2996
Langmuir 1995,11, 2996-3006
Depletion Zones in Polyelectrolyte Systems: Polydispersity Effects and Colloidal Stability Mats A. G . Dahlgren*rt and Frans A. M. Leermakers Department of Physical and Colloid Chemistry, Wageningen Agricultural University, P.O. Box 8038, 6700 EK Wageningen, The Netherlands Received November 18, 1994. I n Final Form: May 22, 1995@ We have used the Scheutjens-Fleer theory for polymers at interfaces to study depletion zone effects in polyelectrolytesystems. The segment density profiles of depletion regions are independent of the chain length, N , and the depletion interaction is often fully repulsive. In polyelectrolyte systems, a depletion zone develops both at adsorbing and at nonadsorbing interfaces, especially at low ionic strengths. Interactions which are repulsive at some surface separations and attractive at others can also be found, e g . , for intermediate ionic strength conditions or for very low charge density in the chain. At very high ionic strength or very low charge density, the classical neutral polymer depletion layers develop, causing attraction between particles in solution. Effects of polydispersity of the polyelectrolytes on the segment density profiles are very small. There is a minor effect on the depletion region at low ionic strengths, provided that there is not a large fraction ofvery short chains present. Only depletion zones near adsorbing surfaces tend to be preferentially populated by low molecular weight polymer. Under these conditions, long chains are still depleted from this filled-up depletion region, which has immediate implications for the equilibration of the adsorbed layer: The diffusion of long chains through this region to the surface is slow. At an adsorbing surface, preferential adsorption of the longer polyelectrolytesover the shorter ones results in a fractionation of the chains at the adsorbing interface, if equilibrium can be reached.
Introduction Polyelectrolytes have important applications in paperpaint and food i n d ~ s t r y and , ~ waste water treatment.4 In recent years, several investigations of polyelectrolyte adsorption have been reported, both from an experimental and a theoretical point of view. Experiments presented include surface force measurement^^-'^ and studies of adsorption to suspended minerals14J6 or cellulose."+'* Theoretical studies include mean field ~ a l c u l a t i o n s ~as J ~well - ~ ~as Monte Carlo simulation^.^^^-^^ In this investiation, we have focused on systems without adsorption energy and with good solvent conditions for
* Author to whom all correspondence should be addressed at the Royal Institute of Technology. + Permanent addresses: Laboratory for Chemical Surface Science, Department of Chemistry, Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden, and Institute for Surface Chemistry, P.O. Box 5607, S-114 86 Stockholm, Sweden. Abstract published in Advance A C S Abstracts, J u l y 15, 1995. (1)Jones, G. D. Uses for Cationic Polymers. In Polyelectrolytes;Frich, K. C.; Klempner, D.; Patsis, A. V., Eds.; Technomic: Westport, CT, 1976. (2) van de Steeg H. G. M. Cationic starches on cellulose surfaces, a study of polyelectrolyte adsorption, Ph.D. thesis, Wageningen Agricultural University, Dept. of Physical and Colloid Chemistry, 1992. (3)Evers, 0. A,; Fleer, G. J.; Scheutjens, J. M. H. M.; Lyklema, J. J . Colloid Interface Sci. 1986, I l l , 446-454. (4)Eriksson, L.; Alm, B. Water Sci. Technol. 1993,28, 203-212. ( 5 ) Dahlgren, M. A. G. Langmuir 1994, 10, 1580-1583. ( 6 ) Dahlgren, M. A. G.; Claesson, P. M.; Audebert, R. J . Colloid Interface Sci. 1994, 166, 343-349. (7) Dahlgren, M. A. G.; Waltenno, A.; Blomberg, E.; Claesson, P. M.; Sjostrom, L.;.&esson, T.; Jihsson, B. J.Phys. Chem. 1993,97,1176911775. (8) Berg, J. M.; Claesson, P. M.; Neuman, R. D. J. Colloid Interface Sci. 1993,161, 182-189. (9) Dahlgren, M. A. G.; Claesson, P. M.; Audebert, R. Nordic Pulp Pap. Res. J . 1993, 8, 62-67. (10)Claesson, P. M.; Ninham, B. W. Langmuir 1992,8,1406-1412. ( W M a r r a , J.; Hair, M. L. J. Phys. Chem. 1988,92, 6044-6051. (12) Afshar-Rad, T.; Bailey, A. I.; Luckham, P. F.; Macnaughtan, W.; Chapman, D. Colloids Su$. 1987,25, 263-277. (13) Luckham, P. F.; Klein, J. J.Chem. SOC., Faraday Trans. 11984, 80. 865-878. @
the polymers. In low ionic strength solutions, polyelectrolytes adsorb in thin layers at an oppositely charged surface. Typically in these situations, the thin adsorbed layer is followed by a depletion zone, where the polymer concentration is low relative to the bulk, when going further into the solution away from the surface. At a n uncharged interface, both polyelectrolytes and uncharged polymers develop a depletion zone. The behavior of such depletion zones for uncharged polymers is w e l l - k n ~ w n , ~ ~ - ~ ~ but little is known for polyelectrolytes under these conditions. The existence of such depletion zones have earlier been suggested from experimental findings on polyelectrolyte de~orption.~'The aim of this study is to (16)van de Steeg, H. G. M.; de Keizer, A.; Cohen Stuart, M. A.; Bijsterbosch, B. H. Colloids Surf. A: Physicochem. Eng. Aspects 1993, 70, 77-89. ( l 7 ) v a n de Steeg, H. G. M.; de Keizer, A.; Cohen Stuart, M. A.; Bijsterbosch, B. H. Colloids Surf. A: Physicochem. Eng. Aspects 1993, 70, 91-103. (18)Tanaka, H.; Odberg, L.; WBgberg, L.; Lindstrom, T. J. Colloid Interface Sci. 1990. 134. 219-228. (19)Israels, R.; Scheutjens, J.M. H. M.; Fleer, G. J . Macromolecules 1993.26. 5405-5413. (20) van de Steeg, H. G. M.; Cohen Stuart, M. A.; de Keizer, A.; Bijsterbosch, B. H. Langmuir 1992, 8, 2538-2546. (21) Bohmer, M. R.; Evers, 0. A.; Scheutjens, J. M. H. M. Macromolecules 1990,23, 2288-2301. (22) vander Schee, H. A.; Lyklema, J.J.Phys. Chem. 1984,88,66616667. (23) Sjostrom, L.; k e s s o n , T.; Jonsson, B. J. Chem. Phys. 1993,99, 4739-4747. (24)Beltran, S.; Hooper, H. H.; Blanch, H. W.; Prasunitz, J. M. Macromolecules 1991,24, 3178-3184. (25) k e s s o n , T.; Woodward, C.; Jonsson, B. J. Chem. Phys. 1989, 91,2461-2469. (26)Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (27) Vincent, B. Colloids Surf. 1990,50, 241. (28) Fleer, G. J.; Scheutjens, J. M. H. M.; Vincent, B. The Stability of Dispersions of Hard Sperical Particles in the Presence ofNonadsorbing Polymer. In Polymer Adsorption and Dispersion Stability; Goddard, E. D.; Vincent, B., Eds.; ACS Symposium Series 240; American Chemical Society: Washington, D.C.,1984. (29) Scheutjens, J . M. H. M.; Fleer, G. J.Adv. Colloid Interface Sci. 1982, 16, 361-380; Erratum in Adv. Colloid Interface Sci. 1983, 18, 309-310. (30) de Gennes, P. G. Scaling Concepts in Polymer Physics, Cornel1 University Press: Ithaca, NY, 1979.
0 1995 American Chemical Society
Depletion Zones in Polyelectrolyte Systems
Langmuir, Vol. 11, No. 8, 1995 2997
since the system is symmetric, only calculations for 0 5 z IM a r e necessary. Each layer is divided into L cubic lattice sites of side length L The volume-to-surface ratio
2 = 0
1
2 3
of the system, VIA,is given by VIA= Md The chemistry of the system is mimicked by monomers, also called segments or (polymer) units. All these monomers are of the same size, exactly filling one lattice site each. Molecules are then made up of one or more unit(s). Every segment in a molecule is given a ranking number, s, which determines its placement in a chain (molecule) (s = 1, 2, ...,Ni; where Ni is the number of segments in molecule i). For monomeric molecules, such as the solvent, only s = 1is present. For chain molecules connectivity information for the chain segments is utilized in the chain statistics, an essential part of the theory used to calculate the equilibrium density distributions of the molecules near the interfaces. In this procedure, bond parameters, Az-zr, are used. Since there are three layers, z’ = z - 1, z, and z 1, to connect to monomer s in layer z from, there are three 2 s . A0 is the apriori probability of reaching segment s in z from another site in z , and 2-1 and A1 are the probabilities of connecting to segment s from layer z - 1 and z 1,respectively. For a simple cubic lattice, which we used in all calculations reported here, A0 = 416 and A-1 = AI = 116. In general, A0 A-1 AI = 1, irrespective of lattice geometry. In effect, the use ofZs implies a Markov approximation in the chain statistics. The mean-field approximation implies that the potential for monomer of kind m is the same throughout layer z and is only effected by the average constitution of layer z and the surrounding layers. Thus, for a monomer of species m the potential in layer z , u&), is of primary interest to calculate (monomers are indexed m , n). The potential u,(z) will be considered in more detail below. For each segment type m a Boltzmann-like weighting factor is defined:
+
Figure 1. Two-dimensionalcartoon of the lattice used with three octamers, two adsorbed (one adsorbed on each wall) and one free. The arrow indicates the reflecting plane. The monomeric species [solvent (water), cations, and anions] are not indicated but they fill up the remaining sites exactly. investigate these regions with lower than bulk densities for polyelectrolytes. We will investigate the depletion zones near adsorbed layers as well as those near nonadsorbing interfaces. Since all polymers used in experiments and in applications are polydisperse to some extent, it is of importance to study the effect of polydispersity. A detailed theoretical analysis of the polydispersity effects in polyelectrolyte systems has not been presented before. Polydispersity is a well-known complication when directly comparing experimental results to those of theoretical investigations. However, it has been claimed that the polydispersity of polyelectrolytes plays only a minor role in experiments, e g . , in surface force measurement^.^,^ We have included polydispersity effects in our modeling to see ifthese claims can be supported by the theory.
Theory In this work we have used the self-consistent-fieldtheory of Scheutjens and Fleel.32-34and the extensions thereofto polyelectrolytes21,22 and to polydisperse systems.35 In this section we review the basic equations and approximations used in this theory. In the Scheutjens-Fleer theory, inhomogeneous polymer systems are modeled as a system where the polymer units and solvent molecules reside on discrete sites defined by a regular lattice. The solution between one surface and the mid-plane between two surfaces is divided into M lattice layers, numbered 1, 2, ..., z , ..., ( M - l),M . In each layer, a mean field approximation is applied. This paper only deals with planar systems, and therefore our system is divided into parallel layers, as depicted in Figure 1. At layers z = 0 and z = 2M, impenetrable walls are situated. In the plane between layers M and M 1, reflecting boundary conditions are chosen. We note that
+
(31)Meadows, J.;Williams, P. A,; Garvey, M. J.; Harrop, R. A,; Phillips, G. 0. Colloids Surf. 1988,32, 275-288. (32)Scheutjens, J. M. H. M.; Fleer, G. J. J . Phys. Chem. 1979,83, 1619-1635. (33)Scheutjens, J. M. H. M.; Fleer, G. J. J . Phys. Chem. 1980,84, 178-190. (34) Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1985,18, 1882-1900. (35)Roefs, S. P.F. M.; Scheutjens, J. M. H. M.; Leermakers, F. A. M. Macromolecules 1984,27,4810-4816.
+
+ +
( K g is the Boltzmann constant; Tis the temperature.)This weighting factor is for a free, unconnected segment directly linked to the density distribution of this segment. In eq 1the reference state is the bulk solution, where uk = 0, and, hence, Gb, = 1(we use the superscript “b”to denote bulk properties). This implies that G&) should be regarded as a measure of the extra affinity monomer species m has for layer z with respect to the bulk, provided it is not connected to other segments in a molecule. In the following,we will denote molecule species by the indices i and j , and treat every chain length of a polydisperse polymer as a different molecule species. A method for handling a set of molecules of different lengths as one species was recently given by Roefs et al.35 In the general case, one considers segments in an N-mer, the segments of which are denoted [l,2, ..., (s - l ) , s, (s 1)...,(N - l),Nl. The statistical weight of segment s in layer z is dependent both on the weight of the chain fragment [l,...,(s - 111 and that of the fragment [(s 11, ..., Nl. In order to calculate the statistical weight of segment s in layer z belonging to molecule i, given that it is connected to segment 1through one chain of segments and to segment N through another chain part, expressions for the weight of s being the last segment in [ l , ...,SI, and one for s being the last segment in [N, ..., SI are needed. These latter quantities, the end point distribution functions, are denoted by Gi(z,sll) and Gi(z,sIN),respectively. The quantity Gi(z,sll) can be calculated using the average weight of the end point distribution functions for segment (s - l),(Gi(z,s-l)ll)), and Gi(z,s)according to
+
+
Dahlgren and Leermakers
2998 Langmuir, Vol. 11, No. 8, 1995
G,(z,sll) = (Gi(z,s-lI1))G1(z,s) where This procedure is carried out for each species in the solution. It is possible to find the overall volume fraction distribution of a segment type m:
s=l
1
1 if segment s of molecule i is of type m (4) 0 otherwise By recursive use of eq 2, the entire chain is gone through until segment N is reached. The initial condition used is Gi(z,lll) = C,GT'lGm(z). In this way, Gi(z,sll) can be calculated from {Gm(z))and, hence, from {u,,,(z)}.Gi(z,sIN) is calculated analogously [Gi(z,sIN)= (Gi(z,s+llN))Gi(z,s) and Gi(zJVIN) = CmSY,NGm(z)l.Now, the statistical weight of segment s being part of molecule i in layer z is given by the composition law
(y=
where the free segment weighting factor Gi(z,s) is the denominator is included to compensate for the doublecounting due to its appearance in both factors of the numerator. For monomeric molecules, eq 5 reduces to
A molecule with at least one segment situated in layer z = 1is said to be adsorbed; a moleculewhich has no segment in layer z = 1is said to be free. It is possible to calculate the density profiles for free and adsorbed molecules straightforwardly.21 From the p's for all species in the solution, the potentials in all layers can be calculated. The potential um(z)is assumed to be made up of three terms. One term which must be included in all systems is the hard core potential, u'(z), which ensures that the condition Cmpm(z)= 1 is fulfilled for all 2. The potential u ' k ) is independent of segment species. The second term present in the equation for u&) is due to short range interactions, described by Flory-Huggins x parameters. The third term is accounting for the long range electrostatic interactions. The total expression for um(z)then reads: n
(12)
where This procedure is carried out for all s: 1 5 s 5 N. For normalization purposes, the chain weighting factor, Gi(NIl), is defined: M
G,(NI 1)= CG,(zflI 1)
(6)
z=1
From the statistical weight the volume fraction of segment s in layer z , pi(z,s), can be calculated according to:
where Ci is a normalization factor which depends on the amount of molecule i in the system. The total amount of molecules of species i is then obtained from summation over all monomers in the molecule and over the whole space: M
M N,
M
z=ls=l
z=1
(13) and the permittivity of layer z, &), for which the density weighted average can be used as an approximation, (14)
(8)
m
s=l
If the bulk concentration, p:, is fixed, Ci is defined as b
vi c.= Ni -
+
NL
ei= C q i ( z )= CCpi(z,s)= CCiCGJz,sl1,iV) z=l
and vm = the valency of segment type m , e = the proton unit charge, and Y(z)= the electrostatic potential in layer z. The termxnn[(pn(z)) -p :] gives the contribution to the segment potential due to contacts between monomer of species m and n (xmm = 0 by definition). The summation is taken over all monomer species in the system, including the surfaces. For the surfaces, a fixed concentration profile applies, in which p"(0) = ps'%2it4 1)= 1 and pa"%) = 0 for all other z . (Superscript "surf" denotes surface properties.) Two quantities which have to be calculated for each layer are the total charge in layer z , q(z),
(9)
(For a monomeric molecule (Ni= l),eq 9 immediately gives Ci = p:.) If the total amount of i in the system, Bi (expressed in number of equivalent monolayers), is fxed, Ci is calculated from
where E,,, is the permittivity of species m (E,,, = E ~ , ~ E o The ). charge q(z) is assumed to be localized in the midplane of the layer. Electrostatically, the layers make up a multiplane capacitor. The capacity of the capacitor formed by layers z and z 1, C(z,z 11, can be expressed as
+
+
Y(z) can then be expressed in terms of q ( z ) , and the adjacent layers' Y and corresponding C's:
Langmuir, Vol. 11, No. 8, 1995 2999
Depletion Zones in Polyelectrolyte Systems
Y(z) = C(z - l,z)Y(z - 1) q(z) C(Z lJ)Y(Z C(z - 17) C(ZJ 1)
+
+
+
+
+ 1) (16)
+
Equation 16 is a descretized version of Gauss's law. With proper boundary conditions, Y(M) = Y(M
+ 1)
where an asterisk refers to the pure amorphous phase and qzi = the volume fraction of segmet type m in pure i. The summation overj is taken over all molecule species. The free energy of the system can be calculated from the density and potential distributions by the following expression:
ei
4
qm(z)um(z)
M z=1 m
k,T
+
4=1 m n
zi
m n
J\
- ~
(17b)
eq 16 can be used to calculate Y(z) iteratively. Note that the use of eqs 16,17a, and 17b ensures electroneutrality for the whole system. The circular definition of q in terms of u, which is itself defined in terms of q, makes it necessary to iterate to find a solution to the set of equations which describes the system. When a set of {u,} is obtained which generates a set of { q m }which is used to find the same set of {u,} (within the desired numerical precision), a self-consistent solution is obtained. Since only equilibrium potentials can be obtained, only equilibrated systems can be described with this model, and dynamic or off-equilibrium aspects are not discussed in this paper. In the self-consistent-field theory thermodynamic data are readily available. The derivations of the equations have been presented in detail by Bohmer et a1.21and by Evers et ~ 1 The . expression ~ ~ for the chemical potential of species i reads (assuming that the electrostatic potential in the bulk is zero)
-- - -1n--CC LkBT ?Nj Gi(NI1)
1o
(17a)
Y(-1) = Yo
A(M)
i\
cp
r=O
where vS,y* = - q ( O ) Y ( O ) if the surface potential is fixed, zero otherwise. The excess energy, Aexc(M),can be calculated from
where the summation is taken over all molecules which are in full equilibrium with the bulk solution. By calculating Aint(M)/LkJ"= [Aexc(M) - Aexc(=)]/Lk~T for (36)Evers, 0. A.;Scheutjens,J.M. H.M.; Fleer, G. J. Macromolecules 1990,23,5221-5232.
10-9
0
I
I
I
I
10
20
30
40
50
z
Figure 2. Volume fraction profiles on a semilogarithmic scale for poly- and monodispersepolyelectrolytesin the presence of an oppositely charged surfacewith q(0)lL= -0.5 chargdattice site. Solid (coinciding)lines are for Poisson distributed (N, = 500), Schultz-Flory distributed (N, = 500, NJN" = 1.0021, Gaussian distributed ( N , = 501, u = 22.31, and monodisperse (N = 500) polyelectrolyte. The broken line is for block distributed polyelectrolyte. Calculations were done with M = 100, Otct = 0.505; and for all polydisperse samplesNmav = 999, Nmin= 1. For other system parameters, see the text.
systems of varying size (varying MI,a surface force-like interaction curve can be obtained.
Results and Discussion All results in this paper, unless otherwise stated, were obtained with a simple cubic lattice with a lattice side length of L= 0.5 nm and a cation bulk concentration of qialt= 1 x The anion concentration was varied to obtain electroneutrality in the bulk. The relative permittivities, er,i, are 80 for the solvent (water), 20 for the surfaces, and 5 for all ions and polymer segments. The calculations were done in a lattice with M = 100 layers. Where the surfaces carry charges, the surface charge is q(O)/L= q(2M 1)/L = -0.5 chargeAattice site. All x parameters in the system were set to zero. The chain lengths used were N- = 1,N,, = 999 for the polydisperse and N = 500 for the monodisperse polyelectrolytes. The monomers of the polyelectrolytes have a valency of Y = +l per segment. The mathematical expressions for the different chain length distributions are given in the Appendix. Polyelectrolytes Near an Oppositely Charged Surface. In Figure 2 we show typical density profiles for polyelectrolytes interacting with oppositely charged surfaces. No interactions besides the electrostatic are present. A general observation is that at low ionic strength the polyelectrolyte adsorbs to compensate the surface charge. When summing up the total adsorbed amount, there is a slight overcompensation of the surface charge (order of 1%).This is due to (partial) screening of the polyelectrolyte by counterions in the closest vicinity of the surface. The overcompensation causes the electrostatic potential to change signs which on its turn gives rise to a depletion zone. In experimental studies of polyelectrolytedesorption from polystyrene latex, evidence for the existence of a depletion zone next to a layer of adsorbed polyelectrolyte has been found.31 All profiles, = 11, except for the block distributed polyelectrolyte ("ni coincide. This shows that the actual shape of the chain length distribution is not playing any important role in these low ionic strength situations as long as the Nw/N, ratio is not very large ( N , is the weight average chain length and N , is the number average chain length of the polymer). Even in the block distributed sample, where Nw/N, is very large, the density profile in the first few
+
3000 Langmuir, Vol. 11, No. 8, 1995
Dahlgren and Leermakers
a.
10
4
,
I
I
I
I
0
10
20
30
40
z
50
Figure 4. Volume fraction profiles for a block-distributed In order from high p at z = polyelectrolytewith varying "in. 15: Nmin= 1(dotted line), 10 (brokenline), 50 (solid line), 100 (dotted line), 200 (broken line), and 500 (solid line). Other parameters are as in Figure 2.
- 10
0
10
20
30
40
z
50
Figure 3. (a) Volume fraction profiles for a Schultz-Flory-
distributedpolyelectrolyte (N,,= 500)withNJN,, = 1.01(broken line),NJNn = 1.05 (solid line), and NJN,, = 1.10 (dotted line) in the presence of a charged wall. Other parameters are as in Figure 2. (b) Electrostatic potential profile for the system in part a with NJN,, = 1.01. layers coincide with the other samples. However, in this latter case, the depletion zone has vanished. The effect of the spread of the distribution on the depletion zone outside a n adsorbing surface is shown in Figure 3a for a Schultz-Flory distributed polyelectrolyte with N,, = 500. The very small difference in the Q, profile again indicates that the actual shape of the distribution is not ofmajor importance. Again, the least deep depletion is found for the most wide-spread sample (NJN" = 1.101, which, naturally, has the largest fraction of short chains. (Experimentally, NJN,, can be larger than 1.10 in many cases.) This also lends support for the conclusion that it is the shortest chains which give rise to the (small) differences in the q profile seen between polydisperse and monodisperse samples. In Figure 3b we present the Y(z) profile for the system with N,/N, = 1.01 in Figure 3a. Here, the predominance of electrostatics becomes evident: Y(1) < 0, but Y(2) > 0. The potential Y reaches its maximum at z = 3 and is thereafter decreasing until z 2 30, where it starts to oscillate around zero. The entire depletion zone in Figure 3a is within thez range where there is a significant positive electrostatic potential. In this case the layers 2 5 z I5 have a polymer volume fraction higher than that of the bulk. Thus, strictly speaking, the depletion zone starts a t z = 6. In the region z 6 small loops are present, which develop because totally flat adsorption is entropically very unfavorable. Loops and tails are not restricted to layers z < 6, but are also present in part ofthe depletion zone (qfree
