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Langmuir 1999, 15, 1802-1811
Surface Pressure of Adsorbed Polymer Layers. Effect of Sticking Chain Ends C. Barentin and J. F. Joanny* Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, France Received September 8, 1998. In Final Form: November 19, 1998 Using a mean-field theory with two order parameters and scaling laws, we study the properties of adsorbed polymer layers in contact with a pure solvent, at a fixed amount of polymers. We calculate the surface pressure of the polymer Langmuir monolayer, implicitly assuming that the monolayer is insoluble, i.e., that the polymer desorption time is much larger than any other characteristic time, in particular, the equilibrium relaxation time of the layer. This work was motivated by experiments done on poly(ethylene oxide)(PEO) and telechelic PEO adsorbed at the air-water interface that present an adsorbed layer structure at certain densities. Our theoretical results explain quite well the shape of the surface pressure isotherms and the influence of the chain length and of the hydrophobic ends.
1. Introduction Adsorbed polymer layers attract a considerable interest, in particular, in connection with the stabilization of colloidal suspensions.1-4 From a more fundamental point of view, they are also interesting as they provide a good example of coupling between the properties of the surface and the conformation of the polymer chains. Adsorbed polymer layers with high coverages are usually obtained by adsorption from a dilute polymer solution. An adsorbed polymer chain forms small loops close to the adsorbing surface and has two tails extending further away into the bulk solution. Within a mean field approximation, the equilibrium structure of polymer layers adsorbed from dilute solution is classically treated within the so-called ground state dominance approximation. More recently, Semenov and co-workers have shown5 that the ground state dominance approximation correctly accounts for the concentration of monomers belonging to loops but that it ignores the contribution of monomers belonging to the tails at the end of the chains: it is a poor approximation in the outer part of the layer, which is dominated by the tails. To improve this theoretical treatment, they have introduced a mean-field theory of polymer adsorption using two order parameters, ψ(z) and φ(z) associated to loops and tails respectively and they have studied the cases of adsorbed polymer layers in equilibrium with a dilute, a semidilute, or a concentrated solution.6 The effect of the tails was first analyzed by G. J. Fleer et al.,4 using a numerical solution of the full mean-field equations. In other respects, the mean-field theory does not take into account the correlations introduced by the excluded volume interactions. These can be taken into account by using scaling laws based on the physical picture obtained from the mean-field theory. This was done first by de Gennes,7 who ignored the difference between loop and tail monomers and then by Semenov and Joanny.8 (1) Napper, D. H. Polymer stabilization of colloidal dispersions, volume 1; Academic Press: London, 1983. (2) Heger, R.; Goedel W. A. Macromolecules 1996, 29, 8912. (3) Goedel W. A. Progr. Colloid Polym. Sci. 1997, 103, 286. (4) Fleer, G. J.; Cohen Stuart, M. A.; Cosgrove, T.; Vincent, B. Polymer at interfaces.; Chapman and Hall: London, 1993; Vol. 1. (5) Johner, A.; Bonet-Avalos, J.; van der Linden, C. C.; Semenov, A. N.; Joanny, J. F. Macromolecules 1996, 29, 3629. (6) Semenov, A. N.; Joanny, J. F.; Johner, A. Theoretical and Mathematical Models in Polymer Research; Grosberg, A., Ed.; Academic Press: Boston, MA, 1998. (7) de Gennes, P. G. Macromolecules 1981, 14, 1637.
The pressure-area isotherms of Langmuir monolayers at the air-water interface formed by polymers such as polyethylenoxide (PEO) have been studied in ref 9. This polymer is soluble in all proportions in water, but also forms monolayers. It is important here to stress that our experimental and theoretical work only considers monolayers formed by polymers, for which water is a good solvent, and cannot be generalized to more classical Langmuir monolayers formed by insoluble polymers.2,3 The monolayers of polyethylenoxide are stable below a critical surface density. Above this value of the surface density, the PEO desorbs from the monolayer and dissolves into the water subphase.10,11 In a dilute monolayer, the polymer molecules have a flat two-dimensional conformation; at intermediate densities, below the critical density, the polymer forms an adsorbed layer with loops and tails dangling into the water subphase. It is interesting to note that these layers are not in equilibrium with a bulk solution but that the desorption time at those densities12 is much larger than the experimental time and the polymer adsorbed amount (the adsorbance) is imposed by the experimentalist. This type of experimental study could be generalized to any soluble polymer, that presents a good affinity with the air-water interface. Monolayers of polymers endcapped with hydrophobic groups, which stick to the air surface, H2n+1Cn-PEO-CnH2n+1, have also been studied.12,13 For these polymers, the hydrophobic groups are located on the surface. Their sticking energies are large enough to change the loop and tail structure of the adsorbed layer and to allow the formation of grafted layers with stretched PEO chains. The aim of this paper is to calculate the surface pressure of polymer Langmuir monolayers in both cases: where the end points are free and where the end points are strongly sticking to the interface. In this last case, we only consider the low surface concentrations, where polymers form an adsorbed layer and we do not consider the grafted layers at high densities, which have been described in detail in ref 12. This problem has been considered within the framework of a mean-field theory (8) Semenov, A. N.; Joanny, J. F. Europhys. Lett. 1995, 29, 279. (9) Shuler, R. L.; Zisman, W. A. J. Phys. Chem. 1970, 74, 1523. (10) Kuzmenka, D. J.; Granick, S. Macromolecules 1988, 21, 779. (11) Kuzmenka, D. J.; Granick, S. Polym. Commun. 1988, 29, 64. (12) Barentin, C.; Muller, P.; Joanny, J. F. Macromolecules 1998, 31, 2198. (13) Kim, M. W.; Cao, B. H. Europhys. Lett. 1993, 24, 229.
10.1021/la981173t CCC: $18.00 © 1999 American Chemical Society Published on Web 02/05/1999
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in the ground state dominance approximation by Andelman and co-workers.14 We here use the more general mean-field theory, which explicitly takes into account the effect of the chain ends. This allows a discussion of the molecular weight dependence of the pressure and of the effect of the sticking of the chain ends on the interface. The paper is organized as follows. In the next section, we introduce the general framework of the mean-field theory; we rederive the concentration profiles and total partition function of an adsorbed polymer. In section 3, we adapt the mean-field theory to study the adsorption of functionalized polymers with sticking chain ends. We calculate the fraction of chains, whose ends stick to the surface and the corresponding partition function. In section 4, we discuss the free energy and the pressure of the polymer monolayer in both cases of sticking and free chain ends, and we compare qualitatively these results with some experiments. Finally, in the last section, using scaling laws, the results are extended to polymer solutions in a good solvent. 2. Mean-Field Theory 2.1. Order Parameters Approach. We consider here an adsorbed polymer layer with Γ irreversibly adsorbed monomers per unit area, in contact with pure solvent. In the mean-field theory of polymer adsorption, the conformation of the chains is described by two order parameters: ψ(z) related to the partition function of adsorbed chain touching at least once the surface, and φ(z) related to the partition function of a free chain. The total monomer concentration is
vc(z) ) ψ2(z) + Bψ(z)φ(z)
(1)
where v is the excluded volume parameter, which we consider throughout as positive (corresponding to good solvent conditions). The first term in the concentration is the contribution of the monomers belonging to loops, cl, and the second one is that of monomers belonging to tails, ct. The constant B gives the weight of the tail concentration and depends on the adsorbance Γ, defined by Γ ) ∫c(z) dz:
B)
2vΓ
(2)
∫
N ψ(z) dz
The concentration of chain ends is
vce(z) ) Bψ(z)
(3)
In the mean-field theory, the monomers feel an average potential: Utot ) Uint + Uw, due to the interaction between the monomers and to the attraction of the monomers by the wall. For good solvent conditions, the interaction potential Uint is directly proportional to the monomer concentration: Uint ) vc(z). The wall attraction is assumed here to be sufficiently short range, that the effect of the local potential can be reduced to an effective boundary condition15
|
1 dψ(z) ψ(z) dz
z)0
)-
1 b
(4)
where b is an extrapolation length, depending only on the wall potential, and inversely proportional to the adsorption strength.16 The two order parameters satisfy Schro¨dinger(14) Aharonson, V.; Andelman, D.; Zilman, A.; Pincus, P. A.; Raphae¨l, E. Physica A 1994, 204, 1. (15) Landau, L.; Lifchitz, E. Physique statistique; Mir: Moscow, 1967.
like equations, directly derived from the Edwards equation for the chain propagator:
0)-
1)-
d2ψ(z) dz2 d2φ(z) dz2
+ (Uint + )ψ(z)
(5)
+ (Uint + )φ(z)
(6)
The boundary conditions are
lim ψ(z) ) 0
(7)
lim c(z) ) 0 and φ(0) ) 0
(8)
zf∞
zf∞
In these equations, we choose the unit length, so that a/x6 ) 1, where a is the monomer size. is the effective adsorption energy per monomer. It depends on the adsorbance Γ and on the chain length N. The basic assumption, which is crucial for all the formalism, is that the binding energy per chain is large enough: N . 1. In this limit, the total partition function of an adsorbed polymer reads
∫ ∫ψ2(z) dz
( ψ(z) dz)2
Z ) exp(N)
(9)
It is important to note that the two-parameter mean-field theory, presented in this paper, is more elaborate than the ground state dominance approximation, in the sense that this theory allows a description of the chain end effects. It is nevertheless a simplified version of the full mean-field theory,4 which is valid in the limit of large molecular weights.5 2.2. Mean-Field Concentration Profiles. The concentration profile in the adsorbed polymer layer can be separated into three regions:7 (i) the proximal region closest to the wall within a distance b, where the effects of the short-range interactions between the monomers and the wall are important; (ii) the central region, in the intermediate range, b < z < λ ) 1/x, where the concentration is high and, where Uint dominates over ; (iii) the distal region, where dominates over Uint and where the concentration decreases exponentially. In the central region, we simplify the propagator equations (eqs 5 and 6) and we neglect the binding energy: Uint + = Uint ) ψ2(z) + B ψ(z)φ(z). Two terms remain in the potential, the first one corresponding to the concentration of monomers belonging to loops, which is dominant close to the wall, and the second one corresponding to monomers belonging to tails, which dominates the outer part of the layer. The loop and tail contributions to the concentration are equal at a distance z* from the wall that depends on the adsorbance Γ. At small adsorbance, in the so-called starved regime, z* is higher than λ, so that the central region is entirely dominated by the loops. However, at higher adsorbance, the crossover between the loop and tail concentrations occurs in the central region, so that for z* < z < λ the interaction potential is dominated by the tails. We suppose first that the density is high enough to have a crossover regime. 2.2.1. Oversaturated Regime. Using dimensional analysis of the eqs 5 and 6, we define a typical length size l(Γ,N) ) B-1/3; it has been shown6,17 that (16) de Gennes, P. G.; Pincus, P., J. Phys. Lett. 1983, 44, L241.
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• close to the wall, for z , l vcl(z) )
x2 (z + b) l 2 . vct(z) ) log 2 3 z+b (z + b) l3
(
)
(10)
l 20 3602l6 . vc (z) ) log l 2 8 z + b (z + b) (z + b)
(11)
• further away l , z < λ vct(z) )
(
)
This asymptotic analysis allows for the calculation of B and l:
l)
( )( N
1/3
x2vΓ
N 1 log 3 x2vΓ
)
1/3
(12)
A numerical solution of the equations shows that z* ) 1.43l. Finally, in the distal regime for z . λ, the concentrations decrease exponentially over a distance λ: vcl = 1/λ2 exp(-2z/λ) and vct = 1/λ2 exp(-z/λ). At this point, it is interesting to compare the two lengths, l (Γ,N) and λ(Γ,N), that must be well separated. To find the thickness of the adsorbed layer λ, it is necessary to calculate Γ beyond the leading term, Γ0 ) 2/vb. The corrections to Γ0 are proportional to powers of 1/N and come from the regions z = l and z = λ. It was shown17 that for large molecular weights the adsorbance Γ depends on N as
vΓ )
2 R β + b l λ
Figure 1. Comparison between the two characteristic lengths l(Γ) (dashed line) and λ(Γ) (solid line), for a polymer with N ) 1000. The lengths cross at Γc, which defines the limit of the starved and oversaturated regime. We choose b ) 1, so that Γ0 ) 2 in all curves.
(13)
A numerical estimation17 gives R ) 3.46 and β ) 7.24. This allows us to express λ as a function of Γ:
λ(Γ,N) )
β 2 R + - vΓ b l
(14)
The comparison of the two lengths scales l and λ in Figure 1 shows that, for each chain length, there is a critical adsorbance Γc(N) e Γ0 defined by l(Γc,N) ) λ(Γc,N). If the adsorbance is smaller than this critical value (Γ < Γc), l is larger than λ, the central region is dominated by the loop concentration. This is the so-called starved regime, which is well described by the classical ground state dominance approximation. There is also an upper limit for the adsorbance Γl e Γ0 defined by λ(Γl,N) = RG, where RG is the Gaussian radius of gyration of the polymer. When the adsorbance is larger than this value, the binding energy per chain is small, so that the basic assumption N . 1 is no longer satisfied. In conclusion, the range of adsorbance, where the loop and tails concentrations cross in the central region, is limited (see Figure 2); we call it the oversaturated regime, because the adsorbance can be larger than the saturation value Γ0. In the oversaturated regime, the partition function is determined by integration of the order parameter ψ and is approximated by
2(log l/b) Z = exp(N) vΓ
Figure 2. Influence of the chain length on the values of Γc (solid line) and Γl (dashed line).
order parameter can be calculated analytically in a closed form:
ψ(z) )
x2 z+d λ sinh λ
(
(16)
)
where d is defined so that the concentration verifies the boundary condition6 at the wall: ψ2(0) ) 2/b2 - 2/λ2. This imposes: d ) b(1 + 1/3b2). The adsorbance is then vΓ ) 2/b - 2/λ, which is equivalent to
λ)
2 2 - vΓ b
(17)
In the starved regime, the partition function can be exactly calculated from the expressions of ψ(z) (eqs 9 and 16):
2(log(1/tanh(d/2λ)))2 = vΓ
Z ) exp(N)
2
(15)
2.2.2. Starved Regime. For Γ < Γc(N), the loop concentration dominates the whole central region and the tails become important only in the distal region. The loop (17) Semenov, A. N.; Bonet-Avalos, J.; Johner, A.; Joanny, J. F. Macromolecules 1997, 29, 2179.
2(log2λ/b)2 (18) vΓ
exp(N)
We do not study here in details the transition between the starved and the oversaturated regimes; that would require a detailed numerical integration of the mean field equations.
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3. Adsorbed Layers Formed by Polymers Bearing Sticky Ends In an adsorbed layer of functionalized polymers, there are three kinds of polymers: the bigrafted polymers of fraction xb, with both ends sticking to the surface, the monografted polymers of fraction xm, and the polymers with two free ends of fraction xf. The various grafting fractions depend on the adsorbance Γ, on the chain length N and on the sticking energy of the ends E kBT. 3.1. Order Parameter Theory. The mean-field theory of adsorbed polymer layers is now extended18 to take the specific adsorption of the ends into account. A new order parameter is introduced, ψ2(z), which corresponds to the partition function of a chain touching the surface only by one of its ends. It satisfies the equation
δ(z - b′) ) -
d2ψ2(z) dz2
+ (Uint + )ψ2(z)
(19)
xb )
with
ψ2(0) ) 0 and lim ψ2 ) 0 zf∞
(20) xm )
and b′ (of order a) is the position of the adsorbed chain ends, which is inversely proportional to the sticking energy of the ends. The total monomer concentration has now three contributions
vc(z) ) ψ2(z) + B′ψ(z)φ(z) + Cψ(z)ψ2(z)
(21)
where the last term is the concentration of monomers belonging to adsorbed tails. B′ is different from B and depends on Γ, N, and E. The concentration of free ends is calculated, as in section 2, by vcef(z) ) B′ψ(z). To calculate B′, we use the conservation of the number of free ends. By integrating cef(z) over all space, we obtain
B′ )
(2xf + xm)vΓ
∫
N ψ(z) dz
(22)
The concentration of adsorbed chain ends is vcea(z) ) Cψ(z), and the conservation of the number of adsorbed ends, leads to
C)
(2xb + xm)vΓ Nψ(0)
(23)
To calculate B′ and C, it is therefore necessary to find the expression of the grafting fractions xi. 3.1.1. Grafting Fractions. In appendix 1, we show that the total partition function of a chain in the layer is22
(ψ(0) exp[E] +
Z ) exp(N)
Figure 3. Influence of the sticking energy on the values of the grafting fractions: xb (solid line); xm (dot-dashed line); xf (dotted line).
∫ψ(z) dz)2
∫ψ2(z) dz
(24)
If we expand the square, the first term corresponds to the partition function of a chain with two adsorbed ends, the second one to that of a chain with only one adsorbed end, and the last one to that of a chain with two free ends. The grafting fractions are then (18) Johner, A.; Joanny, J. F. Macromol. Theory Simul. 1997, 6, 479.
xf )
(ψ(0) exp E)2 [ψ(0) exp E +
(25)
∫ψ(z) dz]2
∫ [ψ(0) exp E + ∫ψ(z) dz]2 2ψ(0) exp E ψ(z) dz
(26)
∫
( ψ(z) dz)2 [ψ(0) exp E +
(27)
∫ψ(z) dz]2
Those expressions automatically imply that
C ) B′ exp E
(28)
which is the equivalent of writing the thermodynamic equilibrium between adsorbed and free ends. The grafting fractions depend strongly on the sticking energy (Figure 3) and also depend on Γ and N through ∫ψ(z) dz; this dependency is however weak, because Γ and N appear in the argument of logarithms. If the sticking energy is high enough, ψ(0) exp E . ∫ψ(z) dz, xb = 1, most of the polymers are bigrafted. 3.1.2. Mean-Field Concentration Profiles. We define a new length: l′ ) (B′)-1/3, which corresponds to the crossover between the loop concentration and the free tail concentration. It would make no sense to define a crossover length from C since, by definition, the adsorbed tails cannot dominate far away from the surface. We suppose, first, that l′ stays in the central region (l′ < λ). In this case, we show, in Appendix 2, that ψ(z) follows the same asymptotic laws and, that Γ has the same expressions (eq 13) as found in section 2. We also find that ψ(0) = x2/b and ∫ψ(z) dz = x2 log(l′/b). This allows us to calculate the grafting fractions xi, the partition function Z, and B′. Using the relation l′) (B′)-1/3, we finally find
l′(Γ,N,E) )
( )( N
x2vΓ
1/3
)
exp E l′ + log b λ sinh(d/λ)
1/3
(29)
An increase of the sticking energy induces an increase of l′, so that the region where loops dominate grows: this is due to the fact that more chain ends tend to adsorb at the surface, and that the tail concentration becomes less important in the central region. As shown in Figure 4, the effect of the sticking of the chain ends is to increase the value Γc of the adsorbance at the crossover between the starved and the oversaturated regimes. The oversaturated
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Barentin and Joanny
c′(0) b )2 c(0)
(35)
The last two equations (eqs 34 and 35) impose the conditions that
R)-
2 3
(36)
and
(
d′ ) b 1 + Figure 4. Influence of the sticking energy on the values of Γc (solid line) and Γl (dashed line), with N ) 1000.
regime only exists for small sticking energy (exp E , N1/2) and high enough adsorbance. Finally, in the starved regime, the results remain identical to those found in section 2, because the chain ends do not play any important role. The only difference appears in the expression of the partition function (eq 24), which now depends on the sticking energy.
4.1. Free Energy. The free energy, per unit area, of an adsorbed polymer layer has the following expression:
∫ vc2 dz
(30)
The first term is the conformational energy of independent chains, the second term corresponds to the mixing entropy, and the third one corrects for the double counting of the interactions between monomers by the mean-field theory. This last term is important for the calculation of the pressure, because it depends on the lengths l and λ, which are functions of the adsorbance Γ. We now estimate this term, in the two asymptotic regimes. In the starved regime, we find
-Γ -
1 2
∫
dz vc2 ) -
1 v
∫
dz
2 1 ∂ψ 2 1 )+ ∂z 3 vb3 3vλ3 (31) 2
( )
with λ ) 2/(vΓ0 - vΓ). In the oversaturated regime, we give an approximated expression of ∫ dz 1/2vc2, neglecting all terms smaller than 1/l3 ∝ 1/N. To find all dominant terms of the integral, coming from the lower bound, it is necessary to expand c(z) to second order.
ψ(z) )
x2 R 1 + (z + d′)2 2 (z + d′)
(
)
(32)
is the general solution of the Schro¨dinger equation (eq 5), with the correct boundary condition. Using the fact that, close to the wall, vc(z) ) ψ(z)2, vc(z) can be written as
vc(z) )
2 +R (z + d′)2
(33)
with the following boundary conditions:
vc(0) )
2 - 2 b2
(37)
which is the same relation found for d in section 2.2.2. Close to the wall, the concentration is
vc(z) )
2 2 3 (z + d′)2
(38)
Finally, we obtain that
-Γ -
1 2
∫ dz vc2(z) ) - 32 vb1 3 - vlR˜ 3
(39)
with
4. Free Energy and Pressure of Adsorbed Polymer Layers
F Γ Γ Γ 1 ) - ln Z + ln kBT N N N 2
)
b2 3
(34)
R˜ )
1 2
∫ dξ (c˜ (ξ) - ξ44)
(40)
where ξ ) z/l and c˜ (ξ) ) vc(z)l2. Using the power laws found in section 2.2, R˜ is a positive constant but whose exact value has to be determined by numerical integrations. For any adsorbance, the free energy is dominated by the interaction with the surface of the monomers, in the vicinity of the surface 2/3vb3. Note that the same results are obtained for chain with sticky ends, if l is substituted by l′. 4.2. Surface Pressure. The surface pressure is calculated by derivation of the free energy:
∂F -F ∂Γ
P)Γ
(41)
We distinguish now the two cases, where the polymer chain ends stick or not onto the surface. 4.2.1. Nonsticky Chain Ends. For a starved layer (Γ < Γc), the pressure is
2 1 vΓ2λ 1 2Γ P ) - (vΓ0 - vΓ)2(2Γ + Γ0) + 3 2λ kBT 3 vb 12 N N ln b (42) with λ ) 2/(vΓ0 - vΓ). The first term in the expression of the pressure is dominant and comes from the interaction of the monomers close to the surface. The second term is independent of the chain length, it is important at small densities, Γ < Γ0, and gives a negative contribution. These two terms have been already calculated by Andelman and coworkers.14 The third term looks like the pressure of an ideal gas of chains. Note however the factor of 2, coming from the dependence of the partition function on the adsorbance. The ideal gas pressure is thus the pressure of a gas of free chain ends, as already noted for several properties. The last term is a small correction but it carries the leading dependence on the chain molecular weight.
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Langmuir, Vol. 15, No. 5, 1999 1807
Figure 5. Pressure of adsorbed polymer layers in the starved regime, for three different chain lengths: N ) 100 (dotted line); N ) 200 (dot-dashed line); N ) 1000 (solid line). The plotted pressure is dimensionless.
Figure 6. Pressure of adsorbed polymer layers in the denser part of the starved regime, for four different chain lengths: N ) 100 (dotted line); N ) 200 (dot-dashed line); N ) 1000 (dashed line); N ) 10000 (solid line). The saturation value of the pressure increases with the chain length.
The chain length N becomes a relevant parameter only close to Γc (Figure 5). An increase of the chain length induces an increase of the pressure but this increase tends to saturate for N = 105 (Figure 6). This observation is in good agreement with the experimental results of Kuzmenka and Granick,10 who measured the pressure on the plateau of PEO isotherms, as a function of molecular weight. The plateau value, starting from 8 mN/m, increases with the molecular weight; it reaches 10 mN/m at Mw = 12 × 104 (equivalent to N = 3000) and then remains constant. For the oversaturated layer (Γ > Γc), the pressure is
2 1 P 2Γ ) + + kBT 3 vb3 N
2Γ
N ln
+
( ) ( )( N
x2vΓ
2Γ N N ln x2vΓ
1-
2
)
3R˜ (43) x2
The first two terms are the same as those found in the starved regime (eq 42). The last two terms become negligible compared to the pressure of an ideal gas for long chain lengths, so that the pressure increases really slowly with the density. In this regime, we observe an inverse effect of the chain length on the pressure. At a given adsorbance, an increase of N tends to decrease the pressure. We have noticed that it is the same effect of N on the pressure as in the case of a polymer brush. This
Figure 7. Pressure of adsorbed layers in the starved and oversaturated regimes, for polymers of chain length N ) 1000. The crossover between the two regimes is not quantitatively described by the presented theory.
suggests that, at saturation, longer polymers expel more monomers from the surface, and relax the constraints more easily. If we plot the pressure in all the regimes of densities, as shown in Figure 7, we observe the same shape as the experimental isotherms,10 in the sense that the surface pressure presents a plateau at the saturation of the layer. 4.4.2. Sticky Chain Ends. The free energy of a polymer with sticky chain ends differs from that of a polymer with non sticky chain ends essentially in the single chain partition function. For simplicity, we consider here only the starved regime and the case, where most chain ends are adsorbed on the surface exp E . log N; in this limit, the oversaturated regime essentially no longer exists. Similar qualitative conclusions are obtained in the oversaturated regime. The single chain partition function in this limit reads
Z)
(
)
2 exp(N) 2λ exp E + ln vΓ d λ sinh (d/λ)
2
(44)
If the sticking energy is not too large (exp E , N log N), the pressure can be approximated by
2 1 1 P ) - (vΓ - vΓ)2(2Γ + Γ0) + kBT 3 vb3 12 0 λ 2Γ vΓ2 (45) N N exp E 2λ + ln b b The comparison with the pressure of adsorbed chains with no sticky ends (eqs 42 and 45) shows that the end sticking increases the pressure, as it decreases the last negative term in eq 42. However, as shown in Figure 8, the effect of the end adsorption on the pressure is small and becomes important only at the highest adsorbance. When the ends stick on the surface, an increase of the chain length N induces a decrease of the pressure P (Figure 9). This result is general; also, in the oversaturated regime, the pressure is increased by the adsorption of the ends, and the effect is significant only close to the saturation. Those results are in agreement with experiments described in ref 12, where the authors compare the isotherms of PEO and telechelic PEO with C12H25 as end groups. The comparison shows, that the effect of the hydrophobic ends is to increase the pressure close to the plateau and at highest densities. They also plot the isotherms of telechelic PEO with different chain lengths and they observe the same effect as mentioned here.
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from the wall, is dominated by the tails. They also calculated the partition function, the concentration profiles of the loops and tails in the two regions. Finally they studied the structure of an adsorbed polymer layer, in equilibrium with a bulk solution. Here, we adapt the calculation to the case of a layer at fixed amount of polymers. 5.1. Scaling Theory of Adsorbed Polymer Layers. 5.1.1. Free Energy and Thickness of the Layer. We now calculate the free energy of an adsorbed layer, and for simplicity, we focus on the starved regime. As shown by Semenov and Joanny,8 the free energy per unit area of an adsorbed polymer monolayer reads: Figure 8. Influence of the end adsorption on the pressure of adsorbed layers with polymers of N ) 1000. The dashed curve corresponds to polymers with sticking ends (E ) 20) and the solid one is the pressure of polymers with free ends (E ) 0).
Figure 9. Influence of the chain length on the pressure of layers composed by polymers with sticking ends (E ) 20): N ) 100 (dotted line); N ) 200 (dot-dashed line); N ) 1000 (solid line).
Finally, if the sticking energy is increased, exp E . N ln N, the variation of the pressure with E saturates and
P 2 1 1 2Γ vΓ2 b2 2 ) (vΓ vΓ) (2Γ + Γ ) + 0 0 kBT 3 vb3 12 N N λ (46) 5. Scaling Theory In this section, we use scaling arguments to generalize the results, obtained in the mean-field approximation to polymers in a good solvent. The equilibrium properties of polymers adsorbed from a good solvent onto a surface have been considered in several papers.8,19 In particular, a scaling argument due to de Gennes7 leads to a self-similar profile and a power law decay of the concentration:
φ(z) ) vc(z) = z1/ν-d
(47)
Here d is the space dimension, and ν is the swelling exponent of a polymer chain in a good solvent; we use here the Flory approximation: ν ) 3/(d + 2). In a threedimensional space, d ) 3, φ(z) ) 1/z4/3; as a check, we will also consider the results obtained when d ) 4, which corresponds to the mean field limit. More recently, Semenov and Joanny8 have shown the existence of a crossover length: z* ∝ N1/(d-1), which divides the space into two regions: the first region, closest to the wall, is dominated by the loops, and the second one, far away (19) Alexander, S. J. Phys. 1977, 38, 983.
F Γ Γ Γ Γ ) f(Γ) + log - Am log N - log Zs2 (48) kBT N N N N where Γ/N is the 2d concentration of chains, Am ) (γ2d 1) ) 3/16, and Zs is the total partition function of a tail. The first term in the free energy (eq 48) accounts for interactions between monomers and between monomers and the wall; it does not depend on N. The second term corresponds to the mixing entropy. The third one comes from the partition function of one chain in the adsorbed layer, which is considered as a 2d melt, it was first introduced by Duplantier.20 Finally, the last term is the contribution of two tails to the free energy. We now need to derive the expressions of f(Γ) and Zs(Γ,N) for adsorbed layers at fixed amount of polymer. It is useful to note that f(Γ) is a negative contribution, which decreases with the adsorbance Γ in the case of a starved layer, because the wall attracts the monomers and adsorption is energetically favorable. The chemical potential per monomer is
1 Γ Am ∂ Γ µ ) -∆µ(Γ) + log log N log Zs2 kBT N N N ∂Γ N (49)
(
)
The first term, which corresponds to the interactions between monomers and between wall and monomers, is dominant for a starved layer. ∆µ(Γ) can be interpreted as the energy necessary to transfer a monomer from the surface into the bulk; the maximum tail or loop size G is obtained, when the energy is of order kbT: G∆µ(Γ) ∼ 1. G is related to the thickness of the adsorbed layer by G ∝ (λ)1/ν. The thickness λ is related to the adsorbance through the conservation of monomers: Γ ) ∫λ0 c(z) dz. The scaling law for the monomer concentration leads to eq 47, and we find
Γ ) Γ0 - c/λd-1-1/ν
(50)
where c is a constant. Thus
-∆µ(Γ) ∝ -(Γ0 - Γ)1/(ν(d-1)-1)
(51)
The free energy is obtained by integration
f(Γ) ) fs + c′(Γ0 - Γ)ν(d-1)/(ν(d-1)-1)
(52)
where fs is the negative constant value of the interaction energy at saturation; it is related to the concentration at the surface by fs ) -φs/b. Due to the so-called proximal effect, the concentration is singular in the vicinity of the surface. De Gennes and Pincus have shown16 that, for a space dimension 3, φs ∝ 1/b; in 4d, there is no proximal (20) Duplantier, B. J. Stat. Phys. 1989, 54, 581.
Surface Pressure of Adsorbed Polymer Layers
Langmuir, Vol. 15, No. 5, 1999 1809
effect, the concentration profile is not singular, and φs ∝ 1/b2. We thus conclude, that:
• in 3d f(Γ) ) -
c1 2
b
+ c2(Γ0 - Γ)6 ) -
c1 2
+
b
c1 λ2
(53)
where c1 and c2 are two constants, not given by the scaling analysis.
• in 4d f(Γ) ) -
c3 b
3
+ c4(Γ0 - Γ)3 ) -
c3 b
3
+
c3 λ3
(54)
As expected, this result agrees with the mean-field theory, in the starved regime presented above (eq 31). The partition function Zs is obtained by integration of the partition function of a tail containing g monomers:
Zs )
∫0G dg Zt(g)
(55)
The partition function of a tail with g monomers, Zt(g), has been calculated in ref 8
Zt(g) ) g[ν(2-d)+γ-2]/2
(56)
where γ is the susceptibility exponent. The total tail partition function then reads
• in 3d Zs(Γ) ∝ λ(γ-ν)2ν ∝ • in 4d
(
(
)
Γ0 Γ0 - Γ
)
(58)
- c2(Γ0 - Γ)5(5Γ + Γ0) 3(γ - ν) Γ2 1 Γ + (59) ν N Γ0 - Γ N
• in 4d π)
xb ) exp(2E)/(exp E + Zs)2 xm )
(62)
2Zs exp E
(63)
[exp E + Zs]2
xf ) Zs2/(exp E + Zs)2
• in 3d vb2
interaction energy f(Γ) is not affected by the existence of the sticking ends. This is the same assumption that was made in the mean-field theory in the calculation of the excluded volume energy. The last two terms correspond to the contribution of the tails to the free energy: the free polymers have two tails, and their partition function is proportional to Zs2 and the monografted polymers have only one tail; the factor of 2 comes from the choice between the two tails. By minimization of the free energy, we find the grafting fractions:
(57)
These results are valid only in the starved regime; close to saturation and in the oversaturated regime, other contributions to the adsorbance (eq 50) must be taken into account. 5.1.2. Pressure. The pressure is obtained by derivation of the free energy:
c1
Ftot AmΓ Γi Γi 2Γb + Γm ) f(Γ) log N + Σ log EkBT N N N N Γf Γm ln Zs2 ln 2Zs (61) N N
3(γ-ν)/2ν
Γ0 ∝ ln(λ/b) Zs(Γ) ∝ ln Γ0 - Γ
π)
gas pressure contains this factor of 2. The dependence of the pressure on the molecular weight N is given by the last two terms, which become important close to saturation. We always find the same dependence as in the meanfield theory; namely, an increase of the chain length induces an increase of the surface pressure. 5.2. Chain with Sticky Chain Ends. 5.2.1. Grafting Densities. We consider now the case where the chain ends can stick to the surface with an energy kBTE. As in section 3, there are three types of polymers: the bigrafted polymers, the monografted polymers, and the polymers with two free ends. The total free energy per unit area of the layer is where Γi ) xiΓ. We suppose here that the
1 2vΓ2 λ 2 Γ 2 vΓ) (2Γ + Γ ) (vΓ + 0 0 N ln λ N 3vb3 12 (60)
Except for the factor of 2 in the ideal gas pressure, the expression of the pressure (eq 60) agrees well with that found (eq 42) in Section 4, from the mean-field theory. The scaling laws do not include more subtle terms in the partition function, which lead to this factor of 2. We believe however, that for the calculation of the ideal gas pressure both in 3 and 4 dimensions, the layer must be considered as a gas of tails (or end points) and that the correct ideal
(64)
In 4d, Zs ) (1/ν) ln(λ/b), so that these expressions reduce to those found with the mean-field theory (eqs 25 and 16). We therefore predict a strong dependence of the grafting fractions on E, but a weak dependence on Γ and no dependence on N. 5.2.2. Pressure. Using the expressions of the grafting fractions, we can write the free energy as follows:
Fa 2Γ Γ Γ Γ Γ ) f(Γ) + log - Am log N - E + log xb kbT N N N N N (65) Comparing to the case of polymers with non sticky ends (eq 49), we see that the two expressions of the free energy only differ by the last term. This term gives the contribution of the free ends to the surface pressure. In dimension 4, we recover the result of the mean-field theory and in dimension 3, we obtain
π)
c1 vb2
- c2(vΓ0 - vΓ)5(5Γ + Γ0) -
3(γ - ν) Γ2 1 ν N (Γ0 - Γ)
( ) ( ( ) ) Γ0 Γ0-Γ
3(γ-ν)/ν
Γ0 exp E + Γ0-Γ
3(γ-ν)/ν
+
Γ N
(66)
The qualitative conclusion is the same as in the meanfield theory: the adsorption of the ends induces an increase
1810 Langmuir, Vol. 15, No. 5, 1999
of the pressure in agreement with the results of the experiments. 6. Concluding Remarks In this paper, we have studied the equation of state of adsorbed polymer layers with a fixed amount of polymers, in contact with a pure solvent. We have used both a meanfield approach and a scaling theory. At small adsorbance, the mean-field theory based on the ground state dominance approximation is sufficient to describe the starved layers, since the structure of the layer is dominated by loops. At higher adsorbance, in the oversaturated regime, the effect of the tails of the chains becomes more important, and we use a mean-field theory with two order parameters. We have determined within these approaches the surface pressure of the layer in two cases, where the chain ends are free or where they can stick to the air-water interface, and we have discussed the variation of the pressure with the chain molecular weight and the sticking energy of the chain ends. If the adsorbance is below the saturation value, in the starved regime, the surface pressure increases with molecular weight and increases as the sticking energy of the chain ends increases. These results are in agreement with recent experiments, in particular those performed on PEO and PEO encapped with hydrophobic alkane chains. One should note, however, that the dependence of the surface pressure on both the molecular weight and the end sticking energy is weak and becomes nonnegligible only in the vicinity of the saturation value of the adsorbance. At higher adsorbance, in the oversaturated regime, where the tails dominate, the effect of molecular weight is inverted and the pressure decreases with molecular weight; this is similar to what is expected for grafted polymer layers. Our results should give a good description of the equation of state of the polymer monolayer at intermediate values of the adsorbance. At a low value of the adsorbance, the adsorbed polymers do not form a continuous layer and can be considered as a two-dimensional semidilute solution. At very high adsorbance, monolayers formed by polymers with free ends are unstable, and the polymers dissolve into the bulk solvent. If the chain ends stick onto the interface, they anchor the chains on the interface and the polymers form a grafted layer (brush). In the meanfield theory, the basic assumption of the theory is that the binding energy is large enough, N . 1. If the adsorbance is too high, this condition is no longer respected and the polymers stretch toward the solvent to form a brush. We were not able to make a quantitative theory of the transition between the adsorbed and the grafted states, which has been already described using bloblike arguments.21 Acknowledgment. We are grateful to P. Muller for his very helpful remarks and suggestions. We also thank A. Johner for his illuminating discussions. Appendix 1. Calculation of the Grafting Fractions Grafting Fractions. The partition function of a chain in the layer starting at z and ending at z′ is Z ) exp(N) ψ(z)ψ(z′)/∫ψ2(z) dz, so that the total partition function Zf (21) Aubouy, M.; Guiselin, O.; Raphae¨l, E. Macromolecules 1996, 29, 7261. (22) This expression implicitly assumes that the sticking endpoint is confined within a distance a/61/2 from the wall. A renormalization of the sticking energy E would allow us to consider a different size of confinement.
Barentin and Joanny
of a chain with two free ends is obtained by integrating twice over all space. The total partition function Zm of a chain with one of its ends adsorbed to the surface is Zm ) exp(N) 2ψ(0)(∫ψ(z) dz)/∫ψ2(z) dz, the factor of 2 coming from the choice between the two ends. Finally, the partition function Zb of a chain with two adsorbed ends is Zb ) exp(N) (ψ(0)2)/∫ψ2(z) dz. This allows us to calculate the total free energy of the layer
Ftot ) kBT
∑Pi log Zi + ∑Pi log Pi - (2Pb + Pm)E
(67)
where Pb, Pm, and Pf correspond, respectively, to the number of bigrafted polymers, monografted polymers, and polymers with free ends. The grafting fractions xi are defined by Pi ) xiP, where P is the total number of adsorbed chains. To calculate the grafting fractions, we use the fact that
x b + x m + xf ) 1
(68)
and that, at equilibrium, the chemical potentials are equal:
µ b ) µm ) µ f
(69)
The equality between the chemical potentials imposes xb/xm ) Zb/Zm exp E, xm/xf ) Zm/Zf exp E, xb/xf ) (Zb/Zf) exp(2E). Using these results and eq 68, we find
xb )
xm )
xf )
[ψ(0) exp E]2 (ψ(0) exp E +
∫ψ(z) dz]2
∫ψ(z) dz [ψ(0) exp E + ∫ψ(z) dz]2 2ψ(0) exp E
∫
( ψ(z) dz)2 [ψ(0) exp E +
∫ψ(z) dz]2
Total Partition Function of a Polymer with Sticking Ends. The total free energy (eq 67) can be expressed as
Zb exp 2E Zm exp E Ftot ) -Pb log - Pm log kBT xb xm Zf Pf log + P log P (70) xf It is easy to show that the arguments of the logarithms are equal, so that the free energy can be written in a compact form
Ftot ) kBT
(
-P log exp(N)
(exp Eψ(0) +
where
)
∫ψ(z) dz)2
∫ψ2(z) dz
+ P log P (71)
Surface Pressure of Adsorbed Polymer Layers
Z ) exp(N)
(ψ(0) exp E +
∫ψ(z) dz)2
∫ψ (z) dz
Appendix 2. Calculation of the Three Order Parameters Oversaturated Regime. The order parameters ψ, φ, and ψ2 satisfy eqs 5, 6, and 19, with the following boundary conditions for ψ2: ψ2(0) ) 0, limzf∞ ψ2(z) ) 0, ψ2 is continuous at b′, and (∂ψ2/∂z)(b′-) - (∂ψ2/∂z)(b′+) ) 1. Looking for power law solutions, we find the following. •For b < z , l′, where the potential is dominated by the concentration of monomers belonging to loops Uint + = ψ2, the order parameters are
ψ(z) )
x2 z+b
δ(l′)3 z4
∝ ψ(z)
where δ is a proportionality constant. •For λ , z, we also find an exponential decrease for ψ2(z) ∝ ψ(z). The results of section 2 can therefore be adapted to the case of sticky chain ends, just by replacing l by l′ in all expressions, so that
∫ψ(z) dz = x2 log l′b vΓ )
(72)
∫ψ2 dz ) b2 + Rl′ - βλ
(73)
This leads to
1 l′ φ(z) ) z2 log 3 z+b
(
ψ2(z) )
2
appears to be the partition function of one chain of any type.
ψ2(z) )
Langmuir, Vol. 15, No. 5, 1999 1811
)
(b + b′)2 b3 b2 1 = 3 3(b + b′) z + b (z + b)
We find the same expression for ψ and φ as in the case of nonsticking ends except for the fact, that l is replaced by l′. •For l′ , z , λ, where the potential is dominated by the concentration of monomers belonging to nonadsorbing tails Uint + = B′ψφ, we find the same power laws for ψ, and φ as in eq 11, and for ψ2
B′ )
(2xf + xm)vΓ
∫
N ψ(z) dz
=
x2vΓ N
b
(
exp E + b log
l′ b
)
(74)
Starved regime. Using a similar approach, we can show that the order parameter ψ2 is proportional to the loop order parameter
ψ(z) ) LA981173T
x2 λ sinh[(z + d)/λ]
