Influence of Charge Lipid Head Group Structures on Electric Double Layer Properties

In this study we derived a model for a multicomponent lipid monolayer in contact with an aqueous solution by means of a generalized classical density functional theory and Monte Carlo simulations. Some of the important biological lipid systems were studied as monolayers composed of head groups with different shapes and charge distributions. Starting from the free energy of the system, which includes the electrostatic interactions, additional internal degrees of freedom are included as positional and orientational entropic contributions to the free energy functional. The calculus of variation was used to derive Euler–Lagrange equations, which were solved numerically by the finite element method. The theory and Monte Carlo simulations predict that there are mainly two distinct regions of the electric double layer: (1) the interfacial region, with thickness less than or equal to the length of the fully stretched conformation of the lipid head group, and (2) the outside region, which follows the usual screening of the interface. In the interfacial region, the electric double layer is strongly perturbed, and electrostatic profiles and ion distributions have functionality distinct to classical mean-field theories. Based purely on Coulomb interactions, the theory suggests that the dominant effect on the lipid head group conformation is from the charge density of the interface and the structured lipid mole fraction in the monolayer, rather than the salt concentration in the system.

: Projections of designated charges of the three lipid head gorups. a) In the case of the rigid rotor, the projection on x-axis refers to the middle charge, b) in the case of exible structure, the additional degrees of freedom require two projections on x-axis, namely of middle x and terminating s charges, and c) in the case of triangle structure the projection of the center of the mass of two charges is computed on the x-axis. The inuence of added salt on the properties of EDL is shown in Fig. S4. In the limiting case of very low salt concentration the results converge to the case of counterions only.

Rigid rotors
In the case of only counterions the conditional probability density shows that the rigid head groups are oriented more perpendicular to the charged surfaces. With increasing salt concentration the tendency of head group orientation is more parallel to the charged surface.
The main reason is the increased screening which eectively increases with increasing salt concentration. The ratio of simple charged head groups (lipids with only phosphate group) and zwitterionic lipids: red -0%, black-25%, blue -50% and green -75%.

Flexible structure
In the case of exible lipid head groups the probability density W depends on two variables x and s. Therefore it is natural to show three dimensional plots. Figs. S5 and Fig. S6 show a three dimensional plots of the probability density W as a function of x and s. The domain of s covers only the interval between 0 and l 1 whereas the domain for s is the interval (0, l 1 +l 2 ).
The peak in the distribution is reached close to x = 0 which means that the middle charge is located close to the charged surfaces.
The second charge is stretched more to the interior of the liquid and W reaches the max- Figure S3: Calculations for four dierent mole fractions of rigid rotors lipids: β = 0 (red curve and squares), β = 0.25 (black curve and circles), β = 0.5 (blue curve and diamonds) and β = 0.75 (green curve and stars). Theoretical conditional probability density proles (a) and MC simulations (b) are shown. The model parameters are a = 0.6 nm 2 , l 1 = 0.5 nm, l 2 = 0.5 nm, l B = 0.7 nm, q 0 = −1, q 1 = 1 and q 2 = −1.
imum at s = 0.5 nm.  Figure S4: Rigid rotors case at low charge regime of the interface for a = 6 nm 2 (σ ≈ 0.03 C m −2 ) and the inuence of added salt. Theory (lines) and MC simulations (squares) are presented for the cases: counterions only (black dashed), 0.05 mol dm −3 (black), 0.1 mol dm −3 (red) and 0.5 mol dm −3 (green) a) Conditional probability density proles W of head groups, b) counterion concentration proles n. The model parameters are a = 6 nm 2 , l 1 = 0.5 nm, l 2 = 0.5 nm, β = 0.5, l B = 0.7 nm, q 0 = −1, q 1 = 1 q 2 = −1. Figure S5: Flexible lipid head groups and couterions only. The conditional probability density W as a function of x and s is shown. The right graph shows the projection of W on the (x, s) plane. The model parameters are q 0 = −1, q 1 = 1 q 2 = −1, l 1 = 0.5 nm, l 2 = 0.5 nm, l B = 0.7 nm and a = 0.6 nm 2 . Figure S6: Flexible lipid head groups with salt. The conditional probability density W as a function of x and s is shown. The right graph shows the projection of W on the (x, s) plane. The model parameters are q 0 = −1, q 1 = 1 q 2 = −1, l 1 = 0.5 nm, l 2 = 0.5 nm, l B = 0.7 nm, 0.1 mol dm −3 salt and a = 0.6 nm 2 .
The inuence of added salt on the properties of exible lipid head groups is shown in  Figure S7: Flexible structure monolayer at low charge regime of the interface for a = 6 nm 2 (σ ≈ 0.03 C m −2 ) and the inuence of added salt) Conditional probability density proles W x of head groups, b) Conditional probability density proles W x of head groups and c) counterion concentration proles n. Inuence of added salt on exible lipid head groups: 0.005 mol dm −3 (blue curves), 0.05 mol dm −3 (black curves), 0.1 mol dm −3 (red curves), and 0.5 mol dm −3 (green curves). MC simulations are presented by squares. In a) and b) the theoretical results are shown by only black lines due to very similar results. Mixture of structured exible lipids and simple negatively charged lipids is β = 0.5. The model parameters are a = 6 nm 2 , l 1 = 0.5 nm, l 2 = 0.5 nm, l B = 0.7 nm, q 0 = −1, q 1 = 1 q 2 = −1.
orientation is more parallel to the charged surface. The main reason is the electrostatic attraction between the positively charged middle groups with the negatively charged surfaces. In the limiting case of very low salt concentration the results converges to the case of counterions only. Triangular Lipid Head Groups S7 Figure S8: Comparison the properties of EDL for the rigid rotor (red line) and the exible structure (green line). a) Electrostatic potential, b) counterions proles, and c) probability which refers to the position of the middle charge W . The results are shown for c = 0.01 mol dm −3 of 1:1 electrolyte, the surface area per lipid molecules is set to a = 0.6 nm 2 and β = 0.5. The model parameters are q 0 = −1, q 1 = 1, q 2 = −1, l 1 = 0.5 nm, l 2 = 0.5 nm and l B = 0.7 nm. Figure S9: Snapshot of triangular lipid head groups attached to the surface. The red spheres depict the negative phosphate group q 0 , while green spheres depict two negative terminating charges. Blue spheres depict mobile counterions. The length of the head groups is l =0.68 nm. the charge numbers are q 0 = −1, q 1 = −1, q 2 = −1. The angle between two terminal charges is α = 30 • Results are given for surface area per lipid molecule a = 0.6 nm 2 . The system is electroneutral. S8 a b Figure S10: Comparison between the Gouy-Chapman model for at the interface and the rigid rotor model presented in this work. Concentrations of counterions nv as a function of distance from the interface x. l is a length of rigid rotor head groups segments, l = 0 represents the solution of the Gouy-Chapman model. a) surface area per lipid molecule a = 0.6 nm 2 , and b) surface area per lipid molecule a = 6 nm 2 . The system is electroneutral.