Multi-scale Modeling and Packing Argument

Cryo-TEM measurements of body washes verify the formation of entangled networks of wormlike micelles. The solutions are viscoelastic and their rheology is similar to that of

entangled polymer solutions. One difference is that wormlike micelles break and recombine rapidly. By combining the theories considering polymer reptation and micellar fast reversible scission, Cates developed a reptation-reaction model to estimate linear micellar characteristic time and length parameters. [Cates and Fielding (2006); Cates and Candau (1990); Khatory et al. (1993); Cates (1987); Granek and Cates (1992)] The frequency-dependent linear rheological moduli predicted by the Cates model in the fast-breakage limit are given by a single-relaxation- time Maxwell model. Zou and Larson (2014) extended the Cates model by including important physics neglected in the previous model and using “pointers” to track relaxation dynamics in ensembles containing thousands of micelles. This “Pointer Algorithm” allows estimates of multiple micellar parameters from fits of the predictions to the rheology of entangled wormlike micelles. [Español and Warren (1995)] We will use the Pointer Algorithm here to extract micelle parameters, including average micelle length, breakage time, etc. from rheological data we obtain for several commercial body washes.

Once the average micelle length is obtained from such simulations, the scission free energy 𝐸𝑠 (i.e., the free energy of creating two additional end caps), can be determined from the dependence of the average micelle aggregation number 〈𝑛〉 on temperature as [Larson (1999)]

〈𝑛〉 ≈𝜌𝑠𝜋𝑑 2_{〈𝐿〉𝑁} 𝐴 4𝑀̅𝑠 ≈ 2𝜒0.5𝑒𝑥𝑝 ( 𝐸𝑠 2𝑘𝐵𝑇 ) (4.1𝑎)

Here, 𝜒 is the mole fraction of the surfactant, and the exponent 0.5 is derived from the law of mass action, assuming activity coefficients of unity for individual surfactant molecules and micellar species. To derive the above expression, which is slightly more precise than the scaling law of Cates and Candau, [Cates and Candau (1990); Vogtt et al. (2017)] which uses surfactant volume fraction 𝜙 instead of mole fraction 𝜒, we note that for cylindrical micelles containing surfactant with surfactant density 𝜌 = 1.1 g/cm3 (which we take from the mass density of the

surfactant crystal) and diameter 𝑑 = 4 nm, 〈𝑛〉 can be expressed in terms of the average micelle length 〈𝐿〉 as: 〈𝑛〉 =𝜌𝑠𝑉𝑠 𝑀̅𝑠 𝑁𝐴 = 𝜋𝑑2_{〈𝐿〉𝑁} 𝐴 4𝑀̅𝑠 (4.1𝑏)

Here 𝑁𝐴 is the Avogadro number and 𝑀̅𝑠 = 345 g/mol is the average molecular weight of the surfactant, which we take as the weight-averaged molecular weight of the components. SLE1S (molecular weight = 346 g/mol) and CAPB (molecular weight = 342 g/mol), with the weight fractions of these two being 9.85/11 and 1.15/11, respectively. We neglect the small

contributions of salt and perfume molecules to the mole fraction of surfactant 𝜒. Substituting the above equation into Eq. (4.1a), yields:

〈𝐿〉 ≈ 8𝑀̅𝑠𝜒 0.5 𝜌𝑠𝜋𝑑2𝑁𝐴 𝑒𝑥𝑝 ( 𝐸𝑠 2𝑘𝐵𝑇 ) (4.2)

Here the mole fraction 𝜒 is obtained as 0.64 for 11 wt. % surfactant solutions (i.e., 𝜒 =

[𝑤_𝑠/𝑀̅_𝑠]/[𝑤_𝑠/𝑀̅_𝑠+ (1 − 𝑤_𝑠)𝑀_𝑤] = 0.64 with 𝑤_𝑠 = 0.11). To limit our analyses to the regime of cylindrical micelles, we apply Eq. (4.2) only over a set of four temperatures separated by intervals of 1 or 2 degrees within a narrow range from 20.0 to 25.0 °C. We note that if the molar volume of the surfactant is set equal to that of water, the simpler expression of Cates is obtained. Similar to the scission energy, we express the temperature dependence of the terminal relaxation time 𝜏 and zero-shear viscosity 𝜂0 in Arrhenius forms involving terminal relaxation time and viscosity activation energies 𝐸_𝑟 and 𝐸_𝑣, respectively: [Cates and Candau (1990)]

𝜏~𝜂𝑠𝑒𝑥𝑝 ( 𝐸𝑟 𝑘𝐵𝑇 ), 𝜂0~𝜂𝑠𝑒𝑥𝑝 ( 𝐸𝑣 𝑘𝐵𝑇 ) (4.3)

where 𝜂𝑠 is the solvent viscosity which itself is a function of temperature. These activation energies can also be extracted from the corresponding rheological curves.

Dissipative particle dynamics (DPD) simulations

To describe the structure of cylindrical micelles in a computationally efficient way, coarse-grained dissipative particle dynamics (DPD) simulations using a soft repulsive potential are applied here. [Español and Warren (1995); Groot and Warren (1997)] In the DPD

simulations used here, three to five heavy atoms are lumped into one quasi-particle or bead, which interacts with other beads via pairwise forces, and obeys Newton’s equations of motion. The pairwise interparticle force contains three contributions,

𝑓𝑖 = ∑𝑗≠𝑖(𝐹𝑖𝑗𝐶+ 𝐹𝑖𝑗𝐷+ 𝐹𝑖𝑗𝑅) (4.4) where 𝐹_𝑖𝑗𝐶 is a conservative force defined by a purely repulsive (harmonic) soft-core potential sensitive to chemical identity, 𝐹_𝑖𝑗𝐷 is a dissipative force and 𝐹_𝑖𝑗𝑅 is a random force. The latter two forces take into account the fluctuation and dissipation of energy and serve as the Langevin thermostat. The conservative force 𝐹_𝑖𝑗𝐶 determines the thermodynamics of the DPD system and the soft potential allows for larger time steps of picoseconds instead of the femtoseconds used in traditional MD simulations. 𝐹_𝑖𝑗𝐶 = −𝑎𝑖𝑗𝜔𝐶(𝑟𝑖𝑗)𝑟̂𝑖𝑗 𝐹_𝑖𝑗𝐷= −𝛾𝜔𝐷_(𝑟 𝑖𝑗)(𝜐𝑖𝑗∙ 𝑟̂𝑖𝑗)𝑟̂𝑖𝑗 𝐹_𝑖𝑗𝑅= −𝜎𝜔𝑅(𝑟𝑖𝑗)𝜉𝑖𝑗∆𝑡−1 2⁄ 𝑟̂𝑖𝑗 (4.5)

Here 𝑎_𝑖𝑗, 𝛾 and 𝜎 are parameters that determine the strength of these forces. The parameters 𝜎, 𝛾 are chosen such that they will obey the dissipation-fluctuation theorem with 𝜎2 = 2𝛾 and 𝜎 = 3. The repulsive parameters 𝑎𝑖𝑗 are dependant on the chemical identity of beads 𝑖, 𝑗. 𝑟̂𝑖𝑗 is a unit vector in the direction of 𝑟_𝑖𝑗 and 𝑟_𝑖𝑗 = |𝑟_𝑖− 𝑟_𝑗|, while 𝜐_𝑖𝑗 = 𝜐_𝑖 − 𝜐_𝑗 is the relative velocity between beads 𝑖 and 𝑗. 𝜉_𝑖𝑗 is a Gaussian distributed random variable with zero mean and unit variance and ∆𝑡 is the integration time step. 𝜔𝐶, 𝜔𝐷 and 𝜔𝑅 are the weight functions:

𝜔𝐶_(𝑟 𝑖𝑗) = 𝜔𝑅(𝑟𝑖𝑗) = { 1 − 𝑟𝑖𝑗 𝑟𝑐𝑢𝑡 𝑖𝑓 0 ≤ 𝑟𝑖𝑗≤ 𝑟𝑐𝑢𝑡 0 𝑖𝑓 𝑟𝑖𝑗 > 𝑟𝑐𝑢𝑡 (4.6𝑎) 𝜔𝐷= (𝜔𝑅)2 (4.6𝑏)

where 𝑟𝑐𝑢𝑡 is the truncation distance.

Since its introduction by Hoogerbrugge and Koelman in 1992, [Hoogerbrugge and Koelman (1992)] DPD has been improved significantly by Español and Warren [Español and Warren (1995)] and then by Groot and Warren. [Groot and Warren (1997)] Recently, Liyana Arachchi et al., [Liyana-arachchi et al. (2015)] introduced a revision of the Groot and Warren DPD equation of state (rGW-EOS) that improves it for low bead number densities and allows for beads of different number densities to be used in a simulation. Different bead number densities correspond to different effective sizes of beads. This formalism does not change the cutoff distance 𝑟𝑐𝑢𝑡; however the conservative repulsion parameter 𝑎𝑖𝑖 for beads of the same type 𝑖 now depends on the chemical species associated with 𝑖.

Liyana Arachchi et al. [Liyana-arachchi et al. (2015)] also provide for a more systematic determination of parameters 𝑎𝑖𝑗 for distinct types 𝑖, 𝑗 determined from infinite dilution transfer free energies as determined for instance using COSMOtherm.

In this Chapter, we used a water model lumping three water molecules into one bead with a volume of 90.05 Å3. We set the bead number density of water molecules to 5.0. This results in a definition of the DPD length scale of 𝑟_𝑐𝑢𝑡 = 7.6 Å. We set the repulsion parameter between water molecules to 𝑎_𝑖𝑖 = 15.0. The number densities of other beads are determined from comparing their molar volume to the molar volume of the water beads. Molar densities are computed using the empirical model implemented in COSMOtherm. [Klamt et al. (2010); Klamt (2011)] The repulsion parameters 𝑎𝑖𝑖 for like interactions are determined from the respective number density following the rGW-EOS. The repulsion parameters 𝑎𝑖𝑗 for unlike interactions are computed from the infinite dilution transfer free energy approach previously described. [Liyana- arachchi et al. (2015)]

To describe some of the more complex molecules in the simulations described here, multiple beads have to be used. The validity range of the rGW-EOS allows for beads comprising on the order of 3-5 heavy atoms (the definition of the beads used in this work is contained in Table C.1 in Appendix C). To determine the size of the beads, the molar volume is derived from related model compounds by subtracting the molar volume of terminating groups. For instance, the molar volume of a C3 bead describing a propylene functionality is obtained by subtracting the molar volume of ethane from the molar volume of pentane. As another example, some of the isomers found in dipropylene glycol are symmetric in nature. The volumes of the corresponding beads are determined as half of the molar volume of the parent compound.

To determine the interaction parameters for these beads, the weights are assigned to the respective parent compound for COSMOtherm calculations, resulting in only the molecular surface that corresponds to the bead to be considered for the transfer free energies.

For the parameterization of harmonic bond and angles potentials, ensemble distributions were matched to atomistic Monte-Carlo simulations using the COMPASS force field

implemented in Biovia Materials Studio.

To our knowledge, no systematic approach has been described for the parameterization of charged beads in DPD. Given that most ions in aqueous solutions possess a hydration shell, we make the strong assumption that the interactions aside from the electrostatic charge are

dominated by the hydration shell. We hence define the 𝜒𝑖𝑗 parameters describing the interactions of ions with other beads as identical with water interacting with said bead. An ionic bead

interacting with another ionic bead would then have a 𝜒𝑖𝑗 parameter identical to that for a water bead with itself. In addition, integer electrostatic charges are assigned to the beads following their formal electrostatic charges. To prevent singularities in electrostatic forces and energies for the soft core potential, electrostatic charges are described as Gaussian charge clouds centered at the respective bead as suggested by Groot. [Groot (2003)] If the width of the charge distributions are identical for all beads, a standard particle-mesh Ewald scheme can be used to compute the interaction if the real-space cutoff is set to 0.0 (See Appendix C). For these simulations, we used a homogeneous relative dielectric constant of 𝜖₀ = 80 and a standard deviation of 𝜎_𝐸𝑆 = 0.8/√2 for the Gaussian charge distributions.

We used LAMMPS to apply this model in simulations of canonical ensembles at constant pressure and constant temperature with semi-isotropic pressure coupling (𝑝_𝑟𝑒𝑓 = 41.9, 𝜏_𝑝 = 15𝜏) with a time step of 𝜏 = 0.015 DPD units to study the effects on micelle properties of salts in the two formulations BW-1EO and BW-3EO (defined below) and the effects of four PRMs in BW-1EO. All DPD parameters used for the simulations are provided as supplementary materials.

Using a combination of the Pointer Algorithm and the DPD molecular model, we seek to assess quantitatively the changes in rheological properties that result from the addition of salts and PRMs and thereby build a fundamental understanding of the structure-property relationships of surfactant formulations. We used LAMMPS to apply this model in simulations of canonical ensembles at constant pressure and constant temperature with semi-isotropic pressure coupling (𝑝_𝑟𝑒𝑓= 41.9, 𝜏_𝑝 = 15𝜏) with a time step of 𝜏 = 0.015 DPD units to study the effects on micelle properties of salts in the two formulations BW-1EO and BW-3EO (defined below) and the effects of four PRMs in BW-1EO. Detailed DPD parameters and the molecular mapping are listed in Table C.1 in Supporting Information. Using a combination of the Pointer Algorithm and the DPD molecular model, we seek to assess quantitatively the changes in rheological properties that result from the addition of salts and PRMs and thereby build a fundamental understanding of the structure-property relationships of surfactant formulations.

Packing argument and octanol/water partition coefficient.

A. Packing arguments

At concentrations above the 1st CMC in solution, surfactants self-assemble into diverse structures including spherical, global, and cylindrical micelles as well as ordered phases. [Zana and Kaler (2007); Imae et al. (1985); Larson (1999); Israelachvili et al. (1976); Groot (2003)] These self-assembled structures depend on the concentration and chemical structures of the surfactants, the nature of the counter ions, the presence of salts and/or other surfactants, pH, temperature and pressure. [Zana and Kaler (2007); Larson (1999)] Israelachvili and coworkers proposed a packing argument based on the dimensionless “packing parameter” 𝑝 to predict the shape of the micelles, which is defined as [Israelachvili et al. (1976)]

𝑝 = 𝑉 𝑙𝑐𝑎0

(4.7)

where 𝑉 is the occupied volume of the hydrophobic tail, 𝑙_𝑐 is the tail length, and 𝑎₀ is the area occupied by the hydrophilic heads on the micelle surface.

Generally, the hydrophobic tails are wrapped within the hydrophilic heads, giving them limited access to their environment. Therefore the tail volume 𝑉 and tail length 𝑙_𝑐 are typically taken to be constant, while the head group area 𝑎₀ can change, for example with ionic strength. Using the packing parameter 𝑝, the effects of salts and non-hydrophobic PRMs on micellar properties can be easily explained, at least qualitatively. For example, as salt is added into the micellar solution, the electrostatic repulsions between the surfactant heads are screened out, resulting in a decrease of 𝑎0 and an increase of 𝑝, which flattens the micelle surface, leading, for example, to a transition from spherical to elongated cylindrical micelles. Similarly, addition of a hydrophobic perfume that swells the tails increases 𝑉, also resulting in a similar transition. However, as we shall see, large, very hydrophobic, PRMs can penetrate deeply into the hydrophobic core of the micelle, segregated from the tails and swelling the micelle radius,

effectively changing not only 𝑉 but also 𝑙_𝑐, which can lead, perhaps surprisingly, to a decrease in 𝑝, and a transition towards spherical micelles.

B. Octanol/water partition coefficient

PRMs are essential ingredients for fragrances in body washes formulated to meet consumers’ preferences. The effects of a single PRM are often studied and correlated with the

value of its octanol/water partition coefficient, 𝑃𝑂𝑊, which is the concentration ratio of the PRM in hydrophobic octanol to that in the hydrophilic water phase (See Eq. (4.8)). [Bradbury et al. (2013); Penfold et al. (2008); Saito et al. (2003); Suratkar and Mahapatra (2000); Tokuoka et al. (1994); Zhou and Zhu (2005)] The logarithm, log 𝑃_𝑂𝑊, is often used as a hydrophobicity

parameter. In this paper we use the logP values as computed by the Consensus algorithm implemented in ACD/Percepta version 14.02 by Advanced Chemistry Development, Inc. (ACD/Labs, Toronto, Canada). Approximating the hydrophobic surfactant tail region and the hydrophilic surfactant head region by the octanol phase and water phase, respectively, we obtain

𝑙𝑜𝑔 𝑃𝑂𝑊= 𝑙𝑜𝑔 ( [𝑃𝑅𝑀]_{𝑂𝑐𝑡𝑎𝑛𝑜𝑙} [𝑃𝑅𝑀]_{𝑊𝑎𝑡𝑒𝑟} ) ≅ 𝑙𝑜𝑔 ( [𝑃𝑅𝑀]𝑇𝑎𝑖𝑙 𝑟𝑒𝑔𝑖𝑜𝑛 [𝑃𝑅𝑀]_{𝐻𝑒𝑎𝑑 𝑟𝑒𝑔𝑖𝑜𝑛} + [𝑃𝑅𝑀]𝑊𝑎𝑡𝑒𝑟 ) (4.8)

However, the above approximation neglects geometric constraints, especially in the tightly packed surfactant tail region, which might restrict access of PRMs to the micelle core or cause a significant change of the surfactant packing within micelles.

III. Experiments and Simulation Set Up