Chain molecules at interfaces: A versatile self-consistent lattice model

Colloids and Surfaces, 35 (1989) 151-167 Elsevier Science Publishers B.V.,-Amsterdam - - Printed in The Netherlands

151

Chain Molecules at Interfaces: A Versatile SelfConsistent Lattice Model G.J. FLEER

Department of Physical and Colloid Chemistry, Agricultural University, De Dreijen 6, 6703 BC Wageningen (The Netherlands) (Received 31 December 1987; accepted 8 March 1988)

ABSTRACT This paper discusses the background of the Scheutjens-Fleer theory for chain molecules in a concentration gradient and gives a number of illustrative examples. The model is introduced starting from the adsorption of monomers and dimers at an interface. For chain molecules, all the possible conformations have to be taken into account. Any segment is assigned, in a mean field approximation, a weighting factor that depends on the local concentrations. Chain conformations are treated as step-weighted walks in a field that is determined by all other molecules. The model can be applied to a variety of situations. Several examples are given: adsorption of homodisperse and polydisperse polymers, interaction between polymer covered surfaces, terminally anchored polymer, polyelectrolye adsorption, surfactants at solid and liquid interfaces, and the structure of surfactant micelles.

INTRODUCTION

The behaviour of chain molecules at interfaces is of great importance for a variety of applications. Polymers are used extensively to modify properties of interfaces and to stabilize or destabilize colloidal dispersions. In some cases homopolymers are used, which are nearly always heterodisperse, especially if the polymer is a commercial sample. The presence of chains of different lengths leads to preferential adsorption effects that may influence the structure of the adsorbed layers considerably. Quite often the polymer chains carry charged groups, either strong or weak, that also affect the adsorption and stability. In order to obtain good stabilizing action, polymer hairs may be attached terminally to colloidal particles by a grafting procedure. Alternatively, block copolymers with hydrophobic and hydrophilic moieties are applied to improve colloidal stability, whereby one type of block secures strong binding to the surface and the other type functions more or less like grafted hairs. Also blocky or random copolymers are in use. Surfactants, being essentially block co-oligomers, are commonly used to 0166-6622/89/$03.50

© 1989 Elsevier Science Publishers B.V.

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modify the interfacial properties of many systems. They accumulate easily at surfaces, but can also form self-assembling structures in solution, like micelles and membrane-like bi- and multi-layers. These structures are very interesting from a scientific point of view and have many applications, both in industry and in biological systems. It turns out that a recently developed theoretical model due to Scheutjens and Fleer can cope with virtually any system in which chain molecules occur in a concentration gradient. In this paper, a relatively simple introduction to the background of the model is presented and some generalizations are outlined briefly. A selection of results is given for various systems, illustrating the versatility of the model. MODEL

In order to introduce the model in simple terms, we start from the wellknown Langmuir equation for the adsorption of monomers from solution [ 1 ]. Let us consider a flat surface in contact with a bulk solution where the following exchange equilibrium is established: monomer (bulk) +solvent ( a d s ) ~ m o n o m e r (ads) +solvent (bulk)

(1)

The concentrations will be expressed as volume fractions, ¢~for the monomers and ¢o for the solvent. We choose a semicrystalline lattice to describe the solution and number the lattice layers parallel to the surface as i-- 1,2...M, where i = 1 is the surface layer. In this layer the concentrations of monomers and solvent are ¢1 and ¢o, respectively. If the adsorption is restricted to a monolayer, all the other layers have the same concentrations as the bulk solution, ¢. for the monomers and ¢o for the solvent. The mass action law for the above equilibrium is: ¢,¢.o/¢.¢o =~..

(2)

where the equilibrium constant is expressed as a Boltzmann factor containing the dimensionless adsorption energy difference Xs between a monomer and a solvent molecule. The exchange parameter Xs was first defined by Silberberg [2]. For dilute solutions (¢°-~1), Eqn (2) reduces, with ¢1+¢o =1, to the familiar Langmuir expression ¢1/(1 - ¢1 ) = ¢ *exS. For the sake of later generalization we rewrite Eqn (2) as ¢h =P,¢* p, = ( ¢ o / ¢ t ) e z,

(3) (4)

where the quantity Pl can be considered as a weighting factor for monomers in

153

the first layer as compared to the bulk of the solution. It is merely a Boltzmann factor exp (Zs + In ¢~o/¢~o), accounting for the difference in energy (Zs) and entropy of mixing (ln 0 °/¢~o ) for the exchange of a solvent molecule in adsorbed state (i-- 1 ) and a monomer in the bulk solution ( i - - , ) . The Langmuir equation applies only for athermal solutions where the lateral interactions are absent: the sole reason for the accumulation of monomers in the surface layer is the interaction with the surface, and ~i--¢* for i>/2. In terms of weighting factors pi this means thatpi= 1 for i>~2. Only in the surface layer (i= 1) isp~ greater than unity, according to Eqn (4). In general, conditions are nonathermal because monomer-monomer interactions are different from monomer-solvent interactions. In a poor solvent m o n o m e r - m o n o m e r contacts are prefered over monomer-solvent contacts; in a good solvent the reverse situation exists. The solvent quality can be expressed through the well-known Flory-Huggins parameter Z [3]. Now additional energy terms occur in the weighting factors Pi. Moreover, the concentration profile will extend over more than one layer. In a poor solvent (Z> 0) the second layer will have a slightly higher concentration of monomers than the bulk solution because the high concentration of monomers in the first layer leads to some accumulation in adjacent layers (Fig. 1 ). That implies pi ~ 1 for several layers. Generally, we can write for the adsorption of monomers:

Oi=PiO*

(5)

Pi =O°e Oo -zu'/hT

(6)

Equation (5) is the same as Eqn (3), the only difference being the generali-

i !

I

~Pi

I

I I

I

...., i

1i

'ii i

13 I

=

i

Fig. 1. The concentration profile of adsorbing monomers in a poor solvent (Z> 0).

154 zation to all layers. Equation (6) contains the energy Aui for exchanging a solvent molecule in layer i and a monomer in the bulk solution. It can be expressed in terms ofxs and X:

-Aui/kT=xsJl,i + X( ( ¢~ } - ( ~o } )

(7)

where

( ¢ i ) =21~i-- 1 J'-20~i "~-21~i+1

(8)

and 61,i= 1 if i = 1 and zero for i ¢ 1. If X=0, Eqns (6) and (7) reduce to Eqn (4). The extra term in Eqn (7) results from the contact energy under nonathermal conditions (X¢0), accounting for the concentration gradient through the weighted average of ~ and ~o as defined in Eqn (8) [4,5]. A monomer in layer i has a fraction 2o of its contacts with sites in the same layer and a fraction 21 with each of the adjacent layers ( i - 1 and i + 1 ), 20 and 21 being lattice parameters. In a simple cubic lattice ( 6 neighbouring sites), 2 o = 4/6 and 21 = 1/6. Equations (5)-(8) constitute a set of M implicit equations in M unknown @i's. In principle, M tends to infinity because @i> ~* (Pi > 1 ) which leads, through Eqn (8), to ~i+1 > ¢~* (Pi+l > 1 ). In practice, the difference ¢~i- ~* becomes very small after a few layers so that for monomers M = 3 or 4 suffices (see Fig. 1 ). The set of equations can only be solved numerically; for X¢ 0 monomer adsorption cannot be described with a simple analytical expression like the Langmuir equation. For the description of the adsorption of chain molecules, Eqn (5) has to be generalized, taking into account the various conformational possibilities of the chains. Let us first consider dimers, consisting of two monomer units {segments). If the first segment of a dimer is placed in layer i (weighting factor p~), the second segment must be in i - 1 , in i, or in i+1. The probabilities of these conformations are proportional to p~p~_1, P~P~and PiP,+l, respectively. In these products, each of the segments is assigned a weighting factor depending on the layer number where that particular segment finds itself. The weighting factors Pi are still given by Eqn (6). However, an additional weighting factor is necessary to account for the relative number of "parallel" and "perpendicular" conformations. In a parallel dimer configuration the bond connecting the segments is parallel to the surface, in a perpendicular one it crosses into an adjoining layer. In a simple cubic lattice each bond can point in 6 different directions, 4 of which are parallel and 2 perpendicular to the surface. Hence, the bond weighting factors are 20 ( =4/ 6) for each parallel bond (i--.i) and 21 ( = 1 / 6 ) for each perpendicular bond (i-~i_+ 1). Now the conformation probabilities P (i j ) for dimers with the first segment in i and the second in j are p~2pj where 2 = 2 o if i=j and 2=21 if i=j+_ 1. For example, for dimers with at least one segment in the first two layers:

155

--pl,~1P2

P(1,1) =Pl~oPl

P(1,2) = P ( 2 , 1 )

P (2,2) --P2~o P2

P(2,3) -- P(3,2) -----P2~lP3

(9)

Once the conformation probabilities are known, the concentration profile can be computed by summing over all possible conformations. In order to illustrate this we take dimers in athermal solutions (X= 0), making the approximation that 0i is different from 0. only for i= 1,2. This is not strictly true because layer 2 (P2 > 1) has an effect on layer 3 through the conformation (2,3), leading to P3 ¢ 1. In turn, this makes P4¢ 1, etc. However, these effects are small and we assume 0i = 0. for i > 2. Equation (9) gives now all the relevant conformation probabilities. The volume fraction 01 in the first layer is due to one flat conformation (1,1) and two end-on conformations (1,2) and (2,1). The latter two have only half of their segments in layer 1. Since P ( 2,1 ) = P ( 1,2 ): 01/0" ----P(1,1) +2-½ "P(1,2)

(10)

02/0* --2" ½P(1,2) + P(2,2) + 2.½-P(2,3) From Eqns (9) and (10) we find 01 and 02 in terms of the weighting factors Pl andp2. According to Eqn (6) we have, for X-- 0, Pl = eXs(1 - 01 ) / (1 - 0-) and P2 = (1 - 02) / (1 - 0.). Therefore,

01 (1--0* )2/0* ~ ' 0 (1--01 )2e2XsJ¢-~1 ( 1 - 0 1 )(1--02 ) eXs

(11)

02 (1--0.)2/0" =it1 (1--01) (1-02)eX'+20 (1--02)2+~1 (1--02) ( 1 - 0 . ) Hence, for dimers at X=0 the concentration profile follows from two (quadratic) equations in two unknowns. Equation (11) is the equivalent of the Langmuir equation (2) for dimers. For X#0, the profile extends over more than two layers (compare Fig. 1) and a set of simultaneous equations (containing now also exponentials ex¢i) in the unknown 01, 02, 03.... can be easily constructed. Again, this set can only be solved numerically. It is not difficult to generalize Eqns (9) and (10) to chain molecules containing r segments and r - 1 bonds. For any conformation c we define the number r~c of segments in layer i and the number of parallel bonds qc. In Fig. 2 two of the many conformational possibilities are indicated for a chain of 14 segments. Conformation c is characterized by r~c=2, r2~= 1, r3c=3, r4c= 1, r~c=2, r6~= 1, r7~=3, rs~= 1 and q~=6. Conformation d has rid=5, r2d=4, r3d= 3, r4d=2 and qd = 6. Now the conformation probabilities are: P( c ) = ~8~ ~- l-q~ p~'~p~...p~'~.., p~y ~

(12)

because each of the q~ parallel bonds contributes a bond weighting factor ~o, each of the r - l - q ~ perpendicular bonds ~1, and each segment in layer i a segmental weighting factor p~. Note that p~ is still given by the expression for

156

i=

1

2

3

4

5

6

7

8

Fig. 2. Two possible conformations for a chain of 14 segments and 13 bonds.

monomers [Eqns (6) and (7) ] but that its numerical value depends on the concentration profile (which is different for monomers and chain molecules). It is easily verified that Eqn (9) is a special case of Eqn (12). In turn, the concentration profile can be obtained by summing over all conformations, analogously to Eqn (10):

~i = ~ (ric/r) P(c)

(13)

C

The only problem is to evaluate the sum. For long chains the number of different combinations is very large, and writing explicit expressions as in Eqn (10) is extremely cumbersome. Fortunately, a relatively easy matrix procedure [6, 7 ] is available to generate all conformations and do the bookkeeping. For more details we refer to earlier papers [4,5]. Like before we have in Eqns (12) and ( 13 ) a set of M implicit equations in M unknown ¢i's. In principle, M equals at least the number of layers over which the concentration gradient extends. The equations are strongly nonlinear, containing powers of order r and, for %~ 0, complicated exponentials ez<~i>. Therefore, the numerical complications in obtaining convergence of the iteration may be considerable. We will not discuss further details of the numerical procedure but concentrate on the physical background. Once the set of volume fractions ¢~iand, hence, the weighting factors pi have been obtained, the probability of any conformation can be computed. T h a t enables one to calculate several interesting structural characteristics of the absorbed layer, such as the fraction of segments in trains, loops and tails, the average train, loop and tail size, the train, loop and tail size distribution, the average layer thickness, etc. Also the free energy and entropy of the adsorbed

157 layer can be obtained. This feature is particularly interesting for the interaction between two polymer covered surfaces. So far, we have only considered the adsorption of homodisperse homopolymers at a solid surface. The model is, however, versatile enough to allow for several modifications. For example, the structure of a layer of chemically anchored polymer hairs can be computed by including in the sum of Eqn (13) only configurations with the first segment in the surface layer. For the adsorption of polydisperse polymers, an extra summation over chain lengths is necessary in Eqn (13). Sometimes polymers do not absorb so that a depletion layer near the surface develops where, because of entropical restrictions, no polymer is present [8]. The latter situation can be handled by putting Xs equal to zero. In all these cases the expression for the weighting factors remains unchanged. Results can be computed for one surface, but also for two surfaces in the presence of polymer [9,10]. In other extensions a different expression forpi is necessary, by introducing additional terms in the exchange energy Au; as given in Eqn (7). For polyelectrolytes, an electrical term aie~i enters, where ai and ~'i are the degree of dissociation and electrical potential in layer i. Both quantities can be expressed in the volume fractions ~: a~ through the dissociation constant and ~i through the Poisson-Boltzmann equation. To solve the latter equation, an extra iteration step is usually necessary [ 11-13 ]. Many polymers contain more than one type of segment (copolymers). Now one needs more Xs- and x-parameters for the interaction energies. In a system with n different types of segments (including the solvent) and one type of surface, (n-1)Z~-parameters and 0 . 5 n ( n - 1 ) x - p a r a m e t e r s play a role. This leads to more terms in Eqn (7) and different weighting factors for different segment types. These are all of the same structure and give, as a rule, no problems in the numerical procedure [14]. Since through Eqn (12) all possible conformations are considered individually, any type of copolymer (random, block, alternating,.... ) can be treated by specifying for each ranking number in the chain what type of segment (hence, which weighting factor) should be taken. A special case ofcopolymers are surfactants (block co-oligomers). Some types of surfactants are branched, such as lipids, having two hydrocarbon tails. Also branching can be incorporated in the model rather easily. In the above examples, a solid surface is present onto which adsorption takes place. It is also possible to treat adsorption at liquid/liquid surfaces between two solvents that are only partially miscible. Any chain molecules present can permeate through the interface and interpenetration of the two bulk phases (e.g., oil and water) occurs to some extent. Now the X~-parameters are irrelevant and only x-parameters determine the miscibility of the solvents, the solubility of the chain molecules in each phase, and the structure of the interfacial region [ 15]. Finally, it is generally known that surfactants and lipids can form self-as-

158

sembling structures as micelles and membranes. These can be handled in the model by replacing the surface by a symmetry plane. Also here Z~plays no role; the association behaviour depends only on a number of x-parameters (and, for charged surfactants, on electrostatics). For the description of flat membranes a lattice geometry with parallel lattice layers is adequate, but for micelles a different geometry is necessary. Recently, it has been shown how cylindrical and spherical lattices may be used in the model [16]. The most important modification is to define three lattice parameters 2_ 1, 2o and 21, giving the relative number of neighbours in layers i - 1, i and i + 1, respectively, for a site in layer i. In a flat geometry, 20 and 2_ 1= 21 are constants, but for spherical and cylindrical lattices these parameters become a function of the layer number i. SELECTION OF RESULTS

In this section we present a selection of results, chosen such as to illustrate the wide applicability of the model. Obviously, the various topics cannot be treated extensively. For more details we refer to papers that have been published elsewhere or are in preparation. Unless stated otherwise, the following figures were calculated for a hexagonal lattice (20= 6/12, 21= 3/12). Figure 3 gives adsorption isotherms in a dilute concentration region (0<0. < 10 -3) for homodisperse polymers of various chain lengths from a pseudo-ideal or theta-solvent (Z= 0.5) and from an athermal solvent (Z=0). The adsorbed amount 0 is expressed in equivalent monolayers. Monomers (r = 1 ) do not adsorb appreciably at Z~-- 1 unless the concentration is much higher than 1000 ppm. Oligomers (r= 10, 20) have rounded isotherms, resem500(1f

1000

100 f

20I-:

1000 100

10 , 0

10 - k

1,,

I

I 5.10 -~'

I

I

I ~p.

I F:I 10 - 3

Fig. 3. Adsorption isotherms for different chain lengths, for Zs= 1 and two solvency conditions: Z=0 5 (full curves) and x=O (dashed curves).

159

bling a Langmuir-shape (though, obviously, the Langmuir equation does not apply), and long chains show the high affinity isotherms that are characteristic for homodisperse polymers. In poor solvents, the (pseudo)plateau increases with molecular weight (for r > 50, this increase is linear with log r [4 ] ) whereas in good solvents the adsorbed amount is much less and nearly independent of chain length. All these trends are in agreement with experiments on well-defined homodisperse polymers. Figure 4 shows a segment concentration profile for very long chains (r= 104 ) adsorbing from dilute solution. The overall profile is decomposed in the contribution of loops and tails. In accordance with older theories, the segment concentration due to loops decays exponentially as a function of the distance from the surface, but at large distances (i > 25 for this chain length) long dangling tails dominate in the adsorbed layer. Although in this region the segment volume fraction is small (below 1% ), it is of the same order of magnitude as that within a free coil in solution. Tails, that have been neglected in most other theories, may thus have a considerable effect in, e.g., the hydrodynamic layer thickness and in the interaction between polymer covered surfaces. For a more extensive treatment of tails we refer to the literature [5,17 ]. Our next example gives two adsorption isotherms and one desorption isotherm for a heterodisperse sample with a Schulz-Flory distribution (Mw/ M,=1.36, number average chain length r . = 2 0 0 ) . There are several differences between the adsorption of homo- and heterodisperse polymers, all of them related to the preferential adsorption of long chains over short ones [ 18, 19]. This preference causes rounded adsorption isotherms, makes the isotherms depend on the bulk solution volume (at given surface area), and leads i

i

i

i

/'""\'--..~total

lO-~ toop~s~ \\\\\ \ ,

0

I

50

,

"%

I

100

,

I

i

"'".

150

Fig. 4. Semilogarithmie density profile ~, and its build-up in terms of loops and tails, r = 10,000, X~= 1, Z=0.5, ~. = 10 -~.

160

to a difference between adsorption and desorption isotherms. These points are illustrated in Fig. 5. The more rounded shape of the isotherm is due to the gradual replacement of short chains by longer ones with increasing polymer dosage. Since the adsorbed amount increases with increasing chain length (see Fig. 3 ) the adsorption isotherms does not show a pseudoplateau any longer. The effect of solution volume (in Fig. 5 expressed as the number of layers of solution in contact with the surface) is explained along the same lines. If the volume is small, the total polymer dosage required to reach a given (overall) ¢. is also small and the total number of long chains in the system is smaller than at higher volume (and the same ~. ). Consequently, the adsorbed amount is smaller, as can be seen by comparing the isotherms for 5000 and 500,000 solution layers in Fig. 5. The difference between ad- and desorption isotherms touches upon the important issue of the reversibility of polymer adsorption. If at a given point of the adsorption isotherm (see the arrow in Fig. 5) the solution volume is increased at a given total amount of polymer in the system, one crosses from one isotherm (5000 layers) to a higher one (500,000 layers). During this dilution, the long chains on the surface will desorb only to a small extent, similar to homodisperse polymers having an isotherm with an essentially horizontal plateau (Fig. 3 ). Note that in the computation of Fig. 5 full thermodynamic equilibrium was assumed. For polydisperse systems, a desorption isotherm that lies above the adsorption isotherm is merely a consequence of preferential adsorpI 500,000

I

I

i

i

I

..................

layers

l

t5

e

1.0

0.5

I 200

I 400

I 600

' 800

I 1000

I t200

pprn

Fig. 5. Two adsorption isotherms ( - . ) and one desorption isotherm ( ~ ) for a polydisperse sample with a Schuh-Flory distribution. The solution volume is indicated as the number of layers, rn-- 200, r,,./r, = 1.36, X= 0.5, Xs= 1.

161

tion and the overall mass balance, and is not due to irreversibility of the adsorption process. As stated before, the lattice model can also be applied to polymer between two surfaces, a case which is relevant for colloidal stability [9,10]. In Fig. 6 an example is given of the free energy of interaction as a function of the particle separation. The figure applies to a situation where the polymer is trapped between the plates as soon as the interaction starts, e.g., because there is no time to diffuse away during the Brownian collision. It is clear that at low concentrations the particles attract each other. In this situation long dangling tails are caught by the opposing surface to form bridges (bridging flocculation). At high polymer concentrations only repulsion occurs due to entropical restrictions of the dense polymer layers on the two particles (steric stabilization). These trends are generally known from experience, but so far no theory was available that could predict both attraction and repulsion in the same system. A more extensive treatment of polymers between two plates can be found elsewhere [9,10]. Many colloidal systems are stabilized by chemically attached polymeric hairs on the particles. The structure of the polymer layer depends on the grafted amount and on the interaction parameters X and Xs. Figure 7 gives an example 0.02

kT 0.01 10 °;2

10

~

20

M

_o.~i

-0.03 Fig. 6. T h e free energy of interaction per surface site between two plates as a function of the plate separation M, for three polymer concentrations. It is assumed t h a t the polymer c a n n o t leave the gap between the particles, r = 1000, X = 0.5, Xs= 1.

162 0.2 ---~

)

0.1

.5

10

)5 i

Fig. 7. Structure of a hairy layer of relatively short hairs (r--50) at low coverage (0 = 1 ) from a good solvent (x=O).

for relatively short hairs (50 segments) at low coverage (1 hair per 50 surface sites ). If the segments do adsorb (Xs= 1 ) the concentration falls of monotonically and the situation is, at this coverage, comparable to adsorbing non-grafted polymer (see Fig. 4). For non-adsorbing segments (Xs=0) the profile shows a maximum and the tail fraction is much higher, as expected. For higher coverages the profile extends much further into the solution; in that case the effect of)~ is smaller and only significant in a few layers close to the surface [20]. Figure 8 gives semilogarithmic concentration profiles for a strong polyelectrolyte adsorbing on an uncharged surface. Unlike the situation for polymers without charge, the lattice step length has to be specified because it determines the screening of the electrostatic charges by salt. In this example, it was chosen to be equal to the Bjerrum length. The most conspicuous feature of Fig. 8 is the minimum in the profiles for low ionic strength. This is caused by the high potential generated by the adsorbing chains, repelling other molecules from the surface region. At higher ionic strength this minimum becomes weaker due to better screening of the charges, and the adsorbed amount increases. However, even at 3 M salt concentration the screening is not yet complete and the adsorbed amount is still less than for an uncharged polymer. At low ionic strengths, the adsorbed amount is small and the chains lie essentially flat on the surface, having very few loops and tails. Our last examples give some results for surfactant adsorption and association. Figure 9 shows adsorption isotherms of molecules T14H2 (T--tail segment, H -- head segment ) from water (W) on two different surfaces. The headgroup segments have a strong affinity for the solvent (xHW-- --0.5), the tail

163 I

5

10

15

i

i

i

20 1

10 -I

10-4

10-6

10.8

Fig. 8. Concentration profiles of a strong polyelectrolyte near an uncharged surface, at various concentrations of a univalent electrolyte. The lattice step length was assumed to be equal to the Bjerrum length (0.71 nm). r=2000, Xs= 2, X=0.5.

0

I

0

5.10 -~"

10 .3

Fig.9. Adsorption isotherm of T14H2 on a hydrophobic (Xw= 1, XP = 0) and a hydrophilic (Xw = 0, XH = 3 ) surface. The x-parameters are XHw= - 0.5, Xww= X T H = 1.6. s e g m e n t s a r e a p o l a r a n d a r e s t r o n g l y r e p e l l e d b y t h e s o l v e n t ( X T w = 1.6). T h e hydrophilic surface has affinity for the head groups and not for the apolar segments (XsT = l , xsH - - 0 ) . O n a h y d r o p h o b i c s u r f a c e , a " n o r m a l " a d s o r p t i o n i s o t h e r m is f o u n d , w i t h a g r a d u a l l y i n c r e a s i n g a d s o r b e d a m o u n t , levelling o f f t o a k i n d o f p l a t e a u . I n t h i s

164

case the tails are in contact with the surface. On the hydrophilic surface an isotherm with a metastable and a stable part occurs (probably, the metastable part could not be measured experimentally). Because of the few head segments per molecule and the low tendency of tail segments to adsorb, the adsorbed amount is initially rather low. Once a certain adsorption level is reached, the association between the tail segments, which are repelled by the solvent, causes a strong increase in adsorbed amount, ultimately leading to a bilayer structure. S-shaped isotherms like that in Fig. 9 are commonly found for systems with strong lateral interaction. More structural information is obtained by considering the concentration profiles (Fig. 10) corresponding to the arrows in Fig. 9. On the hydrophobic surface (left) both the tail and solvent profiles vary gradually from the surface to the bulk solution, whereas the head groups accumulate at some distance from the surface. As expected, the tails adsorb and the head groups point towards the solution but are distributed over several lattice layers. For adsorption on a hydrophilic surface (right) the association between the tails is so strong that bilayer formation occurs, with part of the heads on the surface and a roughly equal number around layer 10 (but again distributed diffusely) and the tails accumulated in between. Note that this tail association occurs already below the CMC (indicated by an arrow in Fig. 9): apparently the surface promotes the self-association of the surfactant. In Fig. 11 an example is given for surfactant adsorption at an oil/water interface. The surfactant in this case is T17H3, the oil is an octamer (O) of Tsegments and could be taken as octane. In this case Xs-parameters are irrelevant, and Xww=zow = 1.6. The profiles show the expected behaviour, with the tails mixing partly with octane and the heads pointing to the water. Again the profiles are rather diffuse, refuting the simple picture of rigid sticks aligned all I

I

Y

0.5 ~..

~.5

5

5 i

10 i

Fig. I0. Concentration profiles of tail segments, head groups and solvent on a hydrophobic (left) and hydrophilic (right) surface, for the isotherms points indicated by the arrows in Fig. 9.

165 1.1~

octQnei/

~ i~

wGe |r

0.E

J . G

"7 Y\ X K < "-o 20

15

10 i

Fig. 11. Concentration profiles of T17H3 at an octane-water interface at a volume fraction of surfactant of 0.0025 in the water phase. The tail segments have the same properties as the octane segments. XHw =0, ~TW ~_~OS= 1.6, ~HT = 0.2. I.G

taii

0.5

I -10

0

10 i

Fig. 12. Structure of a spherical micelle of TmH2 in water, xHW= -- 1, ~TW=~HT= 1.6.

parallel. As a matter of fact, it is relatively easy to calculate the distribution function of individual segments (for any ranking number) and order parameters, but these points cannot be dealt with in the present context. Finally, Fig. 12 shows the structure of a spherical micelle of a surfactant TloH2 in water. The aggregation number turns out to be about 64 in this case. Also here the head groups spread out over several layers. The distribution of tail segments has considerable overlap with that of the head groups, and some apolar segments on the outside of the micelles have to be in contact with water. The concentration of water in the micelle is comparable to the solubility of water in hydrocarbons. At present, the model probably overestimates this solubility, as a consequence of a deficiency in the Flory-Huggins theory. Work is in progress to incorporate free volume effects by allowing holes as a component. Preliminary calculations have shown that some (but not all!) water in the micelle will then be replaced by holes. The predictions of the model for the critical micelle concentration (CMC)

166 and its dependence on chain length are in excellent agreement with experiment [16]. CONCLUSIONS In this paper it has been shown in several examples how the ScheutjensFleer model can give relevant information for a variety of systems. In adsorbed layers of homopolymers long dangling tails play an important role, not only for the hydrodynamic layer thickness on single surfaces but also for particles in interaction, leading to bridging flocculation in dilute solutions and steric stabilization at higher dosages. In polydisperse systems, preferential adsorption of long chains affects the properties of adsorbed layers drastically. In polyelectrolyte adsorption, the accumulation of charges prevents the formation of thick layers and a minimum in the segment concentration profile may develop. The structure of grafted polymer hairs depends on the surface coverage and (at low coverage) on the adsorption energy Xs, giving a m a x i m u m in the concentration profile unless the coverage is low and Xs high. The adsorption of surfactant molecules on hydrophobic surfaces leads to (sub) monolayers with the head groups protruding in solution, whereas on hydrophilic surfaces the lateral interaction between the tails gives rise to bilayer formation. At liquid/liquid interfaces only x-parameters determine the structure of the interface; the distribution profiles are relatively wide. Finally, the theory leads to realistic descriptions of self-assembling structures like micelles and membranes. ACKNOWLEDGEMENTS The author express his gratitude to J a n Scheutjens, Frans Leermakers and Boudewijn van Lent for their kind permission to use some of their hitherto unpublished results.

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