value of yu is iterated with a Newton-type iteration. After each integration step across a lattice layer, the potential must be examined and, occasionally, the Newton-step length must be reduced to prevent overflows. Calculation ofthe Polymer Profile. Using the Scheutjens-Fleer theory, we followed their line in the evaluation of the polymer
6667
concentration profile, except for the modification that we used In +01 as the iteration variable to find the zero of +! +O,-l, where +I is obtained from 40i using (5). As eq 29 of Roe’s paper’ shows the derivatives of the partition function with respect to &, a term -abi - p)has to be added to the left-hand side of this equation, which is solved, using a Newton-iteration method.
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Single Molecule Gas-Phase Polymerization Kinetics of Vinyl Acetate. Nonsteady Measurements H.Reiss* and M. A. Chowdhury Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: July 23, 1984)
This paper continues our recent work on the kinetics of free-radical chain polymerization in the gas phase. As in the previous work, we avoid having the involatile polymers condense out of the vapor phase by employing a cloud chamber technique which allows us to study the kinetics under conditions such that there are so few polymers growing simultaneously that they cannot encounter one another. We detect the arrival of individual product molecules by having them nucleate macroscopic drops of monomer liquid. The arrival of product molecules is so discrete that signal averaging must be employed in the determination of the rate. As in the previous work, the polymer is poly(viny1 acetate). In the present study we employ a nonsteady technique in which the reaction is initiated by a pulse of photons. Among other things, this approach makes it possible to determine the degree of polymerization of the polymer capable of participating, along with monomer molecules, in the binary nucleation process and therefore makes it unnecessary to rely on a difficult nucleation theory. The monomer supersaturation has been deliberatelychosen to be high, so that high enough rates could be achieved to allow the signal averaging to be performed by hand. As a result, the polymers responsible for nucleation are small, containing only about six monomer units.
1. Introduction
In a recent paper’ a method was introduced, capable of measuring the kinetics of gas-phase chain polymerization reactions by detecting the arrival of single polymer molecules. This method involved the use of an upward thermal diffusion cloud chamber which maintained monomer vapor in a steady, supersaturated state. Ultraviolet photons, admitted to the chamber, produced free radicals and initiated chain polymerization. The polymer chains grew to a critical size at which they were capable of participating in condensation nuclei leading to the formation of drops of liquid monomer. The drops fell through a laser beam, scattered strong light signals, and were counted. The rate of drop formation was identical with the rate of production of polymers of the critical size and was therefore a measure of the kinetics of polymerization. Reference 1 contains a detailed verification of the validity of this method and its underlying mechanism and should be consulted for this purpose. The critical degree of polymerization (necessary for participation in a condensation nucleus) depends on the degree of supersaturation of the monomer vapor. By adjusting the supersaturation, it is possible to “tune” to the arrival of polymer molecules of a specified size. The series of events, outlined above, will only occur faithfully in the manner described, if conditions can be arranged such that each condensation nucleus contains only one polymer molecule. Thus, the number of chains growing simultaneously must be so small that there is little probability of two polymer molecules encountering one another. This, in fact, is the reason for hoping that the gas-phase process can be sustained. In other attempts to study such gas-phase kinetics it has not been possible to avoid polymer-polymer encounters and abundant condensation of involatile small polymer molecules. The enormous sensitivity of the nucleation phenomenon enables one to detect the arrival ( 1 ) Chowdhury, M. A.; Reiss, H.; Squire, D. R.; Stannett, V. Macromolecules 1984,17, 1436.
0022-3654/84/2088-6667$01.50/0
of single molecules and, therefore, to work under conditions (if they can be established) such that growing polymers cannot encounter one another and escape the vapor phase. In ref 1 emphasis was placed on ( 1 ) demonstrating that the observed droplet formation was due to the polymer process just described and (2) showing that conditions could be established such that only the “single polymer” condensation process occurred. As a somewhat secondary goal, the measured data were used to estimate the various kinetic constants. As explained in ref 1, the study was a “bootstrap” one, in which confirmation of the postulated phenomenon depended on the compatibility of experimental observations with one another, as well as with theory. This “bootstrap” character was mandated by the fact that no other method of comparable sensitivity was available to provide independent confirmation of the underlying process. When referred to these goals, the studies of ref 1 may be considered successful. The monomer involved was vinyl acetate, and with this monomer, the results of ref 1 did indicate that conditions for single polymer nucleation had been established. Furthermore, the above-mentioned compatibilities of experiment with experiment and of experiment with theory were observed. The experiments of ref 1 were conducted under steady illumination (under a steady flux of UV photons). Among other things, the critical polymer size had to be deduced from an approximate theory of nucleation.2 The accuracy of that theory could be assessed for the case of homogeneous nucleation of the monomer i t ~ e l f but , ~ one cannot be assured of its quantitative validity in the case of binary nucleation involving both monomer and polymer. Nevertheless, its application indicated that, in ref 1, polymers having degrees of polymerization of the order of 30 were responsible for nucleation. As explained in ref 1 , many improvements can be effected in subsequent studies. Even the degrees (2) Reiss, H.; Chowdhury, M. A. J . Phys. Chem. 1983,87,4599 (3)Chowdhury, M.A. J . Chem. Phys. 1984,80,4569.
0 1984 American Chemical Society
