Langmuir 1999, 15, 7901-7911
7901
Articles Self-Assembly of Model Nonionic Amphiphilic Molecules Claudia B. E. Guerin and Igal Szleifer* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907-1393 Received June 30, 1998. In Final Form: July 12, 1999 The behavior of model nonionic amphiphilic molecules undergoing aggregation is studied using singlechain mean-field theory. The amphiphilic molecules are of the type HxTy where the H (head) monomers like the solvent molecules and the T (tail) monomers are solvophobic. In combination with the mass action model, the theory was used to study the critical micellar concentration, cmc, and the micellar size distribution as a function of head and tail lengths, architecture of the molecule, and temperature. The predictions of the theory are compared with the molecular dynamic results of Smit et al. (Langmuir 1993, 9, 9). Very good agreement is found between the theoretical predictions and the simulations. The theory is shown to predict quantitatively the two different free energy scales responsible for micellization and for micellar size distributions in model systems. The cmc is found to be only slightly dependent on the headgroup molecular architecture. However, the micellar size distribution is found to be quite different when comparing linear and branched headgroups. The structure of the micelle is found, in agreement with earlier theoretical predictions, to have a compact, almost solvent free, hydrophobic core and a wide interface region that includes the headgroups. The hydrophobic region is found to be more compact and larger for longer hydrophobic tails. The hydrophobic region is found to be more compact as the temperature decreases. The molecular organization in the micelles is not very sensitive to changes in the architecture of the headgroups.
I. Introduction Amphiphiles are molecules composed by two moieties: a hydrophobic tail and a hydrophilic headgroup. Its main characteristic is that when they are in solution, under specific thermodynamic conditions, they self-assemble into well-defined microstructures in order to minimize tailsolvent interactions. Self-assembled structures play important roles in many fields, like biological membranes, detergency, and drug delivery, so they have been extensively studied for many years.1-6 In general, experimental measurements of mostly every physical quantity as a function of the amount of surfactant in the solution show sharp changes over a narrow range of surfactant concentration. This is the fingerprint of the spontaneous aggregation process and the concentration at which it occurs is called critical micellar concentration, cmc. Normally, at concentrations close to the cmc, small spherical micelles are formed; however increments in the surfactant concentration can cause changes in the micellar morphology. Spherical micelles are typically composed of two partially overlapping regions: an inner hydrophobic core (rich in the oil moiety) and an outer hydrophilic corona (mostly formed by the hydrated headgroups). (1) Tanford, C. The hydrophobic effect: formation of micelles and biological membranes; Wiley: New York, 1973. (2) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1991. (3) Ben-Shaul, A.; Gelbart, W. M. In Statistical Thermodynamics of Amphiphile Self-Assembly: Structure and Phase Transitions in Micellar Solutions; Ben-Shaul, A., Gelbart, W. M., Roux, D., Eds.; SpringerVerlag: New York, 1994. (4) Israelachvili, J. N. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, 1985. (5) Micellization, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. 1,2. (6) Rusanov, A. I. Advances in Colloid and Interface Science; Elsevier Science Publishers B. V.: Amsterdam, 1993; Vol. 45.
The cmc depends upon the difference between the free energies of unimers and self-assembled amphiphiles, which is of the order of kT per hydrophobic segment. The size distribution is determined by the difference between free energies of amphiphiles in aggregates of dissimilar number which is of the order of a fraction of kT per molecule. The difference in free energy scales associated with micellization and size distributions has been recognized long ago in a variety of experimental and theoretical studies.2,7-9 However, the disparity in the values of these energies makes it difficult to reflect both of them within the same molecular theoretical formalism. A theoretical challenge is to develop a statistical thermodynamic approach that can account for both free energy scales based only on microscopic information, i.e., the model interaction between molecules. Several theoretical approaches have been developed to address self-aggregation. The most detailed molecular study can be done by performing computer simulations.10 Observation of spontaneous aggregation in systems with low surfactant concentration has been reported by several authors using Monte Carlo (MC) and Molecular Dynamics (MD) simulations.11-15 However, the amount of compu(7) Wennerstrom, H.; Lindman, B. Phys. Rep. 1980, 52, 1. (8) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y. J. Phys. Chem. 1980, 84, 1044. (9) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbrich, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80. (10) Larson, R. G. Curr. Opin. Colloid Interface Sci. 1997, 2, 361. (11) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022. (12) Wijmans, C. M.; Linse, P. Langmuir 1995, 11, 3748. (13) Smit, B.; Esselink, K.; Hilbers, P. A. J.; van Os, N. M.; Rupert, L. A. M.; Szleifer, I. Langmuir 1993, 9, 9. (14) Smit, B.; Hilbers, P. A. J.; Esselink, K.; Rupert, L. A. M.; van Os, N. M. Nature 1990, 348, 624. (15) von Gottberg, F. K.; Smith, K. A.; Hatton, A. J. Chem. Phys. 1997, 106, 9850.
10.1021/la980788n CCC: $18.00 © 1999 American Chemical Society Published on Web 09/01/1999
7902
Langmuir, Vol. 15, No. 23, 1999
tational time required by these techniques makes it difficult to study systematic, at the present time, how micellar formation is affected by changes on the solution conditions and type of surfactant molecules. Methods based on thermodynamic theory1,16,17 or in geometric considerations1,18 use phenomenological parameters from experimental data to quantitatively predict cmc and micellar distributions. However, the molecular organization inside the micelle cannot be calculated within these approaches. Statistical thermodynamic theories have been applied to investigate the contribution of the different interactions to the free energy of micellization. Theories based on chain packing19,20 assume a dry core and a sharp core-corona interface and do not include the contribution of the hydrophilic headgroups. Blanckshtein and co-workers use these packing theories within a thermodynamic framework to predict micellar phase behavior.21,22 This approach has successfully predicted several thermodynamic properties of self-assembled systems, including cmc and phase behavior. Their model includes experimental parameters and also several approximations regarding the structure of the headgroups whose accuracy has not been tested yet. In an effort to avoid restrictions on the molecular conformations for tails and heads and the use of empirical parameters, two theories have been developed: the lattice self-consistent field (SCF) theory originally developed by Scheutjens and Fleer23,24 to treat polymer adsorption and generalized by Leermakers25,26 to treat surfactant aggregates. The single-chain mean-field (SCMF) theory, originally developed for dry core micelles20 and then generalized for polymer solvent systems,27 was used by Mackie et al.11 to treat micellar systems in which the position of the modeled molecules are not restricted and self-intersecting conformations are forbidden. The theoretical approach of the Wageningen group has made important contributions to the understanding of key features in micellization, such as the diffuse structure of the core-corona interface, the dependency of the cmc with tail length, and the changes of aggregate morphology.28,29 In particular, the micellization of triblock copolymers with two poly(ethylene oxide) headgroups and a poly(propylene oxide) hydrophobic tail (PEO-PPOPEO), has been correctly predicted.30-32 For short surfactants, SCF has been applied to study the self(16) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (17) Zana, R. Langmuir 1996, 12, 1208. (18) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1976, 2, 1525. (19) Gruen, A. W. R. J. Phys. Chem. 1985, 89, 146. (20) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985, 83, 3597. (21) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (22) Zoeller, N.; Blankschtein, D. Langmuir 1997, 13, 5258. (23) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (24) Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1980, 84, 178. (25) Leermakers, F. Statistical thermodynamics of association colloids. Ph.D. Thesis. Agricultural University, The Netherlands, 1988. (26) Leermakers, F. A. M.; van der Schoot, P. P. A. M.; Scheutjens, J. M. H. M.; Lyklema, J. In Surfactants in Solution; Mittal, K. L., Ed.; Plenum: New York, 1989; Vol. 7. (27) Szleifer, I.; Carignano, M. A. In Advances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; John Wiley and Sons: New York, 1996; Vol. 94, p 165. (28) Leermakers, F. A. M.; Lyklema, J. Colloids Surf. 1992, 67, 239. (29) Leermakers, F. A. M.; Wijmans, C. M.; Fleer, G. J. Macromolecules 1995, 28, 3434. (30) Hurter, P. N.; Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5592. (31) Hurter, P. N.; Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5030. (32) Linse, P. Macromolecules 1993, 26, 4437.
Guerin and Szleifer
aggregation of ionic33 and nonionic surfactants.12 Wijmans and Linse12 used the SCF method of Leemarkers25 to study the aggregation of short nonionic linear surfactant chains in spherical micelles and compared their findings with their own MC simulations on the same system. The theoretical work reproduces qualitatively the molecular organization in the micelles found in the simulations. However, it fails in predicting the right micellar size distribution and cmc, giving larger aggregation numbers and cmc’s much lower, by several orders of magnitude than the ones calculated with the simulations. The lack of agreement may be due to the use of different molecular models in the simulations and in the theory as discussed in refs 11 and 12. Mackie et al. applied the SCMF theory without the assumption of a sharp interface and a dry core for lattice amphiphiles.11 The predictions of the theory were compared with MC simulations and they found excellent quantitative agreement for the cmc. Furthermore, the predictions for the molecular organization in the aggregates were in very good agreement with the simulations. However, the micellar size distributions were not in good agreement with the MC predictions. The MC simulations showed much larger aggregates than those predicted from the theory. Close examination of the MC results showed that the aggregates at the optimal size were cylindrical while the theory was applied to spherical aggregates only. The authors attributed the discrepancy between the two methods to the different shapes of the aggregates. In this paper, we apply the SCMF theory to study the aggregation of simple model nonionic surfactants in spherical aggregates. The specific model that we chose was motivated by MD simulations13 that show spherical micellar formation in off-lattice model chains. Thus, comparisons of the predictions of the theory with the MD results, together with the earlier results of Mackie et al., will enable us to show that the theory is able to predict the two different free energy scales associated with micellization in a quantitative way for lattice and offlattice molecular models. Further, we study the effect of temperature, chain length, and headgroup architecture on the cmc and micellar size distribution. For the headgroup chemical architecture we are interested in looking at whether the approximation usually used in modeling nonionic heads, such as in CnEm surfactants, of replacing the flexible headgroup by a rigid, fixed size sphere confined to a surface21,34 is valid. The paper is organized as follows: section II contains a description of the thermodynamic framework and the molecular model used together with a brief derivation of the SCMF theory. Our findings are described in section III, and concluding remarks are in section IV. II. Molecular Model and Theoretical Approach A. Mass Action Model. The micellar size distribution and the cmc of a solution of surfactants can be predicted by applying the multiple equilibrium model.2,6 Thermodynamic equilibrium requires that the chemical potential of all identical molecules must be the same
µ1 ) µ2 ) µ3 ) ... ) µN ) constant
(1)
where µi is the chemical potential of the molecules within micelles of aggregation number i. For solutions of low (33) Bo¨hmer, M. R.; Koopal, L. K.; Lyklema, J. J. Phys. Chem. 1991, 95, 9569. (34) Schiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618.
Self-Assembly of Model Nonionic Amphiphilic Molecules
surfactant concentration, the aggregate-aggregate interactions can be neglected, and thus
µN ) µN0 +
kT XN ln N N
(2)
XN is mole fraction of the molecules in aggregates of number N, µN0 is the standard chemical potential of the amphiphiles within an aggregate of N molecules, k is the Boltzmann constant, and T is the temperature. µN0 represents the free energy per molecule of an aggregate of size N that is fixed in space (free energy of micellization). The last two equations are combined to give the distribution:
XN ) N{X1 exp[(µ10 - µN0)/kT]}N
(3)
This equation along with the conservation of the total ∞ XN completely defines the solute concentration C ) ∑N)1 micellar size distribution, and from that, the cmc can be obtained. From eq 1 it can be seen that in the ideal solution limit, the only necessary thermodynamic information to describe the micellar size distribution and the cmc is µN0 as a function of N. It is important to emphasize that the ideal solution limit is taken only for the interaggregate interaction. However, to determine µN0 the inter- and intramolecular interactions within the aggregate of size N have to be taken into account. The cmc is determined approximately by the value of µ10 - µM0 where M corresponds to the most probable aggregation number. This difference is found experimentally to be of the order of kT per hydrophobic monomer, i.e., typically 10-20kT per amphiphilic molecule.2 However, the size (and shape) distribution depends upon µN0 - µM0, where N and M are different aggregation numbers, and this difference is of the order of a fraction of kT per molecule.3 The molecular theory presented below is aimed at determining these two very different free energy scales within a single framework. B. Single-Chain Mean-Field Theory. The theoretical approach that we use in this work was originally developed to study the molecular organization of amphiphilic tails in dry core aggregates20 and later generalized to include solvent molecules.27 The implementation of the theory for surfactant self-assembly without the dry core restriction was recently presented for lattice chains by Mackie et al.11 Here we present a brief derivation of the theory for the specific application of self-assembly of model systems in free space. Detailed descriptions of the implications of the approximations used in the derivation of the theory can be found elsewhere.20,27 The main idea behind the SCMF theory is to consider a single chain with all its intramolecular interactions exactly taking into account (within the chosen model to treat the molecules) while the intermolecular interactions are considered within a mean-field approximation. The quantities that are determined by the theory are the probability distribution function (pdf) of chain conformations, P(R), and the distribution of solvent molecules within the aggregate. R (chain conformation) denotes the position of each of the segments of the surfactant, which are given by the bond sequence of the chain, the position of the center of mass, and the orientation of the molecule. Namely, we do not restrict the position of the surfactant molecules within the aggregate but we get the molecular organization as an output of the theory. The pdf of chain conformations and the distribution of solvent molecules are obtained by minimizing the aggregate’s free energy.
Langmuir, Vol. 15, No. 23, 1999 7903
From the knowledge of the pdf and the distribution of solvent any desired average structural quantity and the thermodynamic properties of the aggregate can be obtained, e.g., the distribution of hydrophobic and hydrophilic segments within the aggregate and the standard chemical potential. The theory is applied to study the thermodynamic and structural properties of a single micellar aggregate. The information obtained is then used in the mass action model described above to determine the properties of the solution. The micellar aggregate under study with SCMF theory is therefore isolated from the solution; i.e., we consider the limit of ideal solution of aggregates where there is no interaction among the aggregates or between the aggregates and the monomers free in solution. At this point it is important to remark what the meaning of an isolated aggregate is and how we obtain that in practice. According to the mass action model the chemical potential of the aggregates correspond to those that are isolated in solution. Namely, they are dilute enough that they do not feel any interaction with the other aggregates or unimers that are in solution. In practice, however, any theoretical approach that does not assume a well-defined size and shape of the aggregate will predict the behavior of an aggregate that is in “equilibrium” with some unassociated monomers.25,26 This is indeed the case of our theoretical approach as well. However, what we find is that under the conditions that we are carrying out our calculations the volume fraction of surfactants in contact with the micelles is between 10-9 and 10-12 while the lowest cmc values are of the order of 10-4. Therefore, we can safely assume that the aggregates are isolated. Furthermore, we have calculated the properties of the micelles under conditions in which after a certain cutoff distance, typically of the order of 5 to 10 diameters of a segment, from the tail of the headgroup distribution in the solvent-rich side (see Figure 9 below), we arbitrarily do not allow any more surfactants; i.e., the surfactant concentration is zero. We have obtain identical results even for the largest cutoff distances. The reason being that a volume fraction as low as 10-9 is almost equivalent to zero for all practical purposes. The remainder of this section thus describes the way in which we obtain the properties of a single aggregate for which we have defined the geometry and the total number of molecules. However, how the molecules are organized within the aggregate, i.e., what is the probability of different conformations and the position of the head and tail segments, is obtained as an output from the theory. The chemical potential of the amphiphiles in this aggregate that is fixed in space and does not interact with any other component in the solution is exactly the µN0 required as input in the mass action model, see eq 3. We model the amphiphiles as freely jointed segments (beads) each of volume v0 (see Figure 1). We refer to these molecules as HxTy, where x and y represent the number of segments in the headgroup (H) and tail (T), respectively. Solvent molecules are considered to be single beads of volume v0. The amphiphilic nature of the molecule is defined by the different types of interactions among H, T, and solvent molecules. It is natural in the description of the theory to separate the repulsive and attractive interactions. Thus, the simplest model that we can use is one in which the repulsions are model by hard-core excluded volume interactions and the attractions by square well potentials. In order to keep the minimal amount of parameters possible, and to have our model as close as possible with that used in the MD simulations of Smit et al.,13 we assume that there are only attractive interactions
7904
Langmuir, Vol. 15, No. 23, 1999
Guerin and Szleifer
among T segments. All other interactions being purely repulsive makes the solvent to be a good environment for the H groups. Since there is only one interaction parameter, the T-T well depth , we use it to define the temperature scale; i.e., the reduced temperature used throughout is defined as T* ) kBT/. The range of attraction is rcut ) 1.50532σ, where σ is the hard sphere diameter. The well dimensions are fixed to get the same integral energetic interaction as the one that can be obtained using the Lennard-Jones potentials of Smit et al.13 The intramolecular energy is exactly accounted for in the generation of the set of single chain configurations. The repulsive interaction is ∞ or 0 depending whether there is overlap between any two segments of the chain or not, respectively. If there is overlap, the probability of that configuration is zero and therefore only self-avoiding configurations are included in the set. The intramolecular attractions of a conformation R are denoted by u(R). More specifically u(R) is the number of nonbonded T-T pairs that are at distance r e rcut multiplied by . Thus, the average intramolecular energy is
∑R P(R) u(R)
〈Eintra〉 ) N
(4)
where 〈 〉 denote ensemble averages over the pdf P(R) and N is the total number of amphiphiles in the aggregate. The mean-field intermolecular interactions have repulsive and attractive components. The repulsive (hardcore) interactions among all the different components are accounted for by volume-filling constraints. Namely, no two segments (on average) can occupy the same space
〈φ(r)〉 + φS(r) ) 1
for all r
(5)
where r is the radial distance from the center of the spherical micelle. The volume constraints must be fulfilled for all r which is the only assumed inhomogeneous direction in a spherical geometry. φS(r) is the solvent volume fraction at r and 〈φ(r)〉 is the surfactant volume fraction. 〈φ(r)〉 includes two contributions, one from the hydrophobic tails and another from the hydrophilic heads, i.e., 〈φ(r)〉 ) 〈φT(r)〉 + 〈φH(r)〉, 〈φH(T)(r)〉 ) N(〈nH(T)(r)〉/2πr2)v0, and 〈nH(T)(r)〉 dr is the average number of head (tail) segments, at distance r from the center of the aggregate. The intermolecular attractive contribution is calculated by assuming that the probability of finding a tail segment from another chain at r e rcut from the central chain segment is proportional to the average local volume fraction of tails, 〈φT(r)〉, i.e., a local mean-field approximation.Thus, the intermolecular attractive contribution is written as
〈Fn(r)〉 N 〈φT(r)〉 dr 〈Einter〉 ) 2 v0
∫
(6)
where 〈Fn(r)〉 dr is the average volume at r that the central chain has available for intermolecular interactions. The average tail volume fraction 〈φT(r)〉 is the probability of finding a tail segment of other molecules at distance r from the center. The factor 1/2 corrects the pairs doublecounting. The entropy within the aggregate contains two contributions, configurational entropy of amphiphiles and translational entropy of solvent molecules. Namely
-
S kB
)N
∑P(R) ln P(R) + ∫nS(r) ln φS(r) dr {R}
(7)
with nS(r) dr being the number of solvent molecules at r. Note, that since R denotes bond sequence, orientation, and position of the surfactant, the configurational entropy of the chains include: conformational, rotational, and translational chain contributions within the aggregate. The thermodynamic potential for an aggregate of size N in contact with a bath of solvent molecules is the sameas the relevant thermodynamic potential that corresponds to the surface tension in an interfacial system.35 For our system this corresponds to the difference between the Helmholtz free energy and the solvent chemical potential multiplied by the number of solvent molecules, i.e.
βF ) β[〈Eintra〉 + 〈Einter〉] -
S -β kB
∫ nS(r)µS dr
(8)
where the first two terms are the energetic and entropic contributions, whose sum is the Helmholtz free energy, and µS is the (constant) chemical potential of the solvent molecules. (We take this opportunity to comment that eq 4 of ref 27 has a misprint since the last term there should have a minus sign, as in eq 8 of this work. Further, the last term of eq 14 in ref 27 should not be there. It should be noted, however, that all the calculations presented there are correct, and these errors are typographic mistakes.) As is in general the case in writing thermodynamic equations for a system, we could have started from another free energy, which will be a Legendre transform of F.36 For example, we could derive exactly the same set of equations presented below by starting from the Helmholtz free energy of the system subject to the constraint of constant chemical potential of the solvent. However, the advantage of using the free energy given by eq 8 is that it represents the appropriate thermodynamic potential of our system, and thus all the thermodynamic quantities can be obtained from derivatives of F.35 P(R) and φs(r) are determined by minimization of the free energy (8) subject to the packing constraints (5).The minimization is carried out by the introduction of Lagrange multipliers π(r) to yield for the pdf
P(R) )
[
∫
1 exp -βu(R) - π(r)n(R,r) dr q β [F (R,r)〈φT(r)〉 + 〈Fn(r)〉 φT(R,r)] dr (9) 2v0 n
∫
]
where
q)
[
∑R exp -βu(R) - ∫π(r)n(R,r) dr β
∫2v [Fn(R,r)〈φT(r)〉 + 〈Fn(r)〉φT(R,r)] dr 0
]
(10)
the single-chain partition function, is the normalization constant that ensures ∑RP(R) ) 1. The solvent density profile is given by
φS(r) ) exp{-[π(r) - βµS] - 1}
(11)
The physical meaning of the Lagrange multipliers can be seen from eq 11. The π(r) values are related to the osmotic pressures necessary to keep the solvent chemical potential (35) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (36) Chandler D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987.
Self-Assembly of Model Nonionic Amphiphilic Molecules
constant at all r. Namely, they are the pressure arising from the repulsive interactions among the molecules. Using the explicit expressions for the intramolecular, intermolecular attractive interactions and the entropy,eqs 4, 6, and 7, respectively, with the derived expressions for the pdf and the solvent density profile, eqs 9 and 11, respectively, in the free energy expression, eq 8, we obtain the final expression for the aggregate’s free energy
βF ) -
∫
β N 〈F (r)〉〈φT(r)〉 dr 2 v0 n
2
∫π(r) 2πr v0
dr -
N ln q - NS (12)
The last term is linear in the number of molecules and therefore has no thermodynamic consequences; therefore we drop it having for the free energy
βF ) -
∫
β N 〈F (r)〉〈φT(r)〉 dr 2 v0 n
2
∫π(r) 2πr v0
dr -
N ln q (13) The standard surfactant chemical potential, µN0, needed in the determination of the micellar size distribution and the cmc is obtained by taking the derivative of the free energy with respect to the number of amphiphile molecules
βµN0
∂βF ) ∂N
( )
V,T,NS
) - ln q - n
Langmuir, Vol. 15, No. 23, 1999 7905
the solution is treated within the mass action model with the standard chemical potentials as a function of the aggregates size obtained from the SCMF calculations. We have used both methodologies and have obtained identical results. A detailed description of how to apply the SCMF theory in the grand-canonical ensemble has been recently presented.38 C. Calculation Methodology. To solve eqs 5 we discretize the space in concentric layers of finite thickness δ and we count the number of segments per layer of each generated chain and the available volume for attractive interactions around it. Thus, iδ n(r;R) dr ) n(i;R) ∫(i-1)δ
where n(i;R) is now the number of segments in layer i. This discretization scheme is applied to all the equations of the last section. In particular the equations to be solved to determine π(i), the discrete version of the lateral pressures, are the following set of coupled nonlinear equations,
1)N
v0
∑P(R)[nT(i,R) + nH(i,R)]V(i) + exp{-π(i)} {R}
1eieM
(14)
where n is the amphiphile’s length, the explicit derivation of this expression as well as its thermodynamic significance is shown in the Appendix. The only unknowns to determine the thermodynamic properties and the average conformational properties, through P(R), are the Lagrange multipliers π(r). These are determined by introducing the explicit expressions for the pdf, eq 9, and the solvent density profile, eq 11, into the constraint equations, (5), as explained in detail in the next section. The input necessary to solve the equations is the chain conformations, the number of chains in the aggregate, and the temperature. In principle, one should also need the solvent chemical potential, µS. However, because of the incompressibility assumption one can show that the value of π(z) is defined up to an arbitrary constant.20,27 Therefore, as it has been proven in ref 37 the solvent chemical potential is not needed. In other words, due to the incompressibility assumption there is only one chemical potential in the pseudobinary problem treated here. This is an exchange chemical potential which for the amphiphiles it represents the chemical potential of inserting a chain with n segments replacing n solvent molecules; see Appendix. The application of the theory described above assumed that we know the number of amphiphilic molecules in the aggregate and then we determine the chemical potential. In other words, we are working in the canonical ensemble for the surfactant molecules. An alternative approach is to work in the grand-canonical ensemble, where the chemical potential of the chains is fixed and the number of molecules in the aggregate is obtained as an output of the calculation. It is important to emphasize that the chemical potential used in the grand-canonical application of the theory for the micellar aggregate is not the total chemical potential but the standard chemical potential, since we are treating a single aggregate that is not in contact with the solution directly. As mentioned above (37) Carignano, M. A.; Szleifer, I. Macromolecules 1994, 27, 702.
(15)
〈φT(i)〉 ) N
(16) v0
∑P(R)nT(i,R)V(i) {R}
1eieM 〈Fn(i)〉 )
(17)
∑P(R)Fn(i,R) {R}
1eieM
(18)
where M is the total number of layers. The pdf of chain conformations, P(R), in the discrete version is explicitly given by
P(R) )
1 q
1
{ ∑[
exp -u*(R) -
2T*v0
M i
M
∑i π(i)[nT(i,R) + nH(i,R)] -
]}
v0 Fn(i,R)〈φT(i)〉 + 〈Fn(i)〉NnT(i,R) V(i)
(19)
The above equations (15-17) are a set of 3M coupled nonlinear equations for π(i), 〈Fn(i)〉 and 〈φT(i)〉. Therefore, we have transformed the self-consistent equations for the density profiles into a set of nonlinear coupled equations, which are solved by standard numerical methods; see below. Namely, we do not solve the equations selfconsistently but we have replaced them by the mathematical equivalent form of 3M nonlinear coupled equations for the 3M unknowns. The input necessary to solve the equations is the chain conformations, the number of chains in the aggregate, and the temperature. The outputs are π(i), 〈Fn(i)〉, and 〈φT(i)〉, which correspond to the set of lateral pressures, average number of contacts, and volume fraction of tails, respectively. With these quantities we can calculate any desired average conformational and thermodynamic property of the micellar aggregate using the pdf, eq 18. The chain conformations of freely jointed segments are generated as follows: We start with segments number 1, (38) Szleifer, I. Biophys. J. 1997, 72, 595.
7906
Langmuir, Vol. 15, No. 23, 1999
we place segment 2 at randomly chosen angles over a sphere a radius l centered on segment 1. Segment 3 is then added connected to segment 2, and so on until we reach the chain length desired. For each newly generated segment, we check for self-avoidance against all the other segments in that configuration. If there is overlap between the recently added segment and any of the other ones, that conformation is thrown away and the process starts again from segment 1. In the case of the branched headgroups (see Figure 1), three head segments are randomly placed around the fourth one at distance l; self-avoidance is also required. Once the conformation is generated and its position and orientation are chosen, the internal energy of the chain (u*(R)), the number of segments per layer (nT(i,R) and nH(i,R)), and the number of possible contacts with neighboring chains (Fn(i,R)) are determined. For each model molecule we use a set of 5 × 105 chain conformations. The set is built as follows: We generate randomly 5000 chain configurations (as explained above). Each configuration is rotated randomly 10 times. Every rotated configuration is placed randomly at 10 different radial distances from the center of the aggregate to the edge of the cell. The overall cell has a minimal radius R ) 7δ depending upon the chain length. Increasing the size of the cell that we carry out of the calculations by δ does not change the final results. (Note that in all cases the cell size is larger than the fully extended length of the amphiphile.) In that way we do not impose any structure on the amphiphile’s spatial distribution, beyond the spherical symmetry of the aggregate. The resulting structure of the micelle will be obtained from the theory; i.e., it will be the optimal structure that minimizes the free energy of a single aggregate with that number of molecules. For example, if a chain conformation has all its hydrophobic segments toward the outer part of the aggregate and its position is far from the aggregates center, the probability of that conformation will be extremely small. On the other hand, the conformations that have most of their hydrophobic segments close to the core and the hydrophilic ones toward the solvent will be the ones that contribute the most to the average structure of the aggregates. Note that in the way that the chain conformations are generated the position of the headgroup can be anywhere in the aggregate. The final average position of the headgroup segments, as is the case for all the segments of the chain, will be the one that minimizes the free energy of the whole micelle. Further, in all the calculations we use a number of layers such that at least the two outmost layers have the very low monomeric concentration that corresponds to the chemical potential of the micelle, i.e., 10-9 or below. Once all the chain conformations are generated, we solve eqs 15-17 using standard numerical methods39 or package routines such as IMSL (Visual Numerics) nonlinear equation solver. The convergence criteria that we use is a norm of 10-6; therefore we can be confident of our numerical results up to the fourth significant figure. The typical time required to solve the equations to obtain the three sets of unknowns, i.e., π(i), 〈Fn(i)〉, and 〈φT(i)〉 for all i, is 4.5 min in a SGI O2 workstation. The results presented below did not change if we increase the number of configurations by an order of magnitude. The layer thickness used in most of the calculations is δ ) 1.20l, and it was chosen for convenience. (39) Press, W.; Flannery, B.; Teukolsky, S.; Vetterling, W. Numerical Recipes; Cambridge University Press: New York, 1989; p 269.
Guerin and Szleifer
Figure 1. Schematic representation of the two model amphiphile structures of chemical formula H4T5: linear (left) and branched (right). Open and shaded spheres represent tail and head segments, respectively.
Figure 2. Micellar size distribution of aggregates formed by H4T5 surfactants with branched heads at T* ) 2.2. The total amphiphile volume fraction is 0.08. The circles are MD simulation results from ref 13 and the solid line corresponds to SCMF theory predictions.
We have checked that the results of the calculations do not vary for δ within l and 1.8l. This same conclusion was obtained in many other applications of the SCMF theory.27 III. Results and Discussion The quality of the approximations used in the derivation of a theoretical approach is best checked by comparing the predictions of the theory with full scale computer simulations. The reason is that the simulations represent the exact solution for the same model system. Further, to check the quality of the model system, one compares with experimental observations. Early work has shown that the SCMF theory is able to predict quantitatively the cmc of model nonionic surfactants.11 However, the predictions for the micellar size distribution were not very good. Mackie et al.11 argued that the disagreement between the theoretical predictions and the results of the simulations was due to the fact that the micelles in the simulations were not spherical while the theory was applied for spherical aggregates. In order to check if indeed the SCMF theory is able to predict properly the size distribution for spherical aggregates, we compare its predictions with results from MD simulations. The simulations were carried out for H4T5 amphiphiles with branched headgroups as shown in Figure 1. Snapshots of the simulations shown in ref 13 show that the micelles are very close to spherical, so this is the ideal system to check the quality of the approximations in the SCMF theory. Figure 2 shows the size distribution for branched head H4T5 molecules as predicted from the SCMF theory and the MD results. The agreement between the two calculations is very good. It is important to emphasize that there are no adjustable parameters in the theory. The inputs used to predict the size distribution shown in the figure
Self-Assembly of Model Nonionic Amphiphilic Molecules
Langmuir, Vol. 15, No. 23, 1999 7907
Figure 4. Critical micellar concentration as a function of the temperature for micelles formed by branched H4T5 amphiphiles. The cmc is plotted in logarithmic scale. Figure 3. Standard chemical potential µN0/kT as a function of the aggregation number N, for the same system of Figure 2. The insert shows free energy differences of one-tenth of kT between surfactants in the range of aggregation numbers that determine the size distribution.
are the chain conformations for the same model used in the simulations and the temperature. The very good agreement between the MD simulations and the theoretical predictions, coupled with the excellent predictions for the cmc obtained by Mackie et al.,11 strongly suggest that the SCMF theory is able to predict quantitatively the free energy scales associated with micellization and with the size distribution. The predictions of the theory for the standard chemical potential of the micelles are shown in Figure 3. The cmc is determined mostly by the difference between the chemical potential of the unimer and that of the minimum. This is of the order of 3.2kT while the size distribution is determined by the differences shown in the inset, of the order of 0.1kT and smaller. These free energy scales are of the same order of magnitude as those found experimentally2,40,41 for micellization and size and shape distribution. The fact that the size distribution and the cmc predicted by the theory is in good agreement with the simulations implies that the SCMF theory is accurate in predicting large and small free energy differences within the same molecular system. Therefore, as will be shown in detail below, the theory can be applied to study changes in micellar size distributions and cmc due to small “chemical changes” of the amphiphile molecules. We now look at the effect of temperature for the H4T5 amphiphile. Figure 4 shows the variation of the cmc with temperature. The cmc is defined here as the free monomer concentration at which the number of free monomers in solution is equal to the number of amphiphiles in aggregates. As expected, increasing the temperature increases the cmc due to the dominant role of entropic contributions that favor monomers in solution. We find that at low temperatures the cmc increases exponentially with temperature, and then it levels off. We find that above T* ) 2.3 there is no more aggregation. In the exponential (40) Ambrosone, L.; Costantino, L.; D’Errico, G.; Vitagliano, V. J. Colloid Interface Sci. 1997, 190, 286. (41) Alami, E.; Kamenka, N.; Raharimihamina, A.; Zana, R. J. Colloid Interface Sci. 1993, 158, 342.
region, low temperature, we do not see that the cmc is given by
[ ]
cmc ∝ exp
∆G0 T*
(20)
as it is generally accepted. The reason is that ∆G0 ) µN(0) - µ1(0) is a varying function of temperature in our calculations while eq 20 assumes that it is not. However, the temperature dependence of the cmc is the result of the interplay between the entropic contributions favoring dissociation and the energetic contributions that favor association. Leermakers and Lyklema28 found similar temperature effects applying the lattice SCF theory. The temperature variation of the cmc shown in Figure 4 is qualitatively different than that of nonionic amphiphiles of type EnCm.40,41 The reason is that in our model the solvation of the headgroups is assumed to be temperature independent. This is not the case for ethylene oxide headgroups for which the solubility decreases with increasing temperature.42 One way to improve the theory is to incorporate a better model for the headgroup solvation, as for example used by Linse;32 see section IV. Another way is to improve the model for the solvent to closer resemble the structure and properties of water. The temperature variation of the standard chemical potential suggests that the size distributions of micelles should depend on temperature. This is shown in Figure 5. The general trends are as it would be expected from first principles.3 Namely, for low temperatures the aggregates are large and the distributions are narrow. As the temperature increases the optimal micellar size decreases and the distributions become more polydisperse. Again, this is the direct manifestation of the entropic contribution dominating at high temperatures, more aggregates of smaller size, against the energy dominating at low temperatures, i.e., large aggregates. Similar behavior of cmc and size distribution with temperature has been reported by von Gottberg et al.15 for the micellization of short A2B2 amphiphiles. It has been widely recognized by now that in order to predict properly the behavior of surfactant molecules, the contributions from both the headgroup region and the (42) Karlstro¨m, G. J. Phys. Chem. 1985, 89, 4962.
7908
Langmuir, Vol. 15, No. 23, 1999
Figure 5. Micellar size distributions of branched H4T5 micelles at different temperatures. The solid line corresponds to T* ) 1.7, the dashed line to T* ) 1.9, and the dot-dashed line to T* ) 2.1. The total amphiphile volume fraction is 0.05.
Figure 6. Critical micellar concentration as a function of the number of hydrophobic tail segments of H4Ty micelles. The data correspond to different headgroup architectures: Open symbols represent linear headgroups and filled symbols branched headgroups. The cmc is plotted in logarithmic scale.
tails must be considered.3 In general, the headgroup contributions are considered without explicit account of the conformational degrees of freedom.21 This is particularly important in treating nonionic surfactants. In order to see the effect of the headgroup architecture and internal degrees of freedom on the self-assembly behavior of the molecules, we show in Figure 6 the cmc as a function of the hydrophobic tail length for two types of molecules. One in which the headgroup is branched and the other with linear heads (see Figure 1). The cmc is slightly higher for the branched chains; however, the effect is not dramatic. The figure shows that in both cases the cmc decreases with increasing hydrophobic chain length. The cmc variation with chain length is close to exponential reflecting the additional (∼kT) free energy of micellization per hydrophobic segment. Other theoretical approaches have predicted similar chain length dependence of the cmc for short amphiphilic systems21 and for long polymeric molecules.31,32 However, they did not study the effect of changing the headgroup chemical structure.
Guerin and Szleifer
Figure 7. Micellar size distributions of H4T5 amphiphilic micelles with different headgroup architectures. Dashed line corresponds to linear headgroups and solid line to branched headgroups. T* ) 2.2, and the total amphiphile volume fraction is 0.06.
The different headgroups have a pronounced effect on the micellar size distributions as shown in Figure 7. The molecules with a linear head show larger aggregation numbers than the ones with branched chains. The reason seems to be the higher flexibility of the linear head that, while short, can stretch and accommodate more hydrophobic tails in the micellar core. The branched headgroups however are bulkier, resulting in smaller micelles. While the argument is the same as it would be obtained from the common packing arguments, it is important to realize that, for the case shown in Figure 7, the average radius of gyration of the linear head group (a measure of the headgroup size) is very similar to that of the branched head. Thus, the packing argument based on average bulk sizes will provide the wrong answer. The point is that the linear chains have the ability to stretch. This stretching, while costing in conformational entropy, is favorable to the molecules as a whole due to the energetic gain of packing more hydrophobic tails. Summarizing, the architecture of the headgroup is very important in determining the size distribution. The changes in the structure of the heads induced by the packing must be taken into account for the proper description of the system. Namely, the bulk average properties of the molecules are not enough information to determine the optimal packing geometry. However, if one is only interested in the cmc, then bulk properties of the headgroup may be appropriate to determine it. An interesting question that arises is how the cmc and the micellar size distribution change with the length of the hydrophilic linear headgroups. This is shown in Figure 8. The cmc increases with increasing headgroup length showing the more hydrophilic character of the molecules, i.e., larger solubility. The size distribution shows smaller optimal micellar size as the headgroup length increases, reflecting the more bulkier (repulsive) character of the heads keeping the hydrophobic tail fixed. Note that the slope in the reduction of the cmc with headgroup length is much smaller than the increase with hydrophobic tail increment (Figure 6). This is the result of the gain in energetic interactions of the hydrophobic segment being larger than the entropic loss of the hydrophilic head segment. The last result that we show concerns the structure of the micelle. Figure 9 shows the distribution of head and tail segments as a function of the radial distance from the
Self-Assembly of Model Nonionic Amphiphilic Molecules
Figure 8. (a) Critical micellar concentration as a function of the number of hydrophilic head segments corresponding to linear HxT6 micelles at T* ) 2.2. (b) Micellar size distributions for H4T5 (solid line) and H7T5 (dashed line) amphiphilic systems.
center of the micelle. The two cases shown are for different hydrophobic tail lengths and at the same temperature. The distributions are shown as a histogram which represent the typical discretization that we use in solving the nonlinear equations. All the micelles show a compact core, whose size is determined by the length of the hydrophobic tail, and a relatively broad interfacial region. It is clear from the figure that the longer chain length micelle has a more compact hydrophobic core. Furthermore, the solvent penetration is slightly larger for the smaller micelle. This smaller solvent penetration may be thought as the result of having a more “bulk”-like hydrophobic core in the larger micelle. Note that in the second half of the interface toward the solvent side the headgroup distributions are similar for the two micelles. The very small amount of solvent in the hydrophobic core is in agreement with the recent simulations and SCF calculations of Wijmans and Linse.12 A larger penetration of solvent in the core, as has been found in many SCF applications,28,29 will be found if the interaction parameter used in the calculations is chosen to be much smaller. For example, Figure 10 shows two different micelles formed with the same number of surfactant molecules but at different temperatures. The higher temperature induces a much more floppy micelle as a result of the more dominant role played by the entropy of mixing over the hydrophobic interactions. IV. Conclusions We have shown that the SCMF theory is capable of predicting the micellar size distribution in very good agreement with full scale MD calculations. Further, early work showed that the same theory quantitatively predicts the cmc. Therefore, this theoretical approach is capable of predicting quantitatively the two very different free
Langmuir, Vol. 15, No. 23, 1999 7909
Figure 9. Tail and head density profiles for micelles formed by surfactants of different tail lengths. Solid lines correspond to tail segment distributions and dashed lines to head segments distribution. The left graph corresponds to H4T5 while the right one is for H4T10. The distributions are calculated for the most probable aggregation number at T* ) 2.2, NM ) 38 for H4T5, and NM ) 104 for H4T10. Note that the density at the center of the micelle is close, but it does not reach the value 1. There is always a small amount of solvent inside the core. A zero value for the solvent density will imply an infinity pressure as can be seen from eq 11. Note also the the volume of the region where the tail density is close to 1 is very small. The (almost) dry cores shown in the figures are in agreement with the MC simulations of Wijmans and Linse (see Figure 9 in ref 12).
Figure 10. Tail (circles), head (diamonds), and solvent (triangles) density profiles for micelles formed by H4T5 surfactants with branched headgroups at two different temperatures. Solid and dashed lines correspond to T* ) 3.5 and T* ) 2.2, respectively. The ploted distributions are obtained by a spline of the midpoints of the histograms representing the densities.
energy scales associated with these processes for the model systems studied. The predictions of the theory for the variation of the cmc and the micellar size distribution on temperature
7910
Langmuir, Vol. 15, No. 23, 1999
and length of the hydrophobic tail are in qualitative agreement with experiments and reflect the known interplay between entropic and energetic contributions that determine the thermodynamic behavior of amphiphilic systems.2 Our theoretical results, however, should not be compared directly with the experimental observations on nonionic surfactants of the type EnCm. In that case40,41 the variation of the solubility of the ethylene oxide oligomer in water is different than the model studied here. Linse has incorporated the solubility of EO monomers in water into the SCF approach of Scheutjens and Fleer to treat the micellization of Pluronic triblock copolymers. He found very good agreement with experimental observations for long chains but did not obtain micellization for short molecules.32 We are planing to incorporate the solubility of EO monomers into our molecular model to see if we can predict micellization on short molecules containing ethylene oxide oligomers. However, a proper solubilization of the headgroups will also require a much more realistic model for the solvent (water) than the one we have used in this work. We have studied the effect of changing the chemical structure of the headgroup on the cmc and the micellar size distribution. We found that the cmc’s are only slightly different. However, the micellar size distribution is predicted to be shifted to much larger aggregates in the case of linear heads as compared to branched ones. This result points out the special care that must be used in modeling flexible headgroups by effective spheres. A question that needs to be addressed is what is the validity of models that assume a sharp interfacial region between the hydrophobic tails and the hydrophilic heads. As shown and discussed throughout, the cmc is determined by a free energy difference of the order of kT per hydrophobic monomer. The origin of this free energy is mostly from the transfer of the hydrophobic segments from water to a hydrophobic environment. Therefore, the assumption of a compact core is probably excellent for determining cmc’s, provided that all the relevant interactions, i.e., headgroup and tails, are considered. However, for the micellar size distributions the changes in free energy are very small. The assumption of a sharp interface introduces changes in the free energy of packing that are of the same order of magnitude as the free energy changes that determine the size distribution. Therefore, we believe that for small spherical aggregates the sharp interface approximation is probably not very good if one desires to determine the micellar size distribution. Similar conclusions were obtained in earlier theoretical work using SCMF theory43 for dry core aggregates and lattice SCF methods.25 It is important to discuss the limitations of the work presented here. First, all the systems treated here were considered to be in the ideal solution limit. Namely, that there are no interaggregate interactions, a very good approximation near the cmc. However, if one is interested in the full phase diagram of amphiphile-solvent mixtures, the approach described here has to be expanded or included as part of a more general theoretical framework. Second, the only geometry treated in our study was spherical. While we believe that in most of the cases presented here this is the only relevant aggregate geometry, there may be morphological transformations in the micelles even close to the cmc that may change some of our results. Actually, one of the most interesting aspects of self(43) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1986, 85, 5345.
Guerin and Szleifer
assembling systems is the shape transformations induced by the competition between the different interactions in the system which depend upon the thermodynamic variables as well as the chemical architecture of the amphiphilic molecules; for recent reviews see refs 3 and 44. Therefore, a complete study of different micellar geometries must be carried out. Third, the molecular model used in this study is a very simple one. One of the goals of the work presented here was to compare the predictions of the theory with full scale simulations, and thus, we needed the same model system. The next natural step will be to apply the theory to experimental systems. This has been done for the packing of the chains obtaining very good agreement for conformational properties of the hydrophobic tails as compared with experimental observations.43 However, the theory was not yet applied to treat head and tail aggregation. This is possible for nonionic surfactants where the headgroup is a chain of a kind that has been successfully modeled using SCMF theory.45 Moreover, for ionic surfactants it remains a major theoretical challenge to treat detailed molecular structure and electrostatic interactions in water within the same theoretical framework. Acknowledgment. This work is supported by National Science Foundation Grant CTS-9624268. Partial support was provided by the Petroleum Research Fund, administered by the American Chemical Society. I.S. is a Camille Dreyfus Teacher-Scholar. Appendix In this appendix we derive the expression for the standard chemical potential of the amphiphile chains, µN0. The chemical potential is defined as the derivative of the free energy with respect to the number of amphiphilic molecules, i.e.
βµN0 )
(∂βF ∂N )
(21)
V,T,Ns
We now take the derivative of the right hand side of eq 13
[∫
β ∂βF )〈Fn(r)〉〈φT(r)〉 dr + ∂N 2v0 ∂〈Fn(r)〉 ∂〈φT(r)〉 〈φT(r)〉 dr + N〈Fn(r)〉 dr N ∂N ∂N ∂π(r) 2πr2 ∂ ln q (22) dr - ln q - N ∂N v0 ∂N
∫
]
∫
∫
From the partition function q, eq 10, we can calculate the derivative of the last term to obtain
∫
∂π(r) ∂ ln q 1 ∂q ) )〈n(r)〉 dr ∂N q ∂N ∂N ∂〈φT(r)〉 ∂〈Fn(r)〉 β + 〈φT(r)〉 + 〈Fn(r)〉 2v0 ∂N ∂N ∂〈φT(r)〉 dr (23) 〈Fn(r)〉 ∂N
∫
[
]
The last three terms on the right-hand side of eq 23, after multiplication by -N as needed in eq 22, cancel with the (44) Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1995, 100, 13169. (45) Faure´, M. C.; Bassereau, P.; Carignano, M. A.; Szleifer, I.; Gallot, Y.; Andelman, D. Eur. Phys. J. B 1998, 3, 365.
Self-Assembly of Model Nonionic Amphiphilic Molecules
first three terms of the right-hand side of eq 22. Then we have
∂βF ) -ln q ∂N
∂π(r) 2πr2 ∂π(r) 〈n(r)〉 dr dr + N v0 ∂N (24)
∫ ∂N
∫
which using the constraint condition, eq 5 can be written as
∂βF ) -ln q ∂N
∂π(r) 2πr2 φ (r) dr v0 S
∫ ∂N
(25)
Using the relationship between the solvent volume fraction and the lateral pressures, φS(r) ) exp{-[π(r) - βµS] - 1}, and noting that the solvent chemical potential is independent of the number of surfactant molecules we have that
∂φS(r) ∂π(r) dr ) φ (r) dr ∂N ∂N S
(26)
and the last integral in the right-hand side of eq 25 can be calculated using the constraint equation. Namely, to evaluate ∫(∂φS(r)/∂N)(2πr2/v0) dr, we first take the derivative of the constraint equation, eq 5, and multiply by 2πr2/ v0 to obtain
〈n(r)〉 +
∂φS(r) 2πr2 )0 ∂N v0
(27)
Integrating both sides, as needed for eq 25, we obtain that
Langmuir, Vol. 15, No. 23, 1999 7911
∫
∂φS(r) 2πr2 dr ) - 〈n(r)〉 dr ) -n ∂N v0
∫
(28)
where the last equality arises from the fact that the sum over all the segments of the surfactant molecules gives its total chain length. Thus, the chemical potential is
βµN0 ) -ln q - n
(29)
Several remarks concerning this result should be made. First, note that this is the standard chemical potential of a molecule within an aggregate of size N; therefore it does not contain the ideal (translational) contribution. Second, note that the only relevant term in the chemical potential is the first one, since the chain length is constant. A constant term in the chemical potential has no thermodynamic consequences since it is differences between chemical potentials that determines thermodynamic behavior. (Note that in several presentations of the theory for applications on grafted polymer layers27 we have omitted the constant term due to its thermodynamic irrelevance. Actually, if we do not drop the term Ns in eq 12 we would not get the -n term in eq 28.) Third, it is important to realize that since we have imposed the incompressibility assumption, i.e., that the total volume is occupied either by surfactants or by solvent molecules, toward the end of the derivation, the chemical potential should be understood as an exchange chemical potential. Namely, it is the work related to the “exchange” of n solvent molecules by one surfactant molecule. It is for this reason that the chemical potential of the solvent is not necessary, but the proper interpretation of the thermodynamic quantities, as exchange quantities and not absolute quantities, needs to be given. This is true for the surfactant chemical potential and the lateral pressures. LA980788N
