Stiff A-A bonds

To determine the predictions of the model configurations of different molecu-lar compositions were produced and these were analysed to give their elastic

11.3. Elastic properties of membranes 129

properties. Following standard Monte Carlo procedures the configurations were obtained by minimizing the total interaction energy given by Weinkamer et al. [1998], Glauber [1963], Kawasaki [1972]

U =JX

hiji

σ_iσ_j. (11.40)

The sum extends over nearest neighbour pairs,σ_i is a spin variable (not to be confused with the stressσ!) giving1(−1) if lattice site iis occupied by anA molecule (B molecule, respectively). The choice of the interaction parameter J = 1 >0 led to the favouring of antiferromagnetic order.

Figure 11.5 shows snapshots of the configuration for some composition values. To make the pictures more clear only the strong bonds, connecting A−A pairs are shown. The figure shows how a dense network of strong

Figure 11.5: Three typical configurations for different concentrations of Amolecules: 0.9 (left), 0.66 (middle) and 0.5 (right). Only the strong bonds betweenA-Apairs are shown.

bonds for large concentrations of A molecules with only isolated islands of weak bonds, transforms into a hexagonal lattice (for concentrations of 0.66) and – with still decreasing concentration – the connexion of the existing bonds breaks down, leaving only isolated islands of strong bonds.

To assess the elastic properties of each configuration three different de-formation states were imposed on the sample.



The simulations were done with (pseudo)-periodical boundary conditions in all directions. In contrast to standard periodic boundaries where the posi-tions of the atoms are mirrored, in the presented simulaposi-tions periodic bound-ary conditions in the applied strain field were chosen. This means that the

130 11. Linear Elastic Networks

distance of one atom at the boundary and its mirrored counterpart is held fix. By an appropriate choice of these distance each desired strain state can be imposed on the system. Then the atoms were allowed to relax in their equilibrium position. The resulting forces on the bottom and left boundary were then evaluated and transformed into the appropriate stresses. Know-ing stresses and strains the elastic constants could be easily calculated. The relaxation process of the molecules was done iteratively. One molecule was chosen randomly, its local equilibrium position was calculated with respect to its6 neighbours and then the atom was moved into this new position. This process is now iterated until equilibrium was reached, i.e. until the energy of the system reached a minimum.

The bending rigidity of a membrane as sketched in Figure 11.4 is given by whereE₁,E₂ andν are the3-dimensional elastic moduli of the tail- and head group region, respectively, and the 3-dimensional Poisson ratio. Therefore the change in bending rigidity∆κ caused by the additional hydrogen bonds can be written

∆κ≈ h²δ 2

∆E₂

1−ν² (11.43)

where δ h was used. This expression scales with the square of the mem-brane thickness. So although the increase in bending rigidity stems from an increase in the elastic modulus of the head group region only, this effect is strongly enhanced by the length of the hydrophobic tails similar to a lever arm principle.

To define an order of magnitude for the spring constant k₁ we consider a Lennard-Jones potential with a binding energy of40meV (the strength of a typical hydrogen bond) and an equilibrium spacing of 0.8 nm (the typical spacing of two molecules). This sets

k₁ = 4.5 eV/nm². (11.44) Furthermore it is supposed that the other type of springs between A−B and B−B heads is much smaller – or that there is no binding at all. In the simulations presented in this thesis it was set k₂ :k₁ = 0.0001.

The simulations presented in this thesis are all done on a 2-dimensional membrane, accordingly the 2-dimensional stiffness matrix of the membrane is evaluated. The elements of the stiffness matrix of a corresponding 3-dimensional material can be found by multiplication with the membrane

11.3. Elastic properties of membranes 131

thickness. Also the elastic modulus and the Poisson ratio are defined differ-ently in2 than in3 dimensions:

ν = C₁₂

C₁₁+C₁₂ C₁₁=δC₁₁^3d=δE 1−ν

(1 +ν)(1−2ν). (11.45) Insertion into equation (11.43) leads to the final result

∆κ= h²

2 Y. (11.46)

Since the simulation results showed that the structure shows isotropic symmetry for all concentrations (i.e. C₁₁−C₁₂−2C₆₆ ≈0), the use of only one elastic modulus and Poisson ratio to describe the membrane is justified.

Furthermore it was shown that the Cauchy relation C₁₂ = C₆₆ Born and Huang [1988] does not hold in general. This is not too surprising, since it was shown that the Cauchy relation is strictly only obeyed in a system with purely central forces and if each lattice point is center of symmetry.

Due to the inhomogenities in the structure of the membrane, this symmetry condition is surely violated.

Figure 11.6 shows ∆κ and the Poisson ratio for various concentrations and for two different temperatures: J/kT = 1 and J/kT = 0, i.e. a random configuration of molecules. The length of the tail region was chosen 3 nm Zemb et al. [1999], leading to ∆κ ≈ 22 eV for a material consisting of only strong A−A bonds.

The strong decay of the bending rigidity in a small concentration range is clearly visible. The bending rigidity falls over 4 orders of magnitude from a value of approximately1000kT (which gives extremely stiff, flat membranes) to 0.1 kT, a value which lies well below the room temperature limit, thus resulting in very soft and flexible structures. There is not much difference in the behaviour for both, the ordering case as well as the random structure.

The only difference is that the ordering tendency leads to a more pronounced isolation of atoms than in the random case, which in turn leads to a bit faster decrease in bending rigidity for the ordered than for the random configura-tion.

In contrast to the bending rigidities, which show a very similar behaviour for the two simulation runs, the Poisson ratio exhibits remarkable differ-ences. While the Poisson ratio shows only slight changes for the random configuration (an increase from 0.33 at the end points to approximately 0.4 for concentrations of 0.7), in the ordering case it exhibits a pronounced peak close to 1for concentrations of 0.66. This behaviour is related to the super-structure of the ordered configuration, which forms a hexagonal super lattice

132 11. Linear Elastic Networks

Figure 11.6: The bending rigidity∆κ(top) and the Poisson ratio for various concentrations and two different temperatures.

for atomic compositions of1 : 3 (which can be clearly seen in the middle of Figure 11.5). Since the elastic modulus of a hexagonal lattice is exactly zero (the bonds may follow the elongation by a simple re-alignment, they do not have to stretch), the corresponding Poisson ratio has to take the value of 1 (see equation (11.16)), which corresponds to a deformation without volume change (in 3-D the corresponding value of the Poisson ratio is1/2).