Article pubs.acs.org/JPCB
Effect of Bond Rigidity and Molecular Structure on the Self-Assembly of Amphiphilic Molecules Using Second-Order Classical Density Functional Theory Bennett D. Marshall, Chris Emborsky, Kenneth Cox, and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice University, 6100 South Main, Houston, Texas 77007, United States ABSTRACT: Second-order classical density functional theory is applied to assess the effect of surfactant properties on the interfacial structure and interfacial tension of a planar oil/water interface. Specifically the affect of the relative locations of the hydrophobic and hydrophilic portions, rigidity vs flexibility, and bond angle of the surfactant are investigated. It is found that bond angle and branching significantly affect the tendency of a surfactant to adsorb on the interface and the degree to which the interfacial tension is lowered.
I. INTRODUCTION Amphiphilic molecules such as surfactants contain both hydrophilic and hydrophobic portions. This duel molecular identity drives surfactant self-assembly at fluid/fluid interfaces which in turn affects the thermodynamic and interfacial properties of the system (e.g., reduced interfacial tension). Surfactant selfassembly at the air/water interface has been subjected to extensive study and analysis; however, liquid/liquid interfaces such as water/oil have received less attention. With extensive use of surfactants in several industrial applications and consumer goods such as shampoos, detergents, and enhanced oil recovery, the need for targeted molecular design to maximize the efficiency and effectiveness of surfactant performance in these processes will become increasingly important. Of particular interest is the structure of the water/oil interface in the presence of surfactant. How do surfactant characteristics such as relative numbers of hydrophilic and hydrophobic groups, their arrangement in the molecule, bond angle, and branching affect the structure of the interface, and how does this structure affect interfacial tension? These are fundamental questions whose answers will give scientists the capability to design novel surfactants which are tuned to maximize the efficiency of certain processes. A number of experimental techniques have been developed to probe the structure of interfacial systems such as neuron scattering,1 second-harmonic, and sum-frequency spectroscopy;2,3 however, these experiments can be costly and time-consuming. Alternatively, molecular simulation can be used to study interfacial systems; however, the complex nature of surfactant molecules, and the oil/water interface in general, presents a formidable challenge. Atomistic molecular dynamic simulations can be used to obtain detailed structure of surfactant monolayers at fluid interfaces;4,5 however, due to long diffusion time scales, atomistic simulations which establish bulk surfactant concentrations are extremely computationally demanding and, to the © 2012 American Chemical Society
authors knowledge, have not been performed. Coarse grained molecular dynamic simulations can provide physical insight without full atomistic detail. Smit et al.6,7 was one of the first to study the interfacial properties of surfactant systems at liquid/ liquid interfaces using molecular simulation; they employed the Telo da Gama and Gubbins8 surfactant model at a planar oil/ water interface to study the effect of tail length on surfactant performance. In this model the oil and water molecules are treated as hard spheres with attractions between like species, and the surfactants are constructed by bonding oil and water type spheres together. In these simulations interfacial tensions and density profiles were calculated as a function of the number of surfactant molecules in the system; due to the small size of the system a bulk surfactant concentration could not be established. More recently mesoscale simulation techniques such as dissipative particle dynamics9,10 and that of Shinoda et al.11,12 have allowed larger systems to be studied giving the ability to study the effect of bulk surfactant concentration on interfacial tension. An attractive alternative to molecular simulation is density functional theory (DFT). In density functional theory the grand potential, which is a functional of density, is minimized to obtain the density profiles of an inhomogeneous system.13 Density functional theories retain detailed information on molecular structure, but at a fraction of the computational cost of molecular simulation. Recently Emborsky et al.13,14 applied the iSAFT density functional theory15−17 to the Telo da Gama and Gubbins8 surfactant model at a planar oil/water interface. With use of this model the effect of surfactant length and the relative number and arrangement of hydrophilic/hydrophobic groups on the structure of the interface/interfacial tension was investigated. Received: October 21, 2011 Revised: February 6, 2012 Published: February 7, 2012 2730
dx.doi.org/10.1021/jp2101368 | J. Phys. Chem. B 2012, 116, 2730−2738
The Journal of Physical Chemistry B
Article
In accordance with Traube’s rule,18 it was shown that increasing the length of the oil tail resulted in a surfactant more effective at lowering interfacial tension. Of the previously mentioned studies, none have investigated the effect of surfactant bond rigidity on the structure of the oil/water interface. The iSAFT density functional theory is a first-order perturbation theory incapable of incorporating bond angle dependence. Recently, Marshall and Chapman19 extended the work of Kierlik and Rosinberg20,21 and developed a density functional theory based on Wertheim’s second-order perturbation theory22−26 to account for chain connectivity in linear and branched chains. In Wertheim’s theory statistical mechanics is rewritten in a multidensity formalism which allows for the inclusion of short-range highly directional interactions such as association. Like the single density form of statistical mechanics the application of the theory requires approximation techniques such as perturbation theory or integral equation theory. The first-order perturbation theory TPT1 only accounts for the association between a pair of molecules. Since there are no three body associative terms, the bond angle between association sites on a molecule cannot be specified; at second-order perturbation theory three body associative interactions are accounted for and bond angle, or bond angle distribution, can be imposed in the theory. However, this added information comes at computational expense, in TPT1 only single body integrals need be evaluated while in TPT2 two body integrals must be performed. In this work the Telo da Gama and Gubbins8 surfactant model will be applied to the oil−water interface using the second-order density functional theory, but only certain portions of the molecule will be treated in TPT2 while the rest of the molecule will be treated in TPT1. This treatment will result in surfactants that have rigid portions with a specified bond angle and portions which are fully flexible. With this model the effect of branching, order of hydrophilic and hydrophobic groups and bond angle on the structure of the interface and the resulting decrease in interfacial tension will be investigated for a number of model surfactants. It will be shown that the interfacial properties are strongly dependent on these molecular properties. In section II the second-order density functional theory will be reviewed. Section III will discuss the potential model used and the geometry of the surfactants studied. In section IV we first validate the approach by comparing the theory in its first-order limit to molecular simulations for linear flexible surfactants. Once validated, we perform an extensive study on the effect of surfactant bond rigidity on interfacial structure and surfactant performance.
molecular geometry at any order of perturbation theory. Here we will only consider branched and linear chains in TPT2. The Helmholtz free energy functional is given as β(A[{ρk}] − AR [{ρk}]) m ⎞ ⎛ ⎛ (i ) ⎞ ρo (1) ⎟ i ( ) ⎜ ⎜ = ∑ ρ (1) ln + 1⎟ d(1) − ⎟ ⎜ ⎜ (i ) ⎟ (1) ρ ⎠ ⎠ ⎝ ⎝ i=1
∫
∫ ρ(j)(1) d(1) (2.1)
where β is the inverse temperature, ρ(i)(1) is the density of species i at point 1 in the fluid, ρo(i)(1) is the monomer density at point 1 in the fluid, and AR is the Helmholtz free energy of the reference system, which in this paper will be considered a mixture of hard spheres. The notation 1 represents both position and orientation 1 = {r1⃗ ,Ω1}, where r1⃗ is the position in the fluid and Ω1 is the orientation. The species label in the last term of eq 2.1 is arbitrary; the value of the integral is the same for all segments in the molecule. The segmental density profiles are given as ρ(j)(j) ρ(oj)(j)
m
=
∫
̃ m) ∏ ρ(ϵ)(ϵ) d(ϵ) D2(1...m) F (1... o ϵ≠ j
(2.2)
Here D2(1...m) is the sum of products of reference system correlation functions Y (i,j)(12) and Y (i,j,k)(123), obtained by taking all the ways to partition the molecule into dimer and trimer sections, where (i , j)
Y (i , j)(12) = exp(βφR (12)) g(i , j)(12) (i , j)
(2.3) (j , k)
Y (i , j , k)(123) = exp(βφR (12)) exp(βφR (23)) × {g(i , j , k)(123) − g(i , j)(12) g(j , k)(23)} (2.4)
φR(i,j)(12) (i,j)
The term is the reference system potential between species i and j, g (12) is the reference system pair correlation function, and g(i,j,k)(123) is the reference system triplet correlation function. The term F̃(1...m) contains all of the constraints in the molecule and is defined for a specific case in eq 3.6. The monomer densities are obtained by minimization of the grand free energy Ω[{ρ(j)( r ⃗)}] = A[{ρ(j)( r ⃗)}] −
m
∑ ∫ d r ′⃗ ρ(k)( r ′⃗ )μ(k) k=1 (2.5)
In the systems studied in this paper there is no external potential. The free energy is given as
II. SECOND-ORDER DENSITY FUNCTIONAL THEORY In this section second-order density functional (DFT2) will be reviewed. We begin by considering a mixture of m species of spherical segments with some number of short-range highly directional association sites. No double bonding between segments or sites is allowed. From this mixture complex polyatomic molecules are created by letting the association energy between the segments tend to infinity; this is the limit of complete association. Each location in the molecule is occupied by a certain species of segment. Marshall and Chapman19 extended the results of Kierlik and Rosinberg20,21,27 and showed that if the limit of complete association was taken at an early point in the derivation that a general form of Wertheim’s theory in the complete association limit could be obtained for an arbitrary
A[{ρ(k)}] = Aid [{ρ(k)}] + ACH[{ρ(k)}] + ALR [{ρ(k)}] + AHS[{ρ(k)}]
(2.6)
The free energy contribution due to chain formation ACH[{ρ(k)}] is given by eq 2.1. The ideal free energy is known exactly for a mixture of spheres, βAid [{ρ(k)}] =
m
∫
d r1⃗
∑ β= 1
ρ(i)( r1⃗ )(ln ρ(i)( r1⃗ ) − 1) (2.7)
where we have ignored temperature-dependent terms which do not affect fluid structure. The excess reference free energy 2731
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theory29,30 and split the potential into a reference term and an attractive perturbation
functional for hard sphere repulsion is given by Rosenfeld’s fundamental measure theory28 βAHS[{ρ(k)}] =
(i , j)
∫ d r1⃗ ΦEX,HS [{n(k)}]
Φ EX,HS[{n(k)}] = −no ln(1 − n3) + + −
⎧ (i . j) ⎫ (i , j) uR (r ) = ⎨∞ r < σ ⎬ ⎩ 0 otherwise ⎭
n1n2 1 − n3
3
n ⃗ ·n ⃗ − v1 v 2 1 − n3 24π(1 − n3) n2
1 [{ρ }] = 2
8π(1 − n3)2
m
(2.9)
∑∑∫ | r − r |≥σ(i , j) 1⃗
i=1 j=1
2⃗
⎪
(3.2)
(3.3)
The Lennard-Jones potential is well-known
m (i)
⎪
⎪
⎧ (i , j) (i , j) (i , j) ≤ r < r ⎫ min ⎪ ⎪ uLJ (rmin) − uLJ (rc) σ ⎪ ⎪ (i , j) ⎬ uatt (r ) = ⎨ (i , j) (i , j) rmin ≤ r < rc ⎪ ⎪ uLJ (r ) − uLJ (rc) ⎪ ⎪ ⎭ ⎩0 otherwise
n2(nv⃗ 2 ·nv⃗ 2)
and the {n } are the set of weighted densities. The free energy contribution due to long-range attraction is treated in a mean field approximation and is given by (k)
⎪
The attractive perturbation is given by
2
(k)
A
(3.1)
The reference potential is that of a hard sphere
where
LR
(i , j)
u(i , j)(r ) = uR (r ) + uatt (r )
(2.8)
⎡⎛ ⎞12 ⎛ (i , j) ⎞6⎤ ⎢⎜ σ(i , j) ⎟ σ (i , j) ( i , j ) ⎟ ⎥ uLJ (r ) = ε ⎢⎜ − ⎜⎜ ⎟ r r ⎟ ⎥
(j)
ρ ( r1⃗ ) ρ ( r2⃗ )
(i , j)
× uatt ( r1⃗ , r2⃗ ) d r1⃗ d r2⃗
⎢⎣⎝
(2.10)
⎠
⎝
⎠ ⎥⎦
(3.4)
The potential minimum is located at rmin = 2 σ and the cutoff radius is chosen as rc = 2.5σ(i,j). In the current model there are only oil type and water type spheres, so four energy parameters characterize the system. For attractions between segments of the same type we choose εoo = εww = ε and for attractions between segments of different types εow = εwo = 0. The equations presented in section II are valid for any linear or branched chain in TPT2; however, treating the entire molecule in second-order perturbation theory can be computationally expensive. In this work only certain portions of the surfactants will be treated in TPT2 and the rest of the molecule will be treated in first-order perturbation theory TPT1. The portions treated in TPT2 will be rigid with an imposed bond angle α, and the portions treated in TPT1 will be fully flexible. Here we wish to explore the affects of rigidity, arrangement of water and oil type spheres, branching and bond angle on the structure of the oil−water interface, and interfacial tension. To this end a total of four surfactant structures will be studied and are presented in Figure 1. The blue spheres represent water type spheres, and the gray spheres represent oil type spheres; water type spheres are considered heads, and oil type spheres are considered tails. To isolate the effect of bond angle, the two triatomic surfactants H1T1H1 and T2H1, where H stands for head and T stands for tail, will be studied. They are completely rigid with a specified bond angle and only differ in the arrangement of the water type and oil type spheres. The rod−coil surfactants R3F3L and R3F3B contain rigid and flexible portions, R stands for rigid and F stands for flexible; all water/ water bonds are treated in TPT2 and are completely rigid while oil/oil bonds and oil/water bonds are treated in TPT1 and are fully flexible. The two surfactants differ only in the connectivity of the oil type tail segments. The potential model is symmetric. That is the oil type segments could be considered water type segments and vice versa. From this point of view the rigid section of the rod−coil surfactants could be considered a rigid organic backbone and the flexible section could represent ethylene oxide groups. The triatomics and R3F3L are all chain molecules with a rigid triatomic segment treated in second-order perturbation 1/6 (i,j)
Minimizing the grand potential with respect to the total segment density ρ(j)(j) and using eqs 2.5−2.10, the monomer densities are found to be ρ(oj)(j) = exp[βμ(j) + λ(j)(j)]
(2.11)
where λ(j)(j) =
̃
δD(1...m) (m) ̃ d(1)...d(m) ∫ ρ(1) o (1)...ρo (m) F (1...m) (j) δρ (j)
−
δβALR [{ρ(k)}] δρ(j)(j)
−
δβAREX [{ρ(k)}] δρ(j)(j)
(2.12)
Now eliminating all monomer densities in eq 2.2, the density profiles are obtained as ρ(j)(j) = exp[λ(j)(j) + βμM] I (j)(j)
(2.13)
m Here μM = ∑j=1 μ(j) is the chemical potential of the molecule (j) and I (j) is the molecular integral which is given by
I (j)(j) =
m
∫
̃ m) D2(1...m) ∏ exp[λ(ϵ)(ϵ)] d(ϵ) F (1... ϵ≠ j (2.14)
Equations 2.13 and 2.14 represent a general second-order density functional theory for linear and branched chains. In the next section we will specialize these equations to describe a number of model surfactants.
III. SURFACTANT MODEL In this section the Telo da Gama and Gubbins surfactant8 model is used to model a surfactant/oil/water system. Oil molecules and water molecules are treated as single spheres and the surfactant molecules are constructed from these oil type and water type spheres. For the attractive interactions between segments we use Weeks, Chandler, and Andersen perturbation 2732
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Here θR is the bond angle of the rigid section. The inhomogeneous cavity correlation functions are approximated using a geometric mean approximation that is equivalent to assuming pairwise additivity of the potential of mean force16 ln(y(i , j)( r1⃗ , r2⃗ )) =
1 ln{y(i , j)( r1⃗ ) y(i , j)( r2⃗ )} 2
(3.7)
where the inhomogeneous cavity functions are approximated by homogeneous cavity functions evaluated at a weighted packing fraction16 η( r1⃗ ) =
m
πσ3 6
∑
ρ̅(j)( r1⃗ )
j=1
(3.8)
where ρ̅(j)(r1⃗ ) is the weighted density of component j at position r1⃗ . We use the weighting16 ρ̅(j)( r1⃗ ) =
Figure 1. Surfactant structures studied where blue circles represent water type spheres and gray circles represent oil type spheres. The two triatomic surfactants are completely rigid. For all other surfactants the bonds between water type spheres are rigid while all other bonds are fully flexible. Spheres are labeled for structures for which density profiles are presented.
3
∫|r − r |
