Mean-Field Lattice Calculations of Ethylene Oxide and Propylene

Density profiles of PEO homopolymers across the air/water interface are compared with recent .... lattice model for polymers with internal degrees of ...
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4066

Langmuir 1997, 13, 4066-4078

Mean-Field Lattice Calculations of Ethylene Oxide and Propylene Oxide Containing Homopolymers and Triblock Copolymers at the Air/Water Interface Per Linse*,† and T. Alan Hatton‡ Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden, and Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received December 9, 1996X The adsorption of poly(ethylene oxide), poly(propylene oxide), and triblock copolymers containing ethylene oxide (EO) and propylene oxide (PO) at the air/water interface was modeled on the basis of a mean-field lattice theory for multicomponent mixtures of copolymers with internal degrees of freedom occurring in heterogeneous systems. The surface tension, the surface excess, and the volume fraction profiles across the interface were examined for PEO and PPO homopolymers and for a number of PEO-PPO-PEO triblock copolymers. The effects of the self-assembly of the copolymers in the solution, of depletion of the copolymers in the solution at small volume-to-surface ratios, and of a mass distribution of the copolymers on the adsorption were also considered. Interaction parameters obtained for PEO and PPO in simpler systems were used. Density profiles of PEO homopolymers across the air/water interface are compared with recent neutron reflectivity measurements. The shape and extension of the EO profiles into the water subphase agree well with the results extracted from the experiments, but the predicted surface excess is too large. On the basis of the results from the triblock copolymer systems, the low-concentration break, often experimentally observed when the surface tension is viewed as a function of the logarithm of the concentration, is proposed to be an effect of a depletion of the copolymers in the solution. This new proposal (i) clarifies the unreasonably small surface areas per molecule previously obtained and (ii) brings the predicted reduction of the surface tension made by the copolymers into agreement with experimental data. Finally, the model calculations predict correctly the dependence of the surface tension on the composition of the triblock copolymers. For the polydisperse model systems, the calculations demonstrate that below the cmc the longest and most surface active component dominates the adsorbed amount. Above the cmc, the long components self-assemble preferentially, making the solution depleted of these components and hence reducing the adsorbed amount of the long polymer components at the interface.

I. Introduction During the last decade, the solution properties of watersoluble triblock copolymers of the PEO-PPO-PEO type [PEO and PPO are poly(ethylene oxide) and poly(propylene oxide), respectively] have been extensively investigated. These polymers are amphiphilic in nature and are often referred to by their trademarks as Pluronic, Synperonic, and Poloxamers. Their surface activity and their tendency to self-assembly in solution stem from the fact that the PPO block is relatively hydrophobic, whereas the PEO blocks are more hydrophilic. A number of recent review articles1-4 summarize the experimental literature on the properties of PEO-PPO-PEO triblock copolymers in aqueous solution. Some of the experimental studies5-10 deal with the surface activity of the PEO-PPO-PEO polymers at the †

Lund University. Massachusetts Institute of Technology. X Abstract published in Advance ACS Abstracts, June 1, 1997. ‡

(1) Almgren, M.; Brown, W.; Hvidt, S. Colloid Polym. Sci. 1995, 273, 2. (2) Alexandridis, P.; Hatton, T. A. Colloids Surf. A 1995, 96. (3) Alexandridis, P. Curr. Opin. Colloid Interface Sci. 1996, 1, 490. (4) Mortensen, K. J. Phys.: Condens. Matter A 1996, 8, 103. (5) Prasad, K. N.; Luong, T. T.; Florence, A. T.; Paris, J.; Vaution, C.; Seiller, M.; Puisieux, F. J. Colloid Interface Sci. 1979, 69, 225. (6) Reddy, N. K.; Fordham, P. J.; Attwood, D.; Booth, C. J. Chem. Soc., Faraday Trans. 1990, 86, 1569. (7) Wanka, G.; Hoffmann, H.; Ulbricht, W. Colloid Polym. Sci. 1990, 268, 101. (8) Wanka, G.; Hoffmann, H.; Ulbricht, W. Macromolecules 1994, 27, 4145. (9) Alexandridis, P.; Athanassiou, V.; Fukuda, S.; Hatton, T. A. Langmuir 1994, 10, 2604. (10) Kabanov, A. V.; Nazarova, I. R.; Astafieva, I. V.; Batrakova, E. V.; Alakhov, V. Y.; Yaroslavov, A. A.; Kabanov, V. A. Macromolecules 1995, 28, 2303.

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air/water interface. Surface tension measurement is a common technique to determine the critical micellization constant (cmc), which is a central quantity in applications of solutions of PEO-PPO-PEO copolymers. Ideally, the cmc is extracted from a single break of the surface tension as a function of the logarithm of the copolymer concentration. Throughout the experimental studies, the interpretation of such plots plays a central role. In some cases a single and often wide transition between two linear (or nearly linear) regimes, one with a negative slope and one with a less negative or zero slope, was found.5-7 The broad transition region has been explained as a consequence of a broad molecular mass distribution and of the presence of impurities.6 In other cases, two breaks in the surface tension data could be identified.7-9 At very low concentrations, the surface tension decreases linearly with the logarithm of the concentration, but after the first break, the surface tension is still decreasing, but now with a smaller slope. Finally, after a second break at an even higher concentration the surface tension remains constant. Wanka et al. used the intersection of the extrapolations of the first and third linear regimes to extract the cmc,7 and they explained the existence of the two transitions with the fact that these molecules have broad mass distributions. Later, the same authors made an extended study of several PEO-PPO-PEO polymers at several temperatures.8 The existence of multiple transitions was confirmed and they clearly demonstrated that the extension of the central regime increases with decreasing temperature. When two breaks were present, they used the one at the higher concentration for the determination of the cmc and the slope of the central regime for the evaluation of the area per molecule. © 1997 American Chemical Society

Mean-Field Lattice Calculations of EO and PO

Alexandridis et al. also performed an extended study involving several PEO-PPO-PEO polymers at two temperatures.9 In most cases, two breaks in the surface tension data were also found. A good correlation between the copolymer concentration at the high-concentration break and the cmc estimated from dye solubilization experiments was observed, and hence the second break was suggested to be related to the cmc in the solution. Moreover, after examining several tentative explanations of the low-concentration break, (i) an experimental artifact, (ii) effect of the copolymer polydispersity or of the presence of impurities, and (iii) the formation of monomolecular micelles in bulk aqueous phase, none of them was found likely to give the observed data. Instead, it was proposed that the low-concentration break could originate from a rearrangement of the copolymer molecules, although no direct experimental evidence was presented. They used the first regime for evaluation of the surface area per molecule. Hence, in those cases where three regimes were present, the surface areas obtained by Alexandridis et al.9 were smaller than those by Wanka et al.8 Different properties of the PEO homopolymer at the air/water interface have been investigated for quite some time,11-16 but recent neutron reflectivity measurements17 have added some more direct information on the volume fraction profiles perpendicular to the surface. However, there still remain essential questions such as (i) whether spread monolayers and adsorption from solution produce the same interface (kinetic barriers) and (ii) what the extent of PEO penetration into the air is. One motivation for the present theoretical study is to elucidate the existence of broad and multiple transitions in surface tension data of PEO-PPO-PEO polymers in aqueous solution and to examine if reliable surface excesses and areas per molecules could be extracted from such data. The present contribution is also an extension of our ongoing modeling work on the self-assembly of PEO-PPO-PEO triblock copolymers in solution,18-20 the phase behavior of EO- and PO-containing polymers in aqueous solution,21-23 and the adsorption of PEO-PPOPEO triblock copolymers and related copolymers at solid surfaces.24-26 In these studies, we have applied a meanfield lattice theory, initially developed by Scheutjens and Fleer,27 and later extended in different directions.28 Central to our application of the lattice theory is the inclusion of internal degrees of freedom of the polymer segments. This extension makes it possible to use the two-state model developed by Karlstro¨m29 to describe the (11) Couper, A.; Eley, D. D. J. Polym. Sci. 1948, 3, 345. (12) Glass, J. E. J. Phys. Chem. 1968, 13, 4459. (13) Schwuger, M. J. J. Colloid Interface Sci. 1973, 43, 491. (14) Kim, M. W.; Cao, B. H. Europhys. Lett. 1993, 24, 229. (15) Cao, B. H.; Kim, M. W. Faraday Discuss. 1994, 98, 245. (16) Cao, B. H.; Kim, M. W. Europhys. Lett. 1995, 29, 555. (17) Lu, J. R.; Su, T. J.; Thomas, R. K.; Penfold, J.; Richards, R. W. Polymer 1996, 37, 109. (18) Linse, P. Macromolecules 1993, 26, 4437. (19) Linse, P. Macromolecules 1994, 27, 2685. (20) Linse, P. Macromolecules 1994, 27, 6404. (21) Malmsten, M.; Linse, P.; Zhang, K.-W. Macromolecules 1993, 26, 2905. (22) Linse, P. J. Phys. Chem. 1993, 97, 13896. (23) Noolandi, J.; Shi, A.-C.; Linse, P. Macromolecules 1996, 29, 5907. (24) Tiberg, F.; Malmsten, M.; Linse, P.; Lindman, B. Langmuir 1991, 7, 2723. (25) Malmsten, M.; Linse, P.; Cosgrove, T. Macromolecules 1992, 25, 5434. (26) Schille´n, K.; Claesson, P. M.; Malmsten, M.; Linse, P.; Booth, C. J. Phys. Chem. 1997, 101, 4238. (27) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619; 1980, 84, 178. (28) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993. (29) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962.

Langmuir, Vol. 13, No. 15, 1997 4067

strong temperature dependent EO-water and PO-water interactions. This polymer model originates from quantum mechanical considerations and explains the reduced polymer solubility at elevated temperature by a shift of the polymer conformations toward less polar conformers. The same model has also been employed by Hurter et al. to investigate the self-assembly of PEO-PPO-PEO triblock copolymers in aqueous solution30 and to examine the solubilization of hydrophobic molecules into the core of the aggregates.31 The paper is organized as follows. In section II, we briefly describe the theory basic to the heterogeneous lattice model for polymers with internal degrees of freedom at an infinite interfacial system, including an account of the coupling between the interfacial system with the existence of micelles. We then continue in section III with a presentation of how the PEO-PPO-PEO triblock copolymers are mapped onto the lattice model and our implementation of polydisperse samples. Section III ends with a discussion of the interaction parameters used. After a brief account of the surface activity of the PEO and PPO homopolymers, section IV continues with our predicted surface tensions and structures of the PEO-PPO-PEO triblock copolymers at the air/water interface, including the effects of composition and architecture variation and of polydisperse polymer samples. In section V, we relate our observations to experimental data. One important conclusion drawn is that the low-concentration break of the surface tension curve is likely due to a depletion of the solution of copolymer due to small solution height and very small copolymer concentrations. Hence, the slope of this part of the surface tension data cannot be used to extract the area per molecule at the interface. However, if a central regime exists between the cmc and the depletion range, it could be used for that purpose. II. Theory The Flory-Huggins (FH) lattice theory of homogeneous solutions32 has been extended by Scheutjens and Fleer27 to describe the adsorption of flexible polymers at surfaces. Since then, the heterogeneous lattice theory has been further extended in several directions and applied to a number of different cases.28 We will here give a short account of the theory adapted to block copolymer adsorption at fluid interfaces. For a more detailed description of the lattice theory and nomenclature used, see, e.g., refs 27, 33, and 34. In our application, the polymer solution in the close vicinity of a fluid interface is divided into layers parallel to the planar interface. The layers available to the molecules are labeled i ) 1, 2, ..., M with the first layer well inside one phase and the last layer far inside the other one. The thickness of the layers corresponds to the size of a polymer segment. Within each layer, the BraggWilliams approximation of random mixing is applied, and hence all lattice sites in a layer are equivalent. A hexagonal lattice has been chosen, and thus the number of nearest-neighbor sites, z, is 12. The lattice is completely filled by a mixture of the components (water, polymer, (30) Hurter, P. N.; Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5030. (31) Hurter, P. N.; Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5592. (32) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (33) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221. (34) Linse, P.; Bjo¨rling, M. Macromolecules 1991, 24, 6700.

4068 Langmuir, Vol. 13, No. 15, 1997

Linse and Hatton

and air). (In the FH model gas, or vacancies, is treated as one component with interaction parameters in the usual way.) There are nx molecules of component x and each molecule consists of rx segments (1, rpolymer, and 1, respectively). There are two types of polymer segments, EO and PO. The type of small molecule or polymer segment is referred to as the species. The presence of an interface introduces a unique direction (the normal) in the system. The complete directional degeneration of the segment spatial distribution far away from the interface is thus lifted, and various conformations of the polymer chains can be distinguished according to the different ordering of their segments with respect to the layer numbers. Neglecting at first selfexclusion, the degeneration of a conformation c of component x becomes exp[(rx - 1) ln z + ln ωxc], where the factor ωxc is a product of the number of sites in the layer where the first segment is located and λij factors, one for each segment except the first, which gives the probability of arranging the segments according to conformation c. λij denotes the fraction of nearest-neighbor sites in layer j viewed from a site in layer i. The derivation of the internal state and of the segment distributions starts from the partition function of the model system. After the partition function is replaced by its largest term, the (Helmholtz) free energy of the system relative to the reference state of separated amorphous components may be expressed as

Ω β(A - A*) ) β(Aint - A*int) - ln + β(U - U*) Ω*

(1)

where β ) (kT)-1, k being the Boltzmann constant and T the absolute temperature. The first term on the righthand side (rhs) of eq 1 denotes the contribution arising from the internal degrees of freedom, the second term is the mixing entropy, and the third is the mixing energy. Each state of a species is characterized by an energy term, UAB, and a degeneration factor, gAB, where in both cases A denotes the species and B the state of species A. The total internal free energy arising from the internal states becomes

βAint )

[

]

PABi

nAi∑PABi βUAB + ln ∑i ∑ g A B

AB

(2)

where ∑i denotes the sum over layers, ∑A the sum over species (segment types), and ∑B the sum over the states of species A, nAi is the number of sites in layer i occupied by segments of type A, and PABi is the fraction of species A in layer i which is in state B. There are three contributions to the internal free energy, namely UAB, which is the internal energy of state B of species A; -kT ln gAB, where gAB is the degeneration factor of state B of species A; and kT ln PABi, an entropy term arising from the mixing of the states. In the case of EO (or PO), UAB and gAB describe the equilibrium between the polar and nonpolar states of EO (PO). Since the nonpolar state has a higher internal energy and greater degeneration, UEO,nonpolar > UEO,polar and gEO,nonpolar > gEO,polar. Finally, A* becomes zero by a suitable choice of reference state. When the self-exclusion on a mean-field level is taken into account and the contributions from all conformations of all the components are included, the total conformational degeneration relative to the amorphous reference state, constituting the mixing entropy of the system, is given simply by

ln

Ω

∑x ∑c nxc ln

nxcrx

)Ω*

ωxc

(3)

where nxc denotes the number of chains of component x in conformation c. Within the mean-field approximation, the interaction energy is given by

1 M βU ) L 2 i)1

∑∑ ∑∑∑φAiPABiχBB′〈PA′B′iφA′i〉 A A′ B B′

(4)

where L is the number of lattice sites in a layer and 〈...〉 denotes an average over the actual and adjacent layers according to 〈xi〉 ) λi,i-1xi-1 + λixi + λi,i+1xi+1. In eq 4, χBB′ denotes the Flory-Huggins interaction parameter,32 traditionally defined as χBB′ ) βz[BB′ - (BB + B′B′)/2], where BB′ is the interaction energy on a site-volume basis between species A in state B and species A′ in state B′. In the case of a block copolymer, the reference interaction energy U* in eq 1 is nonzero (see, e.g., eq A.5.2 of ref 34). Minimization of the free energy with respect to {PABi} provides us with the state distribution, which is given by the implicit set of nonlinear equations

PABi )

XAB XAB ∑ B

× (5)

∑ ∑χBB′〈PA′B′iφA′i〉] A′ B′

XAB ≡ gAB exp[-βUAB -

valid for all species, states, and layers. The numerator gives the weight of species A in layer i to be found in state B, the denominator being a normalization factor. As expected, eq 5 shows that a state is favored by high degeneration, low internal energy, and a favoarable interaction (small χ) with neighboring segments. The expression for the segment distributions (water, EO and PO segments, and air) is more complex. Minimization of the free energy, now with respect to {nxc}, and the packing constraint result in an expression for the layer dependent species potential uAi, which denotes the relative free energy of an unconnected segment of type A being located in layer i. The species potential can be divided into two parts, one species independent, u′i, and the other dependent on the species, uint Ai , according to

uAi ) u′i + uint Ai

(6)

If the species potentials are defined with respect to one of the bulk fluids, i.e., if ubA ) 0, then the two terms are given by

βu′i ≡ Ri +

βuint Ai ≡

∑ B

φbx

∑x r

1 +

x

[ (

b b b χB′B′′PA′′B′′ φA′′ ∑∑ ∑∑φbA′PA′B′ A′′ B′ B′′

2 A′

PABi βUAB + ln PbAB

(

)

PABi gAB

βUAB + ln

(7)

-

)]

PbAB gAB

+

b φbA′) ∑ ∑∑χBB′(PABi〈PA′B′iφA′i〉 - PbABPA′B′ A′ B B′

Mean-Field Lattice Calculations of EO and PO

Langmuir, Vol. 13, No. 15, 1997 4069

b where PAB is the fraction of species A in the bulk, which is in state B, as given by a relation similar to eq 5. The species-independent potential u′i, related to the lateral pressure in a continuous model, ensures that by a suitable choice of Ri the space is completely filled at layer i. In bulk, u′ becomes zero. The species-dependent term uint Ai has two contributions: the internal free energy for species A in layer i being diminished by the corresponding quantity in bulk, and the mixing energy for species A in layer i being diminished by the mixing energy for species A in the bulk. In both cases, averages are taken over the relevant state distributions. At distances far from the interface, φAi approaches φbA, and hence uint Ai becomes zero. The second aspect of determining the segment distribution is to take into account the chain connectivity. If only monomers are present, the volume fraction φAi of monomer A in layer i is related only to the bulk volume fraction φbA according to

φAi ) GAiφbA

(8)

where the weighting factor GAi for species A in layer i is given by

GAi ) exp(-βuAi)

(9)

since the species potentials were defined to be zero in the bulk. The matter becomes more complex for polymers. However, using a matrix method, the segment distribution as expressed in terms of nxsi, the number of sites in layer i occupied by segments of rank s belonging to component x, is given by34 s

s+1

∏ s′)r

nxsi ) Cx{∆Ti ‚[

Wt(x,s′)]‚p(x,1)} ∏ s′)2

(Wt(x,s′))T]‚s}{∆Ti ‚[

x

(10)

where Cx is a normalization factor (related to the amount of or to the bulk volume fraction of component x), Wt(x,s) is a tridiagonal matrix comprising elements that contain factors describing the lattice topology and weighting factors GAi for the segment of rank s belonging to component x, and p(x,1) is a vector describing the distribution of the first segment of component x in the layers, with ∆ and s being elementary column vectors. From nxsi the segment volume fractions desired are easily obtained. The species volume fraction φAi, needed in eq 10, is given by

φAi )

1

rx

∑x s)1 ∑δA,t(x,s)nxsi

L

(11)

where the Kronecker δ only selects segments of rank s of component x if they are of type A. Thus, given the species potentials uAi, the species volume profiles φAi are obtained by eqs 9-11, these equations together with eqs 6 and 7 constituting an implicit set of nonlinear equations for the segment distributions. Two central properties in our investigation are the surface tension and the surface excess of polymers. Following Evers et al.33 the surface tension γ is evaluated according to

(aL)γ ) Aσ

(12)

where a is the area of a lattice cell, L is the number of lattice cells forming the interface, and Aσ is the excess

surface free energy given by

Aσ ) (A - A*) -

∑x nx(µx - µ*x)

(13)

with A - A* given by eq A.8.3 in ref 34 and µx - µx* being the difference of the chemical potential of component x in the system and in the reference state given by eq A.7.12 in ref 34. In the following we will use reduced surface tension according to γ* ) aγ/kT. The polymer surface excess Γpolymer is expressed in terms of the equivalent number of lattice layers evaluated using the Gibbs dividing surface obtained from the water profiles. Since the polymer concentration in the interface is much higher that in either bulk phase, Γpolymer is insensitive on the precise location of the dividing surface. In the case of the PEO-PPO-PEO triblock copolymer, aggregates are formed above a critical copolymer concentration referred to as the critical micellization concentration (cmc). Above the cmc, the increase of the chemical potential of the polymers with the concentration is strongly reduced, and this causes a change in the slope of the surface tension as a function of the logarithm of the concentration. The stoichiometric concentration of a polymer component x will be referred to as the total volume fraction of x, φtot x . For an infinite volume-to-surface ratio (which is assumed if nothing else is stated) and below the ) φfree where φfree is the cmc, we obviously have φtot x x x volume fraction of free (nonaggregated) x. Above the cmc, free mic + φmic we have φtot x ) φx x where φx is the volume fraction of component x in the micelles, and we have followed the procedure given in ref 18 for the determination of φfree and φmic x x . In the present calculations and above the cmc, we have used {φfree x } as the bulk volume fractions in the aqueous phase far from the interface (i ) 1) and related the results to the corresponding {φtot x } values. Reflective boundary conditions were applied at i ) 0.5 and M + 0.5. We have employed 30-50 aqueous layers, but all data are presented such that the interface is located at i ≈ 30. III. Polymer Model Polymers. Calculations of PEO and PPO homopolymers and triblock copolymers of the PEO-PPO-PEO type at the air/water interface have been performed. The homopolymers modeled are (EO)35, (EO)91, (PO)11, and (PO)34, corresponding to molecular masses of 1540, 4000, 620, and 2000, respectively. Pluronic polymers (trademark of BASF Corp.) with a broad range of total mass and of EO/PO ratios are available commercially. The synthesis procedure leads, however, to the formation of impurities as homopolymers and diblock copolymers as well as to specific mass and composition distributions of the desired triblock copolymer. Most calculations with the triblock copolymers involved (EO)37(PO)56(EO)37, corresponding to Pluronic P105. Since the main purpose of this investigation is the surface behavior of the commercially available PEO-PPO-PEO copolymers, the effects of composition variation and mass polydispersity have also been considered. We have used a polydispersity model where the polymer sample is represented by several polymer components with different compositions, and we have closely followed the approach taken for modeling the self-association of polydisperse EO- and PO-containing polymers.20 Since we have a triblock copolymer, there is a tridimensional polydispersity if one assumes the number of segments in

4070 Langmuir, Vol. 13, No. 15, 1997

Linse and Hatton Table 2. Internal State Parameters (UAB and gAB) and Flory-Huggins Interaction Parameters (χBB′) of the Theoretical Model (Energy in kJ mol-1) species water EO PO

state

UAB

gAB

polar nonpolar polar nonpolar

0 0 5.086a 0 11.5b 0

1 1 8a 1 60b 1

air

kTχBB′

Figure 1. Number and mass distributions, Pn(x) and Pm(x), respectively, versus x for a polydispersity ratio of Mm/Mn ) 1.2 as obtained from the Schulz-Zimm distribution function. P(x) denotes the probability of x ≡ r/〈r〉n, where r is the number of segments in a polymer (or a block) and 〈r〉n is the numberaverage number of segments of the polymer (or the block). A five-point discrete representation of Pm(x) is also shown (bars). The points are selected with constant spacing in x with 〈x〉n ) 1 and in a manner such that the same polydispersity as for the continuous representation is obtained.20 Table 1. Relative Number of Segments, xj, and Mass Weights, wj, of the Five Polymer Components Describing a Polydisperse Polymer for Mm/Mn ) 1.2a j

xj

wj

1 2 3 4 5

0.317 0.872 1.427 1.981 2.536

0.0482 0.4714 0.3452 0.1114 0.0239

a x ≡ r /〈r〉 , where r is the number of segments of component j j n j j and 〈r〉n is the n-averaged number of segments. wj is the mass weight of component j.

each block to be independent of each other. We have limited ourselves to consider the case with a constant composition (fixed EO/PO ratio); thus there is only a distribution of the mass. As shown in the previous studies,20,26 the polydispersity effects become even more accentuated when a full polydispersity, i.e. mass and composition polydispersity, is included in the model. The calculations of the polydisperse systems were made by assuming a Schulz-Zimm distribution. A width giving Mm/Mn ) 1.2, where Mm and Mn are the mass- and numberweighted mass averages, respectively, was used, and Figure 1 shows the number and mass distributions. We have used the same procedure as given in ref 20 for obtaining the discrete representation of the distribution. Table 1 gives the relative numbers of segments, xj, and the mass weights, wj, of the five components employed, whereas Figure 1 shows graphically the discrete representation. The final numbers of EO and PO segments in the different components are obtained by multiplying xj by 〈rEO〉n and 〈rPO〉n, identified as a and b, respectively, in the formula (EO)a(PO)b(EO)a. By assigning a length to the lattice cell, it is possible to convert our results to real units. The assignment is not unique, but in the present study where one segment in the model is mapped onto one monomeric unit of the real polymer, 4 Å has been found to be reasonable for PEO-PPO-PEO and similar polymers.18,24,26 Interaction Parameters. In order to carry out the calculations, parameters describing the interaction among the species and the state equilibriums have to be specified. The internal state energy, UAB, and the degeneration, gAB, of the states of all species, as well as the Flory-Huggins interaction parameters between all pairs of species in the different states, χBB′, are compiled in Table 2. All of them, except those related to the air, were earlier determined

water EO(polar) EO(nonpolar) PO(polar) PO(nonpolar)

EO(polar)

EO(nonpolar)

PO(polar)

PO(nonpolar)

0.6508a

5.568a 1.266a

1.7b 1.8c 0.5c

8.5b 3.0c -2.0c 1.4b

air 25d 12.5d 12.5d 7.5d 7.5d

a From the fit to the experimental data of the binary PEO/water phase diagram (see refs 29 and 35). b From the fit to the experimental data of the binary PPO/water phase diagram (see ref 34). c From the fit to the experimental data of the ternary PEO/PPO/ water phase diagram (see ref 21). d Determined in this work; see text.

for simpler systems (binary PEO/water,29,35 binary PPO/ water,34 and ternary PEO/PPO/water solutions21) through fitting calculated phase diagrams to experimental ones. These parameters are kept the same as in the earlier investigations concerning the micellization of block copolymers containing EO and PO,18 the effect of polymer impurity19 and of polydispersity20 on the micellization, the phase behavior of PEO-PPO-PEO triblock copolymers in dilute aqueous solution,22 and the formation of ordered phases at higher polymer volume fractions.23 The air-water interaction was selected to give a reasonable air/water surface tension. The choice RTχwater,air ) 25 kJ mol-1 gives γ* ) 2.49 at T ) 298 K. With a lattice cell length d ) 4 Å (and a ) d2 ) 16 Å2), this γ* value corresponds to γ ) 64 mN m-1. There is no advantage to adjust χwater,air more accurately (γexp ) 72 mN m-1). The remaining EO-air and PO-air parameters were adjusted to reproduce experimental surface tension data for aqueous solutions of PEO and PPO at different concentrations and molecular masses (see below). In order to simplify the procedure, no discrimination between the polar and nonpolar states was made, giving only one fitting parameter for each homopolymer system. Finally, we note that in the calculations involving the PEO-PPO-PEO triblock copolymers at the air/water interface, all the interaction parameters are fixed. Hence, no parameters in these calculations are adjusted to experimental data (of corresponding systems). IV. Results and Discussion PEO and PPO Homopolymers. Figure 2 shows experimentally determined surface tensions for relatively short PEO and PPO homopolymers in aqueous solution at 298 K, data taken from Couper and Eley11 (PEO) and Schwuger13 (PPO). More recent investigations show that the surface tension of an aqueous solution of PEO with higher molecular mass displays a more irregular behavior.14,15 As alluded to in the previous section, surface tensions were calculated for corresponding systems by varying the χEO,air and χPO,air parameters, respectively. The best agreement (by inspection) with the experimental data in (35) Bjo¨rling, M.; Linse, P.; Karlstro¨m, G. J. Phys. Chem. 1990, 94, 471.

Mean-Field Lattice Calculations of EO and PO

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Figure 2. Experimentally determined surface tension (γ) for aqueous solutions of PEO (filled symbols) and PPO (open symbols) homopolymers as a function of the polymer concentration (Cpolymer) at indicated molecular weights and T ) 298 K. The horizontal dotted line denotes the experimental surface tension of an air/water interface. [Adapted from ref 11 (PEO) and ref 13 (PPO).]

Figure 4. Calculated volume fraction profiles at the air/water interface for an aqueous solution of (a) (EO)91 and (b) (PO)34 at φpolymer ) 10-3 and T ) 298 K. The polymer profiles are given by solid curves, and the water and air profiles, by dotted curves.

Figure 3. Calculated (a) reduced surface tension (γ*) and (b) surface excess (Γpolymer) as a function of the total polymer volume fraction (φpolymer) for aqueous solutions of PEO (solid curves) and PPO (dashed curves) at the indicated number of segments and T ) 298 K. In (a) the horizontal dotted line denotes the calculated reduced surface tension of an air/water interface.

Figure 2 was obtained with RTχEO,air ) 12.5 kJ mol-1 and RTχPO,air ) 7.5 kJ mol-1. The calculated surface tensions are displayed in Figure 3a, where the same relative γ-window as in Figure 2 is used in order to facilitate the comparison. Such a comparison between the experimental and calculated data shows that the model is able to accurately reproduce (i) the reduction of γ with polymer concentration, (ii) the higher surface activity (larger reduction of γ) of PPO as compared to PEO, and (iii) the reduction of γ with increasing PEO or PPO chain length. The second observation is, of course, related to the choice χEO,air > χPO,air, but also the fact that χEO,water < χPO,water (cf. Table 2) contributes to make PPO more surface active. The trends (i) and (iii) are much less dependent on the values of χEO,air and χPO,air. The close agreement of calculated data with experimental data obtained after adjusting the vertical

scale by varying χEO,air and χPO,air, respectively, is thus encouraging. Figure 3b displays the corresponding surface excess for the aqueous solutions of PEO and of PPO, respectively. The results show that the surface excess (i) is similar for PEO and PPO at not too high concentration (Γpolymer ) 2 corresponds to ca. 1 mg m-2), (ii) increases with polymer concentration, and (iii) increases with chain length. The divergencies of ΓPPO occur to the saturation limits of the PPO homopolymers. Volume fraction profiles across the air/water interface are shown in Figure 4 for φpolymer ) 10-3. For both (EO)91 and (PO)34 the air volume fraction changes very sharply at the interface; φair changes from 0 to 1 over two layers. The water profiles display a more gradual transition. Thus, the polymer surface layer lies on top of the water surface and displays a smooth surface toward air but has loops and tails extending into the water subphase ca. 30 Å from the polymer/air interface. There are no qualitative differences between the PEO and PPO volume fraction profiles. PEO-PPO-PEO Triblock Copolymers. General Information. The calculated surface tensions of aqueous solutions of (EO)37(PO)56(EO)37 as a function of the logarithm of the total polymer volume fraction at two temperatures are given in Figure 5a (solid curves). The surface tension displays a weak convex curvature below the cmc, whereas after the cmc it becomes essentially constant. At increasing temperature, there is a considerable reduction of the surface tension, and the break is shifted to lower concentrations. The corresponding surface excesses are given in Figure 5b (solid curves). The surface excess increases with the copolymer concentration up to the cmc and is approximately constant at higher concentrations. At the higher temperature, we observe a larger surface excess. The weak convex curvature below the cmc in Figure 5a is related to the increase in the surface excess with the logarithm of the total polymer volume fraction. The effects of the temperature on the surface tension and

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Figure 5. Calculated (a) reduced surface tension (γ*) and (b) surface excess (Γpolymer) as a function of the total polymer volume tot fraction (φpolymer ) for an aqueous solution of (EO)37(PO)56(EO)37 triblock copolymer (solid curves) at indicated temperatures. The corresponding results if the micellization is disregarded (dotted curves) and at finite surface-to-volume ratio V/A ) 107 (see text) (dashed curves) are also presented. The arrows denote the location of the cmc’s, and the symbols in (b) represent Γpolymer obtained from γ* using the Gibbs adsorption isotherm, eq 14.

on the surface excess are well in line with the fact that PEO-PPO-PEO becomes more hydrophobic at higher temperatures.2 This change in hydrophobicity is also responsible for the reduction of the cmc with increasing temperature. From the surface excess, the area per molecule at the interface is readily obtained. For example, the value Γpolymer ) 6, corresponds to rpolymer/Γpolymer ) 130/6 ≈ 22 lattice sites or ca. 350 Å2. At sufficiently low polymer volume fractions, the finite volume of a system has to affect the surface tension and the surface excess. Let V/A denote the volume-to-surface ratio. The stochiometric polymer volume fraction at finite tot tot (V/A finite) ) φpolymer (V/A f ∞) + V/A is given by φpolymer tot (V/A)-1Γpolymer, where φpolymer(V/A f ∞) denotes the volume fraction in an infinitely large system or alternatively is viewed as the value far away from the surface in a finite tot (V/A f ∞) can be system. At low concentrations φpolymer neglected, and a low-concentration break of the surface tot (V/A finite) ≈ (V/A)-1Γpolymer. tension occurs at φpolymer Also included in Figure 5 are γ* and Γpolymer (dashed curves) for V/A ) 107, corresponding to a solution height of 4 mm. We indeed find a low-concentration break, and since Γpolymer tot is insensitive to the temperature at low φpolymer , the break caused by the small volume-to-surface ratio is essentially temperature independent. The results in Figure 5 also enable us to relate the surface excess to the dependence of the surface tension on the concentration by using the Gibbs adsorption isotherm. This relation is often expressed as

Γ)-

1 dγ kT d ln C

(14)

where Γ is the surface excess expressed as the number of

Figure 6. Calculated volume fraction profiles at the air/water interface for an aqueous solution of (EO)37(PO)56(EO)37 triblock tot copolymer at (a) φpolymer ) 10-5 (far below the cmc) and (b) 10-1 (above the cmc) with T ) 320 K. The total polymer profiles are given by solid curves with the contributions from EO and PO by dashed curves and the water and air profiles by dotted curves.

molecules per unit area and is related to Γpolymer through Γ ) Γpolymer/rpolymer. In the derivation of eq (14), it is assumed that the activity coefficients approach unity; i.e., only the ideal contribution k ln C contributes to the chemical potential.36 Figure 5b shows that within the model, the Gibbs adsorption isotherm holds very well tot < 0.02. Similar good appearance provided that φpolymer was found for the PEO and PPO homopolymers if tot φpolymer < 0.001 (not shown). Figure 6 shows volume fraction profiles across the interface for the aqueous solution of (EO)37(PO)56(EO)37 at two different volume fractions, one far below and one slightly above the cmc. As for the case of the homopolymers, the surface of the air is sharp but the triblock copolymers extend further into the aqueous phase. There is a clear spatial segregation between EO and PO segments and the PPO layer is located closest to the air/water interface. Since the PPO forms a layer separating air from the solution, it eliminates almost all air-PEO as well as air-water contacts. Only the EO segments contribute to the tail penetrating into the aqueous subphase, and the interfacial region amounts to 40-60 Å depending on the polymer volume fraction. The spatial segregation observed here resembles that for PEO-PPO-PEO and related polymers at hydrophobic surfaces: the air constitutes a sharp surface and there is a clear separation between the different blocks. On the other hand, similar model calculations of block copolymers at the interface between two nonmiscible solvents showed that the two different blocks stretched into the two different phases in a similar way.37 PEO-PPO-PEO Triblock Copolymers. Composition and Structure Variations. Corresponding in(36) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1977. (37) Phipps, J. S.; Richardson, T. M.; Cosgrove, T.; Eaglesham, A. Langmuir 1993, 9, 3530.

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Table 3. Calculated Reduced Surface Tension, γ*, Surface Excess, Γpolymer, and Surface Area per Triblock Copolymer, Apolymer, at Two Different Total Volume Fractions at 320 Ka tot φpolymer ) 10-7

triblock copolymer

γ*

(EO)16(PO)56(EO)16 (EO)25(PO)56(EO)25 (EO)37(PO)56(EO)37 (EO)31(PO)47(EO)31 (EO)25(PO)40(EO)25 (PO)28(EO)74(EO)28

1.35 1.36 1.37 1.45 1.52 1.40

Γpolymer Apolymer 2.60 2.95 3.38 2.96 2.55 3.22

33.8 35.9 38.5 36.8 35.2 40.4

tot φpolymer ) 10-3

γ* 1.00 1.04 1.08 1.14 1.19 1.12

Γpolymer Apolymer 4.71 4.81 5.16 4.55 4.02 5.00

18.7 22.0 25.0 24.0 22.4 26.0

a γ* ) aγ/kT, where a is the area of one lattice cell; Γ polymer is expressed as the equivalent number of monolayers; and Apolymer ) rpolymer/Γpolymer, where rpolymer is the number of segments in the copolymer.

terfacial calculations have also been performed for other members of the Pluronic polymer family at T ) 320 K. In series I, the number of EO segments has been increased at a fixed number of PO segments [(EO)x(PO)56(EO)x, x ) 16, 25, and 37, corresponding to Pluronic P103, P104, and P105, respectively], and in series II the total number of segments is increased at a fixed EO/PO ratio [(EO)25(PO)40(EO)25, (EO)31(PO)47(EO)31, (EO)37(PO)56(EO)37, corresponding to Pluronic P85, P95 (if it would have been available), and P105, respectively]. tot tot ]) and Γ ) f(log[φpolymer ]) relations The γ* ) f(log[φpolymer are similar, and Table 3 displays the calculated surface tensions and surface excesses at two different total volume fractions at 320 K. In series I, γ* is nearly the same at tot φpolymer ) 10-7 and the magnitude of the slope reduces with increasing number of EO segments (more EO segments give higher γ*), whereas in the other series the curves are essentially parallel and shifted by ca. 0.06 (longer polymers give lower γ*). However, in both series the surface excess is reduced. Thus, an increase in the number of EO segments (i) increases weakly the surface tension and (ii) increases the surface excess, whereas an increase in the chain length at constant composition (iii) reduces the surface tension and (iv) increases the surface excess. The qualitative aspects of findings (iii) and (iv) are those expected from the chain length dependence given in Figure 3. On the other hand, observations (i) and (ii) are less obvious. The increase in the EO mass at constant PO mass makes the polymer less surface active, as deduced from (i). This is not at variance with the increase in Γpolymer, and in fact, the number of adsorbed molecules per unit area (Γ) is reduced. However, this reduction is masked since it is weaker than the increase in the total mass of the triblock copolymers in series I. We also note that the surface activity of PEO-PPO-PEO triblock copolymers at the air/water interface decreases with an increasing number of EO segments, although Figure 3a showed that the opposite effect is prevalent when the EO segments reside in PEO homopolymers. We have also considerd an aqueous solution of (PO)28(EO)74(PO)28, a triblock copolymer with the same overall composition as Pluronic P105, but with PEO as the central block. The PPO-PEO-PPO polymer displays a reduced surface activity as compared to Pluronic P105, γ* is shifted +0.03 and Γpolymer is reduced by ca. 0.2 over a large concentration range (see also Table 3). Thus, it is more unfavorable to create a loop of EO segments in the water subphase (the PPO-PEO-PPO case) than to make a loop of PO segments at the interface (the PEO-PPOPEO case). This is expected since the PO layer is longitudinally more compressed than the EO layer (cf. Figure 6). [Given that one end of a chain is attached to a plane and experiences a potential that depends on the

Figure 7. Calculated reduced surface tension (γ*) as a function tot of the total polymer volume fraction (φpolymer ) for aqueous solutions of polydisperse, Mm/Mn ) 1.2, (EO)37(PO)56(EO)37 triblock copolymer (solid curve) at indicated temperature. The corresponding result if the micellization is disregarded (dotted curve) is also given.

distance from the plane, the free energy cost of confining the other end of the chain to the same plane increases with the extension of the chain governed by the external potential.] PEO-PPO-PEO Triblock Copolymers. Polydispersity. Solutions with polydisperse polymers display a modified self-association behavior and adsorption at solid surfaces due to the exchange of different polymer components as the total polymer concentration is changed.20,26 This behavior is also expected to be the case when copolymers are adsorbed at fluid interfaces. Figure 7 shows the corresponding surface tension for a polydisperse representation of Pluronic P105. As described in the previous section, the polydisperse model involves several polymer components. We have used five components and a polydispersity ratio Mm/Mn ) 1.2. A comparison between Figures 5a and 7 shows that the effect of the polydispersity is the following: (i) the cmc is displaced to a lower volume fraction, (ii) the change of the slope at the cmc becomes less prominent, (iii) far below the cmc γ* is reduced by ca. 0.30, (iv) above the cmc the surface tension is slightly reduced, and (v) after the cmc γ* is still decreasing. At volume fractions below the cmc, the reduction of the γ* is a consequence of the larger surface activity of the longer components in the distribution. Figure 8a shows the surface excess and the contribution from the different polymer components as a function of the total polymer volume fraction at T ) 320 K. It is obvious that at low tot φpolymer component j ) 5 dominates the surface excess. At tot increasing φpolymer the relative contributions from the other components increase but the total adsorbed amount is still dominated by component j ) 5. As the cmc is approached in the polydisperse sample, 2 × 10-4 at 320 K, the longest components are withdrawn from the interfacial region and form micelles. Figure 8b shows that the volume fraction of free polymer with the largest mass (component 5) has a maximum close to the cmc. The reduction of the free volume fraction of component 5 makes its contribution to Γpolymer to diminish tot above φpolymer ≈ 2 × 10-4 (see Figure 8a). At this concentration the surface excess of component 4 starts to increase strongly, and this component is the dominating tot > 10-3. However, as for component 5, the one at φpolymer free volume fraction of component 4 is also reduced at increasing total polymer volume fraction (see Figure 8a), making the surface excess of component 4 to peak at tot φpolymer ≈ 3 × 10-3, and at this stage the surface excess of component 3 starts to rise considerably. The successive replacement of longer components with shorter ones at

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Figure 8. Calculated (a) reduced surface excess amount free (Γpolymer), (b) free polymer volume fraction (φpolymer ), and (c) mic micelle volume fraction (φpolymer) as a function of the total tot polymer volume fraction (φpolymer ) for aqueous solutions of polydisperse, Mm/Mn ) 1.2, (EO)37(PO)56(EO)37 triblock copolymer (solid curve) at T ) 320 K. The contributions from the polymer components are also shown (dashed curves, not all curves are visible in (a) and (c)). In (a) the corresponding results if the micellization is disregarded (dotted curves) are also given.

the interface above the cmc is accompanied by a reduction in Γpolymer (Figure 8a, solid curve). We also notice that the polydispersity leads to a considerable increase in the total surface excess over the whole concentration range studied (cf. Figures 5b and 8a). To complete the picture, Figure 8c shows the volume fraction of polymers in micelles and the contributions from the five components. It is clear that at the cmc, the micelles are almost exclusively formed by the longest components (j ) 5). Although the free volume fraction of component tot above the cmc, the majority 5 decreases with φpolymer component (in volume sense) of the micelles is successively tot increases. changed from component 5 to 4 and 3 as φpolymer At very high polymer volume fractions, the composition of the micelles approaches the overall composition of the system.20 As indicated here and further discussed in the previous investigation,20 the properties of the polymers at the higher end of the mass distribution determine the cmc of the model. The micellar volume fraction at the predicted cmc

Linse and Hatton

Figure 9. Calculated volume fraction profiles at the air/water interface for an aqueous solution of (EO)37(PO)56(EO)37 triblock tot copolymer at (a) φpolymer ) 10-5 (below the cmc), (b) 10-2 (above the cmc), and (c) 10-2 (neglecting the cmc) with T ) 320 K. The total polymer profiles are given by solid curves with the contributions from the polymer components by dashed curves (not all curves are visible) and the water and air profiles by dotted curves.

could be far below the normal detection limits of the experiments, and an additional criterion of the cmc has to be provided. Similarly, the model calculations of polydisperse (EO)x(BO)y at a hydrophobic surface showed that the adsorbed amount was dominated by the rare components with extreme composition.26 The results of the present investigation of the adsorption of PEO-PPOPEO triblock copolymers at an air/water interface clearly follow the same pattern. This is in the present system manifested by the fact that the components at the high end of the mass distribution control the surface tension and dominate the surface excess. In Figure 9, the volume fraction profiles across the interface are given for the polydisperse sample. At tot φpolymer ) 10-5 (Figure 9a) the interface thickness is ca. 20 layers (80 Å), which is twice that for the monodisperse model (cf. Figure 6a) and consistent with the difference in the surface excesses. Figure 9a also shows the large tot ) dominance of component 5. Above the cmc at φpolymer 10-2, Figure 9b shows that the interface thickness remains at ca. 20 layers (80 Å) and that the surface excesses of components 3 and 4 are nearly equal (cf. Figure 8a).

Mean-Field Lattice Calculations of EO and PO tot Finally, Figure 9c illustrates the profiles at φpolymer ) 10-2, if the micellization is neglected. The interface thickness becomes larger, which is consistent with the larger Γpolymer ()12.2). As for the case below the cmc, component 5 makes the largest contribution to Γpolymer. In Figure 9, the profiles of the different components are not proportional to each other, similar to what is found for the corresponding volume fraction profiles of micelles20 and adsorbed layers at hydrophobic surfaces.26 The central point is that the EO-PO junctions of the different components are similarly distributed (the distributions of the EO-PO junctions of the different components have their maxima in the same layer i ) imax). That makes the contribution from short components rather centered at imax whereas the chains of the longer components are more stretched in the neighborhood of the junction zone and the components have most of their segments located further away at the air interface and in the water subphase. This is most clearly visible in Figure 9b, where the contribution from component 2 is centered at imax ) 26, whereas the stretching effect makes the volume fraction profile of component 5 bimodal.

V. Comparison with Experimental Data Volume Fraction Profiles of PEO. Lu et al. have obtained segment density profiles of PEO adsorbed at the air/water interface from neutron reflectivity measurements.17 They measured at three different scattering conditions by employing protonated and fully deuterated PEO in the range of Mn ) 17 800-87 000 and at different solvent contrasts. The molecular mass dependence was found to be weak and in the following we consider only PEO with Mn ) 17 800. The evaluation of the volume fraction profiles was accomplished by assuming model profiles and by assessing how well they fit the reflectivity data. They concluded that a uniform layer model did not represent the experimental data and the simplest model which adequately fitted the data was a three-layer model (shown in Figure 10). The surface excess from the threelayer model was determined to Γ ) 0.53 mg m-2. The authors found that their data also fitted well to a sum of two half-Gaussian functions with equal intensity, which also is reproduced in Figure 10. The fact that there is no unique model profile which fits the experimental data is typical. The inset of Figure 10 shows the PEO volume fraction profile from model calculations of (EO)340 at the corresponding bulk volume fraction and temperature. A comparison between the experimental and calculated volume fraction profiles shows that the model overestimates the volume fraction by a factor of 2, but the functional form of the profile extending into the water subphase is similar to the form of the sum of the two Gaussian functions. The range of the extension into the water subphase is in very good agreement, if 4 Å again is invoked as the lattice length. The model calculations also support a flat EO surface against the air, although it is not infinitely sharp as in the model functions used in the evaluation of the experimental data. The calculated surface excess Γ ) 2.5, corresponding to Γ ) 1.2 mg m-2, is too large, which is coupled to the overestimation of the amplitude of the volume fraction. In the comparison, we have neglected the fact that the model calculations give the intrinsic volume fraction profiles, whereas the experimental ones are broadened by capillary waves (ca. 10 Å, as estimated from the surface tension and the capillary wave theory38). An account for (38) Rowlingson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, U.K., 1982.

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Figure 10. Volume fraction profiles of d-PEO at the air/water interface obtained from a three-layer model (solid lines) and a model involving a sum of two half-Gaussian functions (full curve) fitted to experimental neutron reflectivity data from 0.1 wt % d-PEO solution with Mn ) 17 800 and at T ) 298 K (main figure) (data taken from Lu et al.,17) and of (EO)340 obtained from mean-field lattice calculations at φpolymer ) 10-3 and T ) 298 (inset).

this fact does not, however, change the conclusions to any large extent, since the intrinsic width is substantially larger than the contribution from capillary waves. Regarding the overestimation of the surface excess, this could be related to the mean-field approximation of the model. At low bulk volume fraction, the number of EO-water contacts is overestimated, making the chemical potential too high at a given concentration and hence contributing to too large a surface excess. Previous comparisons between results from Monte Carlo simulations and meanfield lattice theories involving self-assembly indicate such effects.39 Thus, the predicted density profile of PEO at the air/ water interface from the mean-field lattice theory (with χEO,water interaction parameters fitted to experimental phase diagrams and χair,water and χEO,water interaction parameters fitted to surface tension data) does compare reasonably well to those deduced from neutron reflectivity measurements, and the calculated profiles support the notion of a flat PEO/air interface. Finally, in a previous attempt, Lu et al. did not succeed in getting a reasonable agreement between the experimentally obtained density profiles and those obtained by applying the same meanfield theory but with no internal degrees of freedom and with a different set of interaction parameters.17 Surface Tension and Area of Pluronic P105. The surface tension of aqueous solutions of several Pluronic triblock copolymers at different temperatures and at different polymer concentrations was recently measured by Wanka et al.8 and by Alexandridis et al.9 An intriguing feature found in many of the experimental surface tension measurements is the existence of two breaks (Figure 6 in ref 8 and Figure 2 in ref 9). In Figure 11 we reproduce data for Pluronic P105 from Alexandridis et al., and at the lower temperature the two breaks and three linear, or nearly linear, regions are visible. The dependence at low and high concentrations will be referred to as regimes I and III, and if a central regime is present, it will be denoted by II. By applying the Gibbs adsorption isotherm to data in regime I, Alexandridis et al. calculated the area per molecule at the interface. In particular, for Pluronic P105 at 25 °C this procedure gave 99 Å2.9 A similar calculation using regime II gave a 3-fold increase in the molecular area, viz. 300 Å2. The present mean-field lattice calculations give a molecular area of 38.5 and 25 at low and high volume fractions, respectively, and the use of a lattice (39) Wijmans, C.; Linse, P. Langmuir 1995, 11, 3748.

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Figure 11. Experimentally determined surface tension (γ) for an aqueous solution of Pluronic P105 as a function of the polymer concentration (Cpolymer) at indicated temperatures. The arrows denote the location of the cmc’s obtained from dye solubilization. The solid lines indicate regions with linear dependence and they are further explained in the text. (Adapted from Alexandridis et al.9). Table 4. Area per Pluronic P105 Triblock Copolymer air/water interface experimental lattice model calculations

99 Å2 a 300 Å2 b 38.5d (620 Å2)g 25e (400 Å2)g

lamellar liquid crystal 240 Å2 c 23.6f (380 Å2)g

a From surface tension measurements and application of the Gibbs adsorption isotherm using regime I (cf. Figure 11) at 25 °C. b From surface tension measurements and application of the Gibbs adsorption isotherm using regime II (cf. Figure 11) at 25 °C. c From small-angle X-ray scattering of a P105/water system with 75% tot polymer at 25 °C.40 d From model calculations at φpolymer ) 10-7 tot e and 320 K (this work). From model calculations at φpolymer ) 10-3 and 320 K (this work). f From model calculations of a P105/water system with 50% polymer at 320 K (unpublished data obtained from the model and procedure described in ref 23). g Value in parentheses is obtained by using a lattice cell length of 4 Å.

size of 4 Å gives a predicted area per copolymer in the range 400-620 Å2. Related to the present discussion are also the areas of Pluronic P105 in lamellar lyotropic liquid crystals. Recent small-angle X-ray scattering measurements40 gave 240 Å2, and an area of 380 Å2 was obtained from mean-field lattice modeling. All areas are compiled in Table 4, which also provides further details. Alexandridis et al. suggested that the low-concentration break “is believed to originate from rearrangements of the copolymer molecules on the surface at complete coverage of the air/water interface” after rejecting other alternatives, whereas the high-concentration break “occurred at concentrations that correspond to the cmc”. The latter proposal was supported by dye solubilization techniques (cf. arrows in Figure 11) and is fully consistent with the results of the present investigation. However, from the stability criteria ∂c/∂µ > 0 in solution and ∂Γ/∂µ > 0 at the interface, where µ is the chemical potential, we obtain ∂Γ/∂c > 0 and hence ∂A/∂c < 0 (A is here the area per molecule). Thus, the interpretation of the lowconcentration break as a molecular rearrangement on the surface is not consistent with the stability criteria used. As described below, this interpretation of the lowconcentration break also implies some other unrealistic relations. In order to resolve those, we here suggest that in regime I a depletion of the copolymer in the solution affects the surface excess, and hence the central region (regime II) should be used to evaluate the surface areas in such experiments. In the following we will give support of our suggestion. (40) Alexandridis, P.; Zhou, D.; Khan, A. Langmuir 1996, 11, 2690.

(I) The use of the experimental surface tension data in regime II gives an area per copolymer (300 Å2) that relates to the experimentally measured area in the lamellar phase (240 Å2) in the same way as the areas in the model calculations do (Aair/interface g Alamellae, which is independent of the assumption of the lattice cell length). The use of regime I leads, however, to an area per copolymer at the air/water interface that is a factor 2.5 smaller (hence the molecules are predicted to be more stretched) than the corresponding area in the lamellar phase. Since the molecular arrangement is similar in these two systems, the observed difference is unexpectedly large. The experimental data of the lamellar phase from X-ray scattering were evaluated without any essential model assumptions and should be regarded as trustworthy. (II) At finite volume-to-surface-ratio and at sufficiently low concentration, depletion of the copolymer in the solution has to affect the surface tension. The surface tension measurements by Alexandridis et al. were accomplished by using a solution height of 1-2 cm,41 and the lowest concentration was 10-7 kg dm-1. Figure 5a illustrated the results of the model calculations at a finite V/A ratio, and if a lattice cell length of 4 Å is used again, the conditions in Figure 5a would correspond to a solution height of 4 mm. However, the predicted volume fraction at the break occurs at a concentration considerably lower than that observed. But it is conceivable that adsorption also takes place on the surfaces of the glassware in the experimental case, which shifts the effects of depletion on the surface tension to occur at higher concentrations and at larger V/A ratios (solution heights). Thus, the model calculations support the notion that the experimental surface tension at low concentrations, and hence the areas extracted, are likely to be affected by a depletion of the copolymer in the solution. (III) Alexandridis et al. also observed that the concentration at the break between regimes I and II was insensitive to the temperature (25 and 35 °C) and the copolymer selected (six different PPO-PEO-PPO triblock copolymers polymers, a PPO-PEO-PPO triblock copolymer, and a random EO-PO copolymer were considered).9 Such an insensitivity is more likely due to some external factors (such as the V/A ratio) than to some intrinsic properties of the system (as the copolymer composition). We have also examined other possible explanations of the occurrence of two breaks as (i) the validity of the Gibbs adsorption isotherm and (ii) the effects of mass polydispersity. Our conclusions from the model calculations are, however, the Gibbs adsorption is perfectly valid for the monodisperse copolymer sample in the concentration region of interest (Figure 5b). The effects of a mass polydispersity are that the break in the surface tension at the cmc becomes less sharp (Figure 7), but it does not strongly violate the area per molecule extracted from the surface tension (data not shown). Similar evaluation of the surface area was accomplished by Wanka et al. for Pluronic P104.8 At low temperatures, where three regimes were present, they used regime II to evaluate the areas, whereas at higher temperatures, where only two regimes appeared, they used regime I. (This procedure was not explicitly given, but inferred by us using data presented in Table 2 and Figure 6 in ref 8.) At 25 °C, where two breaks were present, they obtained an area per molecule of 312 Å2, whereas at 40 °C, where only one break occurred, they found only 62 Å2. According to our suggestion, this mixed use of different regimes gives a too large temperature dependence on the area per molecule. However, 62 Å2 is close to 67 Å2 as obtained by Alexandridis (41) Alexandridis, P. Personal communication.

Mean-Field Lattice Calculations of EO and PO

Langmuir, Vol. 13, No. 15, 1997 4077

Table 5. Reduced Surface Tension, γ/γ0,a for PEO-PPO-PEO Triblock Copolymers at the Air/Water Interface triblock copolymer P103 P104 P105 P85

expetlb (EO)16(PO)56(EO)16 (EO)25(PO)56(EO)25 (EO)37(PO)56(EO)37 (EO)25(PO)40(EO)25

0.47 0.49 0.50 0.55

γ/γ0 model calcsc 0.43 0.45 0.47 0.51

a Ratio of the surface tension with (γ) and without (γ ) copolymer. 0 γ0 ) 72 mN m-1 and γ0* ) 2.32 were used. b At C ) 10-3 kg dm-3 tot and 25 °C (from Table 2 of ref 9). c At φpolymer ) 10-3 and 320 K (from Table 2).

et al. for the same conditions (Table 2 in ref 9), also using regime I. Thus, the two investigations are consistent with each other and our conclusion also affects some of the areas given by Wanka et al.8 Finally, we notice that regime II is not always linear. Often, a weak, and sometimes quite strong, convex curvature appears, see e.g., Figure 1 in ref 7, Figure 6 in ref 8, and Figure 2 in ref 9. Such convex curvature also appeared in the model calculations (Figure 5a). In the model system, the curvature is due to an increased adsorption with increasing polymer concentration and this is most likely the case in the experimental cases as well. To summarize, the present study suggests that the existence of two breaks of the surface tension of PEOPPO-PEO triblock copolymers at the air/water interface experimentally observed is due to a depletion of copolymer in the solution caused by small concentrations in connection with small solution heights. This conclusion was made by using results from mean-field lattice model calculations which showed similar low-concentration breaks at finite volume-to-surface ratios. The reevaluation resolves the unlikely small areas obtained and gives surface areas of PEO-PPO-PEO triblock copolymers at air/water interfaces, in good agreement with those found in lyotropic liquid crystalline lamellar phases. Since, our model calculations resort to a mean-field model, we cannot, however, exclude the argument that molecular rearrangement occurs as originally suggested. But from thermodynamic arguments, such rearrangements should give rise to a convex, not a concave, break. As a consequence of our conclusion, precaution is needed when only one break appears in the experimental surface tension plots. The situation could be that (i) the concentration range covers regimes I and II, (ii) regimes I and III have merged due to a small cmc and/or a small solution height, or (iii) the concentration range covers only regimes II and III. In (i) a correct cmc cannot be evaluated whereas an area could be extracted using the high-concentration regime, in (ii) a correct area cannot be obtained whereas the cmc should be trustworthy, and finally, in (iii) the area and the cmc are both available in the normal way. Surface Tensions for Different Triblock Copolymer Compositions. After having identified the region (if present) between the break at the cmc and the one at lower concentration (at which the depletion affects the surface tension) as the interesting low-concentration regime, we shall now compare experimental and calculated surface tensions for different PEO-PPO-PEO copolymers at concentrations below the cmc. For that reason, we will use the surface tensions reported at C ) 10-3 kg dm-3 and 25 °C.9 The surface tension data are compiled in Table 5, and for facilitating a comparison between experimental and model data, they are divided by the surface tension of a bare air/water interface, γ0. Initially disregarding the variation among the PEOPPO-PEO copolymers, a comparison between experi-

mental and modeling data shows that the magnitude of the reduction of the surface tension, ca. 50%, agrees very well. Since the interaction parameters involving the air component were fitted to experimental data of PEO and PPO at an air/water interface with good agreement, it is not unexpected that similar good agreement for PEOPPO-PEO copolymers is present. On the contrary, such agreement is required if the assumption that the central region is the relevant low-concentration region. A closer inspection of the data in Table 5 shows that the increase of the surface tension with increasing EO mass at constant PO mass (P103-P104-P105) and the increase of the surface tension with decreasing total mass at constant EO/PO composition (P105-P85) are excellently reproduced by the model calculations. This confirms the usefulness of the model for describing surface tension of block copolymers containing EO and PO, and hence also supports our use of the model for the reinterpretation of the experimental data accomplished above. VI. Summary The adsorption of homopolymers and block copolymers containing EO and PO at the air/water interface was modeled on the basis of a lattice theory for polymer solutions in heterogeneous systems. The inverse temperature behavior occurring in the phase diagrams was taken into account by allowing the polymer segments to adopt different states depending upon the temperature and concentration in the solution. All interaction parameters employed for the copolymer systems were derived independently from previous fitting using simpler systems, and in the case of PEO, the model was tested on experimental results not used in the parameter fitting. The simplicity of the lattice model and the rather coarse approximations involved, suggest that such a model should only be used for qualitatively describing aspects of polymer solutions. However, the present and previous investigations on EO- and PO-containing block copolymers in solution have shown that semiquantitative results can also be obtained,18-26 and in the present contribution some of the conclusions rely on a comparison between experimental and modeling results. In this context, a quantitative discrepancy between experimental and modeling data should not refrain one from using the model for qualitative studies to see the effects of changing conditions in the system. The major conclusions made from this study are (1) An extensive set of surface tension data for PEO and PPO homopolymers were well represented by the model after only adjusting one parameter, respectively. (2) Calculated volume fraction profiles for PEO at the air/water interface compare favorably with recent profiles deduced from neutron reflectively measurements. The penetration into the water subphase and the flat interface toward the air are well described, whereas the predicted magnitude of the surface volume fraction profile and the surface excess are overestimated by a factor of 2. (3) Surface tension data for PEO-PPO-PEO triblock copolymers display a temperature dependent highconcentration break due to self-assembly. (4) The model calculations demonstrate that at low volume-to-surface rations, a low-concentration break also appears. After comparison with previous experimental data, such observed low-concentration breaks are suggested to be a consequence of a depletion of polymer in the solution and an explanation involving rearrangements of the copolymers need not be invoked, but could not be excluded. (5) The model calculations suggest a clear separation between EO and PO at the air/water interphase. The PO

4078 Langmuir, Vol. 13, No. 15, 1997

segments are located in a layer ca. 20 Å thick with little water penetration and exposes a flat surface toward the air, whereas the EO layer is ca. 40-50 Å wide and extends into the water subphase. (6) The predictions of the concentration and temperature dependence of the surface tension on the PEO-PPOPEO triblock copolymer composition and size reproduced the experimental data excellently. (7) A polydisperse block copolymer sample leads to a less clear break in the surface tension data, a known effect occurring in polydisperse systems. (8) Below the cmc, the high-end mass fraction of a polydisperse sample governs the surface tension and the excess amount in a similar way as in previous model

Linse and Hatton

studies of the adsorption at hydrophobic surfaces and on the self-assembly in solution. (9) Consistency of the directly calculated surface excess and that derived from the Gibbs adsorption isotherm was obtained. Acknowledgment. P. Alexandridis is gratefully acknowledged for numerous discussions throughout this work and valuable comments on the manuscript. This work was supported by the Swedish Research Council for Engineering Science (TFR) and the Swedish National Science Research Council (NFR). LA9620871