Severin, , Nikolai: Molecular Dynamics Simulations of Polymers and Micelles at Interfaces

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Chapter 4. MD simulation of cylindrical and half-cylindrical micelles in water and at solid liquid interfaces.

4.1. Introductory remarks.

In the second part of this work MD calculations of aggregates of amphiphilic molecules at sold-liquid interfaces are presented. Amphiphilic molecules by definition are composed of two parts, hydrophobic and hydrophilic. These opposing properties in the same molecule are the origin of micro-phase separation of amphiphilic molecules and solvent when dissolved in a polar solvent like water. It is well known that amphiphilic molecules form a wide variety of different aggregates like spherical micelles, bilayers or even more complicated structures of interpenetrating networks of amphiphilic molecules and solvent18. As amphiphilic molecules are very important in industry and biology, self aggregation of amphiphilic molecules and the structure of their aggregates was investigated intensively. Most of experimental data lead to a qualitative picture of a micelle. A few experimental techniques give as well structure of a micelle quantitatively. But due to a limited resolution some aspects of the structure of micelles are not clear.

While the self aggregation of amphiphilic molecules in aqueous solution is fairly well understood, it is not clear how these aggregates are affected by the presence of a solid surface. The surface force apparatus provide quantitative measures of adsorption, but little information on the aggregate structure. Neutron reflection is a very powerful tool to probe the structure of adsorbed aggregates. Substituting of hydrogens from different parts of amphihpilic molecule by deuteriums allows to obtain high resolution in the direction perpendicular to the substrate surface. The drawback of neutron reflection is that it does not resolve the adsorbed structure in the plane parallel to the surface. Another powerful tool for the investigation of structures formed by amphiphilic molecules on different substrates is scanning force microscopy (SFM). In contradiction to the neutron reflection SFM provides good resolution in the plane parallel to the surface and relatively poor resolution in the direction normal to the substrate surface. A combination of these two experimental methods allows to propose a picture of the adsorbed structure.

MD simulation is another powerful mean to obtain detailed insight into the structure and molecular behaviour of amphiphilic aggregates. The advantage of the MD method is that it allows to track the position of each atom in the system with time, so that it allows to simulate and predict the exact picture of micelles. At present time MD simulation has several limitations. First, the MD method allows to simulate the evolution of a micelle on a very short time period,


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usually several nanoseconds. Second, MD allows to simulate only relatively small molecular systems, consisting of thousands of atoms. In order to simulate the evolution of micelles on longer time scales some simplifications could be made. Starting from the models with a unification of atom groups like CH2 or H2O and ending with the models that simulate just main properties of amphiphilic molecules - hydrophobic and hydrophilic parts. Simplifications allow to simulate larger amphiphilic systems on much longer times scales, even selfaggregation of amphiphilic molecules form aqueous media. But, of course, each simplification leads to the loss of some information about a simulated molecular system. That is why in order to get a detailed picture of an amphiphilic aggregate one should use an atomistic MD simulation.

Neutron magnetic resonance (NMR) measurements of correlation times in the micelles show that fast correlation times are in the picosecond range. That is why it is supposed that a micelle should equilibrate its structure during the MD simulation on several nanoseconds. But as the characteristic times of self aggregation are much longer, it is clear that the simulated structure would depend highly on the starting configuration.

Atomistic MD simulation was performed in order to obtain the detailed structure of the aggregates formed by trimethyl ammonium bromide (TAB) molecules on different substrates. Structures of TAB aggregates on hydrophobic and hydrophilic substrates suggested by SFM were adopted to use as the starting structures for MD simulation. Half cylindrical micelles on gold and paraffin and cylindrical micelle on gold was simulated. Cylindrical micelle in water environment was also simulated for two reasons: first to compare the simulation data with experimental data in order to establish the reliability of the simulation, and second to compare the structure of the cylindrical micelle in water with adsorbed aggregates in order to find out how the presence of the substrate affects their structure.


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4.2. MD simulation

4.2.1. Amphiphilic molecules

The separation of two immiscible molecular components from their liquid mixture is described by the fundamental thermodynamic equations of self-assembly8. Equilibrium thermodynamics requires that in a system of molecules forming aggregated structures in solution like micelles the chemical potential of all identical molecules in different aggregates should be the same. For aggregates of the same type but with a different total number M of molecules this may be expressed as:

M=1,2,3...N... eq.1

where µM is the mean chemical potential of a molecule in an aggregate of aggregation number M and XM is the concentration of these molecules in the aggregate. The first term of this formula

is the standard value of the chemical potential (the mean interaction energy per molecule), the second term is the loss of entropy by aggregation. Combining of these equations for M=1 and M=N yields:

eq.2

This equation together with the conservation relation for the total concentration in the solution defines the system totally.

When remains constant for aggregates of different size ( = ) equation 2 becomes . Since X1<1 we must have XN<<X1. Hence most molecules will be in the monomer state. Aggregates will be formed when there is a difference of cohesive energy for the molecule in the dispersed phase and in the aggregated structures. For sufficiently low monomer concentration X1 such that is less than unity we have essentially the dispersed phase of molecules. With increasing monomer concentration micelle formation becomes more favourable. The monomer concentration (X1)crit at which this occurs is called the critical micelle concentration (CMC).


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The self aggregation of amphiphilic molecules is also described by the fundamental thermodynamic equations of self-assembly. The fundamental thermodynamic equations of self-assembly describe the equilibrium of all types of aggregates including phase separation. On the other hand the dual nature of amphiphilic molecules does not allow macroscopic phase separation. The head groups prefer to stay in contact with a polar solvent, while tail groups tend to minimise contact with solvent. This leads to the formation of aggregates where head groups are mostly concentrated on the surfaces of aggregates and tails pack in the aggregate’s core.

It is well known that amphiphilic molecules may aggregate in a wide variety of different aggregate types like spherical micelles, lamellas, bilayers or even more complicated structures of interpenetrating networks of solvent and amphiphilic aggregates (Fig. 10, the figure is adopted from the book of Israelachvili8).

Figure 10 Examples of structures formed by amphiphilic molecules.

In most amphiphilic molecules an alkyl chain is used as hydrophobic tail. Fully saturated alkyl chains are very flexible. Each C-C bond in a fully extended


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alkyl chain adopts a trans state but each bond can also exist in two other conformations, the gauche+ and gauche- states, which can be formed with the amount of 0.8 - 1.0 kT energy at room temperature. Therefore, a fully saturated alkyl chain possesses a large number of accessible states. On the other hand n-alkanes are found to be in a crystalline state at room temperature. One may consider that in an aggregate the chains could also pack in an array of parallel all-trans chains, and the head groups are concentrated on two of six faces of a crystalline block and on four faces hydrocarbon chains would contact with water. Hence, it is impossible to form such an array without energetically expensive hydrocarbon-water contact. Therefore we should expect that the alkyl chains in the interior of a micelle are conformationally disordered (liquid like). This expectation is supported by NMR measurements of fast correlation times and order parameters for alkyl chains in micelles4. The motions of alkyl chains in a micelle can be classified by two types: fast local motions in the alkyl chain and slow motions, ascribed to tumbling of the whole aggregate and/or diffusion of the amphiphilic molecules in the micelle. The measurements reveal fast intra-chain motions with correlation times around 10-11 s, a value in close agreement with the correlation times calculated from similar measurements on liquid n-alkanes.

It is important to describe more detailed NMR experiments, by comparing results of MD simulations with the correlation times measured by NMR. NMR measures spin relaxation of the nuclei in the external magnetic field and provides information on the rotational motion of the atoms. In MD simulations it is better to express atom rotation in terms of rotations of the bonds, for example rotation of C-H bond in the alkyl chain. This rotation is characterised by the auto-correlation function of bond orientation: , where is the bond vector. In general a time relaxation function C(t) can be represented as a sum of exponentials: , where each decay term represents different types of motion. For example in the case of amphiphilic molecules in a micelle orientation changes of C-H bonds can be due to transition between trans and gauche conformations in the alkyl chain (fast modes) or due to rotational motions of the micelle as a whole and diffusion of the heads of the amphiphilic molecules on the micelle surface (slow modes). Fast modes are in the range 1-100 ps and slow modes are in the nanosecond time regime. It is also interesting that it is possible to distinguish whether carbon atoms in the methyl group or in the alkyl chain. It is even more important that it is possible to distinguish carbon atoms in alkyl chain i.e. how far they are from the terminal group. Measurements of fast correlation times in amphiphilic micelles show that correlation times for the terminal head groups of alkyl chains are almost the same as the correlation times for the head groups of the same amphiphilic molecules in the dispersed state. The values of the fast correlation times are


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increasing considering carbons closer to the amphiphilic head group. The alkyl chains are anchored to the bulky head groups and therefore carbon atoms closer to the head groups loose their mobility.

The very low solubility of water in the bulk n-alkane phase (about one water molecule for each 2x104 CH2 groups5) allows to predict a negligible water penetration into the aggregated micelle hydrophobic core. Indeed this point of view was supported by NMR relaxation measurements6. Claims of experimental evidence for extensive water penetration in the core were shown to be unfounded7. NMR may distinguish between CH2 group in contact with water molecules and CH2 group in the hydrophobic core composed of alkyl chains. It was shown that there is extensive contact between CH2 groups and water molecules, that is why it was concluded that there is extensive water penetration in the micelle core. On the other hand it was shown later that NMR results may be explained by fact that water molecules should contact with CH2 groups from the amphiphilic molecules in dispersed phase and in addition CH2 groups on the edge on the micelle contact with water molecules. It was shown on the basis of NMR experiments that water does not mix with alkyl chains, but there is no evidence that water can not be found in the centre of the micelle. However neutron diffraction studies8 give the evidence that water is totally excluded from the micelle core.

Accepting the above arguments, lead to the picture that the micelle core can be considered as a non-polar liquid droplet consisting (almost exclusively) of fully saturated alkyl chains. The packing density of the chains is determined by the Van der Waals forces and chain etropy. As the chains are chemically identical to n-alkane chains, we should expect a packing density almost identical to that found for bulk n-alkane.

The standard picture of amphiphilic micelles now may be summarised in three simple points9:

  1. on average, each amphiphil molecule associated with a micelle has almost all of its hydrocarbon chains in the micelle core;
  2. the ionic head-groups, ions and water are almost completely excluded from the micelle core;
  3. the amphiphil chains are conformationally disordered (liquid like) and fill the core at approximately liquid n-alkane density throughout.

The standard picture implies that the chain-water interface is sharply defined (with an average roughness in the order of a few Å) and that the head-groups are just outside the micellar core in a layer, again with a roughness of a few Å.


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While self aggregation of amphiphilic molecules in free solution is well understood. It is not clear how these aggregates are affected by the presence of a solid boundary surface. Recently the pioneering work of scanning force microscopy (SFM) imaging of surface aggregates formed by tetradecyl trimethylammonium bromide (TAB) molecules was performed2:

The SFM method gives detailed resolution in the plane parallel to the surface but can not resolve the thickness of an adsorbed layer. The surface force apparatus8 allows to determine the thickness of the adsorbed layer with 0.5 Å accuracy. Hence combination of SFM imaging with surface force apparatus measurements allowed to define the shape of the structures formed by TAB molecules on different substrates. It was suggested that TAB molecules self-assemble in aggregates like spherical micelles, half cylindrical micelles or cylindrical micelles depending on the substrate.

The SMF images of the structures formed by TAB molecules on the hydrophobic substrates (graphite) showed parallel stripes, whose widths are 4.7±0.3 nm - roughly twice the length of an amphiphilic molecule and oriented perpendicular to the substrate symmetry axis. Hence alkyl tails that are supposed to be packed perpendicular to the half-cylinder symmetry axe are oriented parallel to substrate symmetry axes. The height of this structures is roughly the length of TAB molecule. These results are in a good agreement with model of half-cylindrical aggregates. The substrate symmetry axes are defined as lines connecting nearest neighbour atoms from different simple crystalline cells.

The SFM images of the structures formed by TAB molecules on hydrophilic substrates like mika show meandering stripes, which are separated by a distance slightly more than twice the length of TAB molecules, fore axample 6.0±0.5 nm on mika. Stripes did not follow consistent orientation relative to the underlying substrate lattice. The height of these aggregates determined with surface force apparatus was roughly twice the length of fully extended TAB molecule. Full cylindrical micelles model was proposed. It was supposed that cylindrical micelles may be flattened at the substrate bottom to allow a better contact area between head-groups and the surface. Flattening of the cylinders may also explains why the spacing between full cylinders is slightly higher than the spacing between half cylinders.

The adsorption of amphiphilic molecules from solution onto metals received some attention in the past as a method of controlling electrochemistry1


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at electrode surfaces and a way of limiting corrosion of metal surfaces. AFM scanning of structures formed by TAB molecules and TAOH - where the Br- ion is substituted by OH- group - on gold was performed9. It was suggested that TAB molecules form full cylindrical micelles on gold substrate, while TAOH molecules form half cylindrical micelles. This phenomenon was explained by the presence of halide ion in TAB molecule. It is well known that gold can strongly adsorb both organic molecules and halide ions from aqueous solution. It was supposed that during adsorption of TAB molecules on the gold surface halide ions adsorb preferentially on the gold surface producing a hydrophilic surface. Then the TAB molecules can form full cylinders to bring the head groups in contact with adsorbed Br- ions.

The SFM method provides valuable information on the shape of the adsorbed aggregates of amphiphilic molecules, but it does not provide any information either on the detailed structure i.e. the organisation of the amphiphilic molecules in these aggregates nor on the dynamic properties. There are few experimental techniques which provide information on structures adsorbed on the surface. Neutron reflection experiment provide high resolution in the direction perpendicular to the adsorbate surface but do not yield any structural information in the plane parallel to the surface. It is interesting to combine neutron reflection with SFM results because SFM on the other hand provide very good resolution in the plane of the surface but very poor resolution in the direction normal to the surface. Neutron reflection experiments were performed on the structures formed by TAB molecules on silica. It was suggested that TAB molecules form bilayers with 32±1 Å thickness, while the length of fully extended TAB molecule is 25 Å. That is why the model of a bilayer with interpenetrating tails was proposed. It was also found that about 10 % of the bilayer is occupied with water. The SFM imaging of structures formed by TAB molecules on silica showed circular structures. Hence the model of spherical aggregates was proposed on the basis of SFM data. Combining neutron reflection with SFM imaging one may propose spherical aggregates very pressed toward the surface. Experimental methods like nuclear magnetic resonance or neutron scattering provide information on bulk materials but can not determinate signal from the thin layer of amphiphilic molecules at the solid-liquid interface beyond noise.


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4.2.2. Model

The present limitation of MD simulation is the length of simulation time. The characteristic times for aggregation of amphiphilic molecules are in the seconds time range or even longer. MD simulation allow to trace the time evolution of a system on a nanosecond range only. It is therefor impossible to see the aggregation of amphiphilic molecules from a polydispersed phase. That is why one should create these molecular systems in conformations which are closely related to experimentally established molecular structures. The SFM experiments on imaging of structures formed by TAB molecules on different substrates were taken as the starting point for computer simulation.

Four conformations were constructed and considered as starting structures for MD simulation: 1. a cylindrical micelle of TAB molecules in water environment, 2. a half cylindrical micelle on gold substrate in water environment, 3. a half cylindrical micelle on paraffin substrate in water environment and 4. a cylindrical micelle on gold substrate in water environment.

1. Cylindrical micelle of TAB molecules in water environment:

Figure 11 Starting structure for cylindrical micelle in water environment.


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The diameter for the starting structure of a cylindrical micelle was chosen to be the same as the diameter of a half cylindrical micelle of TAB molecules on a graphite substrate2, i.e. 4.8 nm. The head groups of TAB molecules were placed on the micelle surface, while the tails of the TAB molecules were completely stretched toward the centre of micelle. The cylindrical micelle constructed this way exhibits a low density at the periphery and a high density in the centre, due to the initially stretched tails. The average density of the TAB molecules in the micellar system was set to 0.9 g/cm3 resulting in 0.81 g/cm3 average density of the micelle core composed of alkyl chains. The length of cylindrical micelle was set to 25 Å with the use of periodic boundaries. The micelle was placed in water environment with distance of the micelle edge to the simulation cell border of 5 Å and with nearest distance of micellar periodic image of 10 Å (Fig 11.). It is important to notice that during MD simulation with constant pressure condition size and hence density of the system could change. The pressure was controlled as stress with diagonal elements of stress tensor equal to pressure and zero non-diagonal elements. Cell sizes could vary independently during MD simulation.

The cylindrical micelle of TAB molecules in water environment was subjected to energy minimisation and then MD simulation was applied. For the first 100 ps of MD simulation NVT (Number of particles, Volume and Temperature were constant) conditions were chosen. It was important to choose constant volume, because it allowed to equilibrate the high density in the centre of the micelle. Constant pressure conditions has lead to an instability of MD simulation - high energy deviations. Then the structure was equilibrated at the temperature 300 K for 200 ps under NPT conditions (Number of particles, pressure and temperature were constant) and subject to simulated annealing for another 100 ps at a temperature of 500 K. A pressure was set to 1 bar. After annealing of the cylindrical micelle MD simulation was done with NPT conditions at a temperature of 300 K for 1.6 ns. In total the cylindrical micelle was simulated over 2 ns.

2. Half cylindrical micelle of TAB molecules on the gold substrate

The diameter for the starting structure of the half cylindrical micelle was set to 4.6 nm, which is the same as the experimentally established width of half cylindrical micelles formed by TAB molecules on the graphite substrate2. The height of the half cylindrical micelle was set to half of the value of the micelles width. The gold substrate was constructed with (111) contacting surface, and was 10 Å thick. The head groups of TAB molecules were placed on the micelle surface, while tails of TAB molecules were stretched toward the micellar centre. The average density of the amphiphils in the micelle was set to 0.9 g/sm3. The surface of the half cylindrical micelle, in contact with the gold substrate, was


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constructed as the layer of TAB molecules. The molecules were placed parallel each other and parallel to the surface of gold. Alkyl tails of TAB molecules were oriented parallel to a gold symmetry axe. The system of gold substrate and half cylindrical micelle was placed in water. Distance between the top of the micelle and periodic image of the gold surface was set to 15 Å (Fig 12.).

Figure 12 Starting structure for half cylindrical micelle on gold substrate in water environment.

The half cylindrical micelle on the gold substrate in a water environment was energy minimised and then a MD simulation was started. The system was equilibrated at a temperature of 300 K for 50 ps under NVT conditions and for the next 50 ps under NPT conditions. The half cylindrical micelle was then annealed for 100 ps at a temperature of 500 K. After annealing of the molecular system a MD simulation was performed with NPT conditions at a temperature of 300 K and 1 Bar pressure for 1.8 ns.

3. Half cylindrical micelle of TAB molecules on paraffin substrate

In order to save CPU time used for the equilibration of the molecular system the output structure after a 500 ps of MD simulation of the half cylindrical micelle of TAB molecules on the gold substrate was used to construct a half cylindrical micelle on a paraffin substrate. The gold substrate


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was replaced by paraffin substrate. The (100) crystallographic plane of paraffin was chosen to contact with the half cylindrical micelle. The paraffin substrate was placed with matching of its symmetry axe with the symmetry axe of the gold substrate. The water molecules were not changed.

The half cylindrical micelle on the paraffin substrate was energy minimised and then treated by MD simulation. The structure was first equilibrated at a temperature of 300 K for 100 ps and then simulated annealing at 500 K temperature was done for 100 ps time. After annealing of the molecular system MD simulation with NPT conditions at the temperature 300 K was performed for 1.3 ns. Paraffin substrate was kept fixed.

Cylindrical micelle of TAB molecules on gold substrate in water environment.

In order to save CPU time the atomic coordinates after a 500 ps of MD simulation of the cylindrical micelle of TAB molecules in water was used to construct a cylindrical micelle on the gold substrate. For that purpose the water environment was partially removed and replaced by the gold layer with a (111) contacting crystallographic plane as contact surface. The cylindrical micelle was put about 3 Å above the gold surface. The layer of gold was constructed with 10 Å thickness. An additional layer of water of 5 Å thickness was added in between the cylindrical micelle and the periodic image of gold in order to prevent contact of cylindrical micelle with the periodic image of the gold substrate.

Cylindrical micelle on the gold substrate in water solution was energy minimised and then treated by MD simulation. The molecular structure was equilibrated at a temperature of 300 K for 100 ps of time and then annealed at 500 K for 100 ps. After annealing the molecular system was subjected to MD simulation with NPT conditions at 300 K for 800 ps.


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4.2.3. Results

Cylindrical micelle in water environment.

A snapshot of the structure of the cylindrical micelle of TAB molecules in water environment after 2 ns of MD simulation is presented in figure 13.

Figure 13 Cylindrical micelle in water environment after 2 ns of MD simulation.

The initial cell size of the system of was 60x60x25 Å and the micelle had a diameter of 49 Å and was 25 Å long. The length of the micelle matched with the cell size in z direction. The micelle diameter was determined by the overall distance of the bromine ions. The MD simulation did not change cell sizes significantly; the cell size fluctuated on opposite positions around initial values, and after 2 ns of MD they were 59.8x59.8x25.1 Å. The diameter of the micelle also did not change drastically; after 2 ns of MD it increased from 49 Å to 50 Å. It is easy to define the diameter for the starting structure of cylindrical micelle. However it is difficult to measure it after MD simulation since the surface roughness of the micelle is of order of several Å, and can thus lead to uncertainties of the micelle diameter in the order of 1-2 Å. Interestingly the change from 49 Å to 50 Å results in a change of the micelle core density


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composed of the hydrophobic alkyl chains from the initial value of 0.81 g/cm3 to 0.74 g/cm3, as compared to the density of liquid N-alkanes which is 0.63 pentane, 0.7 octane and density of amorphous polyethylene is 0.855 g/sm3. While the shape of the micelle and size did not change drastically, the micelle structure changed significantly. Initially stretched alkyl chains adopted a random orientations during the minimisation and first several hundreds pico-seconds of MD simulation. The order parameter that characterise the regularity in the structure of the micelle core will be introduced later.

The most striking feature of the structure of the MD simulated cylindrical micelle is the inhomogeneous density of the hydrophobic micelle core composed of the hydrophobic alkyl tails. The alkyl chains did not fill completely a centre region of the micelle of about 4±1 Å in diameter. A plot of the micelle core density versus the distance from the micelle centre is presented in figure 14a. The density profiles were averaged for structure snapshots made each 25 ps over the last 500 ps of MD trajectory.

Figure 14 Density distribution functions in cylindrical micelle in water plotted vs the distance from the micelle centre: a) full-square : density of the hydrophobic core composed of alkyl chains; b) full-delta : head groups ( ), c) - density of the Br- ions; d) • - density of the water and e) --- - density of the total molecular system of the cylindrical micelle of TAB molecules in water.

Clearly during 2 ns of MD simulation water did not diffuse into the micelle core (Fig 14a and 14d). Instead the water formed a perfect shell around the micelle. No water tentacles or single water molecules were penetrating into the micelle core. The roughness of the water shell corresponds to the roughness of the surface of the cylindrical micelle which is roughly 2-3 Å. The head groups and


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Br- ions sit between the hydrophobic core and water (Fig 5b and 5c respectively). Br- ions stay attached to the surface of the micelle (Fig 14b and 14c).

Figure 15 Pair distribution functions of the Br- and N+ ions in cylindrical micelle. a) solid line: pair distribution function for initial configuration of the subset of Br- and N+ ions (Br - Br, N - N and Br - N pair distribution functions), the first maximum is the Br - N distance in the dipole and second is the Br - Br or N - N distance, hence inter-dipole distance; b)-e) pair distribution functions after 2 ns: b) dotted line: Br - N pair distribution function; c) dashed line: N - N pair distribution function; e) dashed-dotted line: Br - Br pair distribution function.

Initially the micelle was constructed with Br- - N+ dipoles forming regular rows oriented along the micelle axe. After 2 ns of MD simulation the rows of Br- - N+ ions still can be recognised on the surface of the micelle. Figure 15 demonstrates the arrangement of Br- - N+ dipoles. The curve 6a presents the pair distribution functions of Br- and N+ ions for the initial configuration. The first maximum corresponds to the Br- - N+ dipole distance and the second to the inter-dipole distance in a row. The curves 6.b-e present the pair distribution functions of Br- and N+ ions for the micelle after 1 ns of MD simulation. We may see that both maximums can still be recognised well demonstrating the clustering of Br+- N- dipoles. In order to obtain the pair distribution function for the MD simulated structure, snapshots of the half cylindrical micelle on paraffin were saved every 0.5 ps and then pair distribution functions for each snapshot were averaged over the last 500 ps of the MD trajectory. Diffusion coefficient for the Br- - N+ ions is lower then 0.08 Å2/ns. The diffusion coefficient was determined from the mean square displacements of the Br- and N+ ions: , where D is the diffusion coefficient and are the spacial positions of the ions.


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To compare the dynamic of alkyl chains in the micelle core from NMR measurements of fast correlation times4, , fast correlation times were estimated from the auto correlation functions of C-H bond orientation: , where is the vector of C-H bond orientation. From the definition of the auto-correlation function it follows directly that , where alpha(t) is the angle between the bond vector at time zero and the bond vector at time t. For simple systems like a solvent of small molecules as for example water. The bond auto-correlation function displays a single exponential decay form , where tau is the characteristic time for the average rotational motion of water molecules. In general, C(t) decays as a sum of several exponentials, where each term represents different types of motion of the C-H bonds. For example in the case of amphiphilic molecules packed into a micelle the C-H bond can loose its initial orientation: 1) due to the changing of C-C-C torsional angle in the alkyl tail (trans-gosh transitions) 2) due to the diffusion of the head groups on the surface of a micelle and the rotation of the micelle. The characteristic time for the first type of motion is in the pico-second time regime - fast correlation times, while the second type is in the nanosecond time regime - slow correlation times. Because of the large difference between slow and fast correlation times we may determine a fast correlation time by fitting the correlation function on the time of 50-100 ps using a single exponential, where A represents slow motions and is almost independent on time scale of pico-seconds.

Figure 16 Correlation functions of the terminal methyl groups that are picked: a) from the centre of the micelle and b) from the micelle surface. The dashed line is the best fit of the curve a) with decay law , and realxation time tau 2.4±0.2 ps.


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The main problem in the interpreting fast correlation times arises because in NMR experiments fast correlation times are plotted for the different carbon atoms vs their positions in the alkyl hydrophobic tails. The motion of carbon atoms in amphiphilic molecules in a dispersed phase depends on the position of the carbon atoms with respect to the bulky head group, carbons close to the head group move slower. The motion of carbon atoms in amphiphilic molecules packed in a micelle depends not only on their position in the alkyl chain but also on their position in the micelle. Parts of alkyl tails move faster in the centre of a micelle and slower, when they are close to the micelle surface, where their motion is hindered by the bulky head groups, which are mostly concentrated on the surface. Of course, the ends of alkyl chains have a higher probability to be in the centre of the micelle but they could be found as well on the surface of the micelle. Figure 17 presents correlation times for the carbons of the terminal methyl groups picked from the centre of the micelle 17.a and from the surface of the micelle 17.b. The position of carbon atoms in the centre or on the edge of the micelle was determined visually.

Figure 17 Correlation function for the terminal methyl groups that are picked a) randomly and b) from the centre of the micelle. The dashed line is the best fit of the curve a) with single exponent and the decay term tau (relaxation time) is 7.9±0.8 ps.

One can observe indeed that the relaxation times for these two cases are quite different: for the carbon atoms of the alkyl tails in the micelle centre tau=2.4±0.2 ps and for the case at the micelle surface where it is difficult to determine the correlation time exactly. The relaxation time is obviously much longer than 2.4 ps. The correlation function of carbons from all terminal methyl


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groups C(t) is an integral of the correlation function C(t,r) over all positions r of the carbons in the micelle with a weight given by the probability to find the carbon on the given distance from the micelle centre: . It is clear that C(t) cannot be described just by one decay exponential. Figure 18 presents the correlation function for the terminal methyl groups picked randomly within the volume of the micelle; the correlation function for the terminal methyl groups from the centre of the micelle is drawn for comparison. The relaxation time for the correlation function of the terminal methyl groups picked randomly is tau=7.9±0.8 ps.

NMR experiments give just one averaged relaxation time for each carbon atoms in the alkyl tail. Therefore in order to compare the MD simulation data with NMR data a single relaxation time was computed from the correlation functions by fitting it with a single exponential. Each correlation function was averaged over more than 30 randomly picked carbon atoms and averaged over the last 200 ps of the MD trajectory. The relaxation times were computed from the correlation functions by fitting them with a single exponential using the data from trajectories of 50 ps length. The correlation functions were calculated for different carbon atoms in the alkyl tails. It was possible to evaluate relaxation times just the last 5 carbon atoms in the alkyl tail, because carbon atoms close to the head group move too slow to determine their relaxation times on the time scale of the MD simulations (Fig. 18).

Figure 18 Relaxation times. Filled squares (full-square): correlation times for MD simulated cylindrical micelle of TAB molecules. Open circles (o): experimental relaxation times for PPALM lamella ( ). Open up-triangles (Delta): experimental relaxation times for PPALM spherical micelle.


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The order parameter of the cylindrical micelles also could be compared with experimental results. The order parameter is defined as , where theta is the angle between the surface normal and the C-H bond vector in an alkyl chain. NMR experiments yield the dependence of the order parameter on the position of a carbon atom in the alkyl chain. The modulus of the order parameter micelle is plotted on Fig 19 for MD simulated cylindrical. Experimental order parameters for spherical micelles and lamellas are added on the same figure.

Figure 19 Order parameters S open symbols - experiments, closed symbols - simulated data. Filled squares (full-square): order parameter from MD simulation of cylindrical micelle of TAB molecules. The dashed line is a line guide for eyes. Open circles (o): experimental order parameter for the lamella phase of SOBS molecules . Open triangles (Delta): experimental order parameter for the micelle phase of CTAC molecules

The dependence of the order parameter on the carbon number was calculated as on averaged over 10 snapshots of the last 300 ps of MD simulation. There are no experimental data for the order parameter of structures of amphiphilic molecules adsorbed on different substrates available. We have modified the order parameter for the MD simulated structures adsorbed on different substrates. It allows to show the influence of the substrate on the aggregated structures. The angle ominus was defined as the angle between the C-H bond and the normal to the substrate surface and it was calculated as the function of the distance from the surface. Figure 20 presents the modified order


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parameter Sm for the cylindrical micelle in water defined for the structures adsorbed on the different substrates. The substrate surface normal was defined as a vector perpendicular to the cylinder symmetry axis. Because of the isotropy of the vector normal to the cylinder surface, the order parameter was averaged over different orientations of the vector.

Figure 20 Modified order parameter Sm for a cylindrical micelle of TAB molecules plotted vs the distance from an edge of the micelle.

The small-angle neutron scattering technique (SANS) can provide information on the cross-section of structure of micelles in terms of a radial scattering density distribution, which is proportional to the averaged scattering lengths <b> of all elements in the volume involved in the averaging. The radial scattering density was calculated from MD simulations for cylindrical micelle by averaging of scattering lengths over atoms that fall in a volume element

, and then it was normalised to the volume

, which is proportional to r, the distance from the micelle centre. The plot of <b> versus the distance from the centre of the micelle is displayed in figure 21, for protonated micelle (a) and deuterated micelle (b). Each curve was averaged over more than 30 snapshots from the last 500 ps of MD simulations. The difference between radial scattering densities form protonated and deuterated micelles is explained by the large difference between the scattering lengths of hydrogen and deuterium: bH=-0.374 and bD=0.667 . The scattering length of carbon bC=0.665 is almost twice the scattering length of hydrogen and has the opposite sign. That is why scattering density of CH2 groups is almost zero and the scattering density of a protonated micelle is determined mostly by distribution of the terminal methyl groups. The scattering length of deuterium is positive and the scattering density of deuterated


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micelle is much higher than for a protonated one and better reflects the actual micelle mass density.

 

CH2

CH3

CD2

CD3

|<b>|

0.083

0.457

1.999

2.666

Figure 21 Radial scattering density distribution function for a cylindrical micelle of TAB molecules simulated in water. a) rhomb - micelle with hydrogens, b) full-square - all hydrogens were substituted by deuteriums. Both curves where scaled so that curve b) would be unity at its maximum.

The calculated scattering density for a protonated cylindrical micelle is highly fluctuating. The reason is almost zero scattering density. In order to demonstrate that the scattering density of a protonated micelle is determined mostly by the distribution of terminal methyl groups, the scattering density of a protonated micelle and the distribution of terminal methyl groups are plotted in one figure (Fig. 22). Just one experimental paper was found where the scattering cross section of a protonated cylindrical micelle was deduced from a SANS experiment14. The scattering cross section of the cylindrical micelle formed by egg yolk lecithin molecules was calculated from neutron scattering intensities. The structure of the lecithin molecule is quite complicated: it has a complicated head group, two alkyl tails with C15 and C17, which is comparable to the length of the alkyl tail in TAB molecules. The fact that lecithin molecules form cylindrical micelles allows to suggest that the structure of the hydrophobic core of the cylindrical micelle formed by TAB molecules and by lecithin molecules


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should be compared. That is why to figure 22 also the scattering cross section of a protonated cylindrical micelle of lecithin molecules was added.

Figure 22 a) rhomb- radial scattering density for MD simulated cylindrical micelle, b) •- radial distribution of the terminal methyl groups for MD simulated cylindrical micelle and c) - - radial scattering density deduced from neutron scattering from cylindrical micelles formed by egg yolk lecitin14. All profiles were scaled in order to fit to one graph.

It is interesting to notice that the scattering cross section of the MD simulated cylindrical micelle and the experimental scattering cross section of the cylindrical micelle formed by lecithin molecules grow from the edge of the micelle to the centre. The MD simulated scattering cross section peaks at about 5-6 Å from the micelle centre and drops to zero scattering density at the micelle centre, while the experimental scattering cross section bends at the distance 5-6 Å from the centre and has a maximum at the micelle centre.

SANS measures the distribution of terminal methyl groups in the micelle, which is defined as the probability density to find a terminal methyl group at a given distance r from the micelle centre P(r). If the probability to find the terminal methyl group on the r distance from the micelle centre is ?(r), then the volume weighted probability density for a cylindrical micelle is P(r)=rC?(r), where C is a normalisation constant. The probability density for the terminal methyl groups for a cylindrical micelle of TAB molecules calculated for a MD simulated structure is plotted in figure 23. The data were averaged over 7


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snapshots from the last 800 ps of MD trajectory. Averaging over more snapshots did not make sense as the distribution of the terminal methyl groups does not change significantly over 100 ps.

Figure 23 Volume weighted probability density P(r) versus the distance from micelle centre r. Full circles connected by the dotted line are the data of a MD simulation of a cylindrical micelle of TAB molecules. The solid line is the FFT smoothing.

Experimental data for P(r) of cylindrical micelles, could not be found, only data for spherical micelles formed by LDS molecules ( ) were available. The radii of a LDS micelle and a MD simulated TAB cylindrical micelle are comparable: the mean radius of LDS micelles is 18.9 Å, while the mean radius of a MD simulated cylindrical micelle is roughly 20 Å. In order to compare results of MD simulation for a cylindrical micelle with experimental results for the spherical micelles, the data of the MD simulation were corrected for the shape of the micelle and for the polydispersity of the micelle sizes in the experiment. It was established that the experimental data could be explained taking into account the polidispersity of the micelle sizes approximated by a gaussian distribution. In order to correct the distribution of terminal methyl groups of the MD simulated structure for the polydispersity of micelle sizes it was assumed that the distribution of terminal methyl groups in micelles of different sizes could be scaled as , where R0 is the mean micelle radius and R is the current micelle radius and is the distribution for the micelle with the radius R0. Integration over all radii R with a weight given by


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a gaussian function gives the distribution of terminal methyl groups measured with the SANS experiment. The volume weighted probability has square dependence on the distance from the micelle centre for the spherical micelle and a linear dependence for the cylindrical micelle. The data of MD simulation with polydispersity correction were multiplied by the distance from the micelle centre to compare MD data for a cylindrical micelle with the experimental data on spherical micelles. MD simulation data of a cylindrical micelle of TAB molecules with polydispersity correction and shape correction is presented in figure 24.

Figure 24 Volume weighted probability density P(r). • - MD data corrected to polydispersity add to shape. ···· - FFT smoothing of MD data. - - the experimental data for micelles of LDS molecules ( ) with the radius of the hydrophobic core 18.9 Å. (For comparison the radius of the hydrophobic core of the MD simulated cylindrical micelle is roughly 20 Å.)

Half cylindrical micelle on the gold substrate in water environment.

As mentioned above, in order to save computational time and to avoid possible artefacts the substrates were fixed. In the case of half a cylindrical micelle on gold, the substrate was involved in the MD simulation for the first 200 ps. It stayed more or less unperturbed for 100 ps and then started to form domains, where in each domain crystallographic planes of gold were slightly rotated with respect to the same crystallographic planes in the other domains. Bulk gold was simulated to compare it with the thin layer of the gold substrate.


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The simulation of bulk gold was carried out for 100 ps and bulk gold was obviously much more stable, hence the instability of the gold substrate was induced by the influence of the amphiphilic aggregate from one side and water from the other side. After 200 ps gold substrate was replaced with the gold from the initial configuration and kept fixed.

The initial size of the amphilic aggregates on different substrates was set as close as possible to the experimentally measured sizes. The exact matching with experimentally established size was impossible because the cell size was dictated by the substrate crystalline structure. Fixing of the substrates fixed as well the two cell size dimentions defined by substrate, yet the third dimension was not fixed. For simplicity unfixed dimension will be called the z - axis. The initial value of the z cell size of the half cylindrical micelle on gold was set to 50 Å and it did not change significantly with time, after 1.6 ns of MD simulation it was 50.1 Å. The height of the half cylindrical micelle changed from 20.6 Å to 24±1 Å, with the error bar defining the fluctuation of the micelle height. The height was defined as the distance from the micelle atom that is the closest to the gold substrate to the micelle atom that is most far from the surface. The shape of half cylindrical micelle did not change significantly.

Figure 25 Snapshot of a half cylindrical micelle of TAB molecules on a gold substrate in water environment after 1.5 ns of MD simulation.

It is possible to definitely conclude that water molecules did not diffuse into the hydrophobic core of the half micelle composed of the alkyl chains, and


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that the alkyl chains did not leave the micelle core (Fig 26.a and 26.d). The head groups of the TAB molecules formed the surface of the half micelle (Fig 26.b) and the Br- ions stayed attached to the surface of the micelle, i.e. they did not diffuse into the surrounding water (Fig 26.c).

Figure 26 Density distribution functions for a half cylindrical micelle on a gold substrate plotted vs the distance from the geometrical centre of the first adsorbed layer of TAB molecules: a) full-square - density of the hydrophobic core composed of alkyl chains, b) full-delta - head groups ( ), c) - Br- ions, d) • - water and e) --- - total density of the system half cylindrical micelle of TAB molecules on gold substrate and water.

Unlike for the cylindrical micelle, Br- - N+ dipoles randomised their positions and dipole moments. In the pair distribution function of the Br- - N+ subset (Fig 27) the maxima that characterised inter-dipole distances and the distance between Br- and N+ ions at the starting configuration can not be recognised at all. The curve a) presents the pair distribution function of Br- and N+ ions for initial configuration. The curve b)-d) presents the pair distribution functions of ions for the MD simulated half cylindrical micelle. In order to obtain pair distribution function for the MD simulated structure, snapshots of the half cylindrical micelle on gold were saved every 0.5 ps and then pair distribution functions for each snapshot were averaged over the last 500 ps of the MD trajectory.


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Figure 27 Pair distribution functions of Br- and N+ ions in half cylindrical micelle on the gold substrate. a) solid line: initial configuration of the subset of Br- and N+ ions (Br - Br, N - N and Br - N pair distribution functions), the first maximum reflects the Br - N distance in the dipole and the second the Br - Br or N - N distance, hence an inter-dipole distance; b) dotted line: Br - N pair distribution function; c) dashed line: N - N pair distribution function; e) dashed-dotted line: Br - Br pair distribution function.

The reason for the randomisation of the Br- - N+ dipole positions during 1.6 ns in the half cylindrical micelle on gold in comparison to the cylindrical micelle in water being much higher mobility of the ions in the half cylindrical micelle. The diffusion coefficient for the Br- and N+ ions in the half cylindrical micelle is 1.3±0.3 Å2/ns. It was determined from the mean square displacements of the Br- and N+ ions: , where D is the diffusion coefficient and are the coordinates of the ions. Here it should be noticed that it is hard to define a diffusion coefficient for the Br- and N+ ions in a half cylindrical micelle on the gold substrate, because of the fast motion of the whole half cylindrical micelle. The motion of Br- and N+ ions consists of their internal motion in the micelle and motion of the micelle itself. As the bottom of the micelle stayed attached to the gold surface, the micelle motion should not contribute to the diffusion coefficient, but highly influences the mean square displacement of the Br- and N+ ions.

An explanation for the high mobility of Br- and N+ ions could be that the Br- ions lost their ionic bonds with the N+ ions and instead got bound to the gold surface. Since the Br- ions would have relative freedom along the surface of gold, and since the bulk dipoles are destroyed, the Br- ion and the charged head


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group move much faster. In this case the mean square displacements of Br- and N+ ions would be independent for the ions close to the surface and the mean square displacement of all ions also would be quite different. However the comparison of the mean square displacements of Br- ions and N+ ions show that they are almost identical. This lead to the conclusion that all ionic bonds are still active and the supposed explanation for the increased diffusion coefficient of Br- and N+ ions is not suitable.

The TAB molecules in the layer close to the substrate stayd moreover parallel to the surface of gold. The molecules further away from the surface of gold adopted more random configurations. It is interesting to notice that the hole in the centre of the half micelle was not formed, unlike in the cylindrical micelle in water.

In order to characterise the influence of the gold substrate on the structure of the half cylindrical micelle, the order parameter was calculated. The order parameter for the structures adsorbed on the surfaces was redefined in comparison to the order parameter for a cylindrical micelle in water. It was defined as:

, where the ? is the angle between the C-H bonds and the normal to the substrate surface. The order parameter was calculated as a function on the distance from the substrate surface (Fig 28).

The orientation of the TAB molecules parallel to the surface does not necessarily result in the orientation of the C-H bonds parallel to the surface normal and Sm=1. The TAB molecules are forced by the substrate to orient parallel to the substrate symmetry axes and adopt ideally all trans conformations. The alkyl chains with all trans conformation can lie on the substrate either perpendicular or parallel to the surface which lead to the two possible angles between C-H bond and the substrate normal: roughly 35° and 55°. The initial conformation was constructed with a 35° angle between the C-H bonds and the surface normal resulting in an order parameter of 0.46 for the first layer of TAB molecules. This is the maximum possible order parameter for molecules which are close to the surface and oriented parallel to it. The order parameter Sm was normalised to the maximum order parameter Smax=0.46 for gold substrate. The order parameter was averaged by calculating order for structure snapshot every 25 ps from last 400 ps of MD trajectory. The standard deviations from the mean or fluctuations of the order parameter are shown as error bars.


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Figure 28 Order parameter Sm/Smax for the half cylindrical micelle of TAB molecules on the gold substrate. The error bars denote the standard deviation from the mean. It is more important to display the standard deviation instead of the standard error of the mean determination as the standard deviation defines actual fluctuations in the system. The error of the mean determination is about four times lower than the fluctuations.

A layer of well ordered molecules close to the surface of gold can be detected. High fluctuations of the order parameter at a distance of 2-4 Å away from the gold substrate indicate a spacing between the first layer and the rest of the micelle. It seems possible to define a second layer of alkyl chains at a distance about 5 Å from the gold surface.

The maximum order parameter for the molecules close to the substrate surface is not a strictly defined parameter. Molecules close to the surface did not move far from the initial configuration. That is why the maximum order parameter for the first layer of alkyl chains is determined the starting configuration taking as a reference state. In general, alkyl chains close to the substrate surface may like to adopt conformations different from the starting conformations. That is why the order parameter is introduced, where ominus is defined as the angle between the CH bonds and a substrate symmetry axis. Again, Ss depends on the starting configuration. But in the case of a half cylindrical micelle on hydrophobic substrates it was established in the SFM work that half cylindrical micelles are oriented perpendicular to the substrate symmetry axes and hence the alkyl tails of the


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TAB molecules are oriented parallel to the symmetry axis. That is why it is reasonable to characterise the orientation of the alkyl chains with respect to the orientation of the substrate symmetry axes (Fig 29).

Figure 29 Order parameter Ss for a half cylindrical micelle formed by TAB molecules on a gold substrate.

The order parameter Ss for a chain oriented parallel to a substrate symmetry axis is -0.5. The order parameter for chains close to the substrate surface is -0.4 and 0.8 from the maximum order parameter. This is in excellent agreement with the Sm/Smax order parameter for chains close to the substrate surface. It is also important to notice that the dependence of Ss on the distance from the substrate shows the same features as the dependence of Sm/Smax on the distance from the substrate.

Half cylindrical micelle on paraffin substrate in water environment

Since the paraffin substrate was kept fixed, two cell dimensions of the half cylindrical micelle on gold did not change during MD simulation. The z cell dimension has changed from the initial value of 51.2 Å to 53 Å. The height of the half cylindrical micelle has changed from its starting value of 22.2 Å to 24.2±0.2 Å. The height was defined as the distance from the micelle atom that is the closest to the paraffin substrate to the micelle atom that is most far from the surface.


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Figure 30 Snapshot of the half cylindrical micelle of TAB molecules on the paraffin substrate after 1.2 ns of MD simulation

The structure of the half cylindrical micelle on the paraffin substrate is qualitatively the same as the structure of the half cylindrical micelle on the gold substrate. The water did not diffuse into the micelle core composed of hydrophobic alkyl tails (Fig 31 a and 31 d) . The hydrophobic tails did not leave the micelle core either. The Br- and N+ ions formed the surface of the half micelle. Br- ions did not diffuse into the water surrounding (Fig 31 b and 22 c). A hole (vacuum) in the centre of the half micelle was not formed, even though the density of the micelle core is still not homogeneous (Fig 31 a). The density dependencies are plotted vs the distance from the geometrical centre of the first adsorbed layer. The densities were averaged over half spherical slices with width given by the resolution of the curve. When the density of water was calculated it was taken into account that water cover half micelle partially and just parts of half spherical slices filled with water were taken into account. The layer of TAB molecules close to the paraffin surface stayed parallel to the


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substrate, while alkyl tails far form the surface adopted more random configurations.

Figure 31 Density distribution functions in half cylindrical micelle on paraffin substrate plotted vs the distance from the geometrical centre of the first adsrobed layer of TAB molecules: a) full-square - density of the hydrophobic core composed of alkyl chains, b) full-delta - density of the head groups ( ), c) nabla - density of the Br- ions, d) • - density of the water and e) --- - total density of half cylindrical micelle of TAB molecules on paraffin substrate and water.

The pair distribution function of the subset of Br- and N+ ions of the half cylindrical micelle on the paraffin substrate exhibit the same characteristics as the pair distribution function for the half cylindrical micelle on the gold substrate (Fig. 32). The maxima that characterise inter-dipole distances and the distance between Br- and N+ ions at the starting configuration are very broadened (Fig 32.b-d).


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Figure 32 Pair distribution functions of the Br- and N+ ions in a half cylindrical micelle on a paraffin substrate. a) solid line: pair distribution function for initial configuration of the subset of Br- and N+ ions (Br - Br, N - N and Br - N pair distribution functions); the first maximum is the Br - N distance in the dipol and the second is the Br - Br or N - N distance, hence an inter-dipole distance; b) dotted line: Br - N pair distribution function; c) dashed line: N - N pair distribution function; e) dashed-dotted line: Br - Br pair distribution function.

In order to obtain the pair distribution function for the MD simulated structure, snapshots of the half cylindrical micelle on paraffin were saved every 0.5 ps and then the pair distribution functions for each snapshot were averaged over the last 500 ps of the MD trajectory.

The order parameter for the half cylindrical micelle on the paraffin substrate was defined exactly in the same way as the order parameter for the half cylindrical micelle on the gold substrate:

the theta was defined as the angle between the C-H bonds and the normal to the substrate surface. The calculation of the maximum order parameter Smax for alkyl chains close to the paraffin substrate was based on the assumption that alkyl chains would tend to adopt a crystalline structure of paraffin (Fig 33).

It is interesting to notice that molecules close to the paraffin substrate have the same normalised order parameter as molecules on a gold substrate. However the layer of alkyl chains is smeared out as indicated by the width of the first maximum. The order of the half cylindrical micelle fluctuates highly on the


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paraffin substrate. The fluctuations of the order parameter are indicated with error bars. From the high fluctuations and the smearing out of the first layer on the paraffin substrate it is reasonable to conclude that TAB molecules are relatively strongly bound to the gold substrate.

Figure 33 Ration of order parameter to the maximum order parameter for the half cylindrical micelle of TAB molecules on the paraffin substrate. The error bars show the standard deviation from the mean. The error of the mean determination is about four times lower than the deviations.

The order parameter Ss is defined exactly in the same way as for the half cylindrical micelle on the gold substrate (Fig. 34).

Figure 34 Order parameter Ss for half cylindrical micelle formed by TAB molecules on paraffin substrate.


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It should be noticed that the order parameter Ss exhibit in general the same features as the ratio of the order parameter Sm/Smax. The order of alkyl chains close to the surface is -0.4 or 0.8 times the maximum order parameter -0.5. The first layer of alkyl chains is smeared out.

Cylindrical micelle on the gold substrate in water environment.

The gold substrate was kept fixed, causing that two cell dimensions of the system of a cylindrical micelle of TAB molecules on a gold substrate did not change during MD simulation. The z cell dimension has changed from the initial value of 69.7 Å to 68.1 Å. The height of the cylindrical micelle changed from 48.6 Å to 48 Å, the width changed from 49.7 Å to 51.4 Å. The height and width were defined by the positions of bromine ions.

Figure 35 Snapshot of a cylindrical micelle of TAB molecules on a gold substrate in water environment after 1 ns of MD simulation.


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The structure of the cylindrical micelle on the gold substrate was qualitatively the same as the structure of a cylindrical micelle in water. The water molecules did not diffuse into the hydrophobic core and the hydrophobic alkyl tails did not leave the micelle core (Fig. 36.a and d). The Br- and N+ ions formed the surface of the cylindrical micelle (Fig 36.b and c).

Figure 36 Density distribution functions for cylindrical micelle on a gold substrate plotted vs the distance from the micelle center of mass a) full-square - density of the hydrophobic core composed of alkyl chains, b) full-delta - density of the head groups ( ), c) nabla - density of the Br- ions, d) • - density of the water and e) --- - total density of the system half cylindrical micelle of TAB molecules on the paraffin substrate and water.

The pair distribution function of the subset of the Br- and N+ ions is presented in Fig. 37. The first maximum on the pair distribution function which characterises the Br- and N+ distances still can be recognised (Fig. 37.b), while the N - N pair distribution function that characterise the inter-dipole distance is smeared out (Fig. 37.c).


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Figure 37 Pair distribution functions of the Br- and N+ ions in a cylindrical micelle on a gold substrate. a) solid line - pair distribution function for initial configuration of the subset of Br- and N+ ions (Br - Br, N - N and Br - N pair distribution functions), the first maximum is the Br - N distance in the dipol and the second is the Br - Br or N - N distance, hence inter-dipole distance; b) dotted line - Br - N pair distribution function; c) dashed line - N - N pair distribution function; e) dashed-dotted line is the Br - Br pair distribution function.

In order to obtain the pair distribution function for the MD simulated structure, snapshots of the half cylindrical micelle on paraffin were saved every 0.5 ps and then the pair distribution functions for each snapshot were averaged over the last 500 ps of the MD trajectory.

The order parameter (Fig. 38) for the cylindrical micelle on gold substrate was calculated in the same way as the order parameter for the half cylindrical micelle on the gold substrate:

where the theta is the angle between the C-H bonds and the normal to the substrate surface (Fig. 38). The error burs show the standard deviations from the mean or fluctuations of order parameter.


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Figure 38 Order parameter for the cylindrical micelle on the gold substrate. The error bars show the standard deviation from the mean.

4.2.4. Discussion

The simulation of the cylindrical micelle of TAB molecules in water was performed in order to establish the reliability of the MD simulation of structures formed by TAB molecules. Therefore first of all the MD simulated structure of the cylindrical micelle in water shell be compared to experiments.

Qualitatively the MD simulated cylindrical micelle is in good agreement with the standard picture of the ionic micelle in aqueous solution, which was described in the introduction. From the density profiles of the alkyl tails, ions and water (Fig. 14) it is clear that the boundary between the hydrophobic core and the water is sharp. Ions and head groups sit between the hydrophobic core and water. From the same figure and from visual inspection it is clear that the alkyl chains are packed into the micelle core and do not mix with the surrounding


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water. The Br- ions and the head groups are totally excluded from the micelle core. The overall density of the hydrophobic core is a little higher than the density of n-alkanes: 0,74 and 0.62-0,7 g/cm3 respectively. The conformational disorder of the alkyl tails requires, however a more quantitative comparison.

As it was described in details in the results section, the comparison of the correlation times from the MD simulation with the correlation times given by NMR experiments is questionable.

While it was not possible to find any measurements of the order parameter for cylindrical micelles, one would expect that the order parameter for a cylindrical micelle would be in-between the order parameter for the lamella case and for the spherical micelle. Indeed, the order parameter for the cylindrical micelle is in good agreement with the experimental order parameters for the spherical micelle and the lamella cases (Fig. 18). The order parameter is defined such that it gives -0.5 when the C-H bonds are oriented parallel to the micelle surface, 1. when the C-H bonds are perpendicular to the micelle surface and 0 for complete disorder. The orientation of the alkyl tails perpendicular to the micelle surface automatically means that the C-H bonds are oriented parallel to the surface, hence the order parameter is -0.5. The orientation of the alkyl tails parallel to the surface does not necessarily result in the orientation of the C-H bonds perpendicular to the surface, but the C-H bonds can be oriented randomly resulting in an order parameter 0. Fig. 18 presents the modulus of the order parameter in order to compare experiments with the MD simulation. Without taking the modulus, the sign of the order parameter is always negative indicating that the alkyl tails are oriented preferably perpendicular to the micelle surface. The parts of alkyl tails that are close to the head groups have a more pronounced orientation perpendicular to the micelle surface, while the ends of the alkyl tails are conformationally almost disordered.

The modified order parameter Sm is more helpful to explain the spatial dependence of the order parameter in the cylindrical micelle (Fig. 19). Sm for the cylindrical micelle in water varies from -0.1 at the edges to slightly positive values in the centre. The negative order parameter at the edges indicates that the alkyl chains have a slightly pronounced orientation perpendicular to the surface close to the surface of the cylindrical micelle. Random orientation of the chains in the plane parallel to the surface results in an order parameter 0, but when the alkyl chains are oriented parallel to the surface and perpendicular to the cylinder symmetry axis the C-H bonds have just two dimensional disorder resulting in an order parameter of 0.25. The slightly positive order parameter at the centre of the curve indicates this (Fig. 19).


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It is interesting to compare the pair distribution functions of Br- and N+ ions for the cylindrical micelle in water and for different structures adsorbed on the surfaces. The main difference is the well pronounced second maximum on the pair distribution function for the cylindrical micelle and the complete absence of the second maximum of the pair distribution functions for the adsorbed structures. Let me remind that the second maximum characterises the inter-dipole distances. It was experimentally established that TAB molecules form spherical micelles in water and cylindrical micelle in water with high salt concentration11. On the other hand it was also found that dimerized TAB molecules:

form cylindrical micelles in water for s equal to 2-3 and for s=4, and larger cylindrical micelles converge to spherical ones12. The chemical bond between two head groups forces the head groups to stay at on the given distance from each other. This experimentally established fact allows to suppose that the cylindrical micelle in water with a narrow distribution of inter-dipole distances is a metastable aggregation of TAB molecules.

The most striking feature of the MD simulated cylindrical micelle in water is the hole in its centre with a radius of roughly 2-3 Å and an inhomogeneous density of the hydrophobic core. A good agreement between the simulated cylindrical micelle and the experimental data does not help to understand whether the formation of the hole is an artefact of the MD simulation or not. One of the explanations could be that the starting structure was too dense and this resulted in the formation of the shell. The simulation of the cylindrical micelle was performed under constant pressure and pressure was controlled anisotropically. This means that the length of the cylinder could change independently of its diameter and this could help to relieve the high density by elongation of the cylinder. On the other hand the process of cylinder elongation could take place on times much longer than several nanoseconds. That is why an additional simulation was performed, where the initial structure contained 7.5 % less molecules within the same dimensions of the cylindrical micelle. The hole of roughly the same size formed as well.

The overall density of the hydrophobic core of the cylindrical micelle of 0.74 g/cm3 is in-between the density of amorphous polyethylene (0.855 g/cm3) and the density of liquid n-alkanes 0.62 g/cm3 for pentane and 0.7 g/cm3 for octane. The difference in density between liquid n-alkanes and PE is caused by the high concentration of terminal methyl groups in liquid n-alkanes and almost


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none of them in polyethylene. The hydrophobic alkyl tails in TAB molecules are terminated just at one end by methyl groups; therefore it is reasonable that the overall density of the hydrophobic core is in-between the density of liquid n-alkanes and amorphous PE. The hole in the centre of the micelle does not affect the overall density as the hole of the radius 2 Å in the core with the radius 20 Å takes just 1% of the volume. The surprisingly inhomogeneous density can be also explained on the basis of the distribution of the terminal methyl groups. They are mostly concentrated in the centre of the micelle, which causes a lowering of the density in the centre of the hydrophobic core. The hydrophobic core density close to the head groups should be comparable to the density of amorphous PE . Because of the high order of alkyl chains close to the head groups, the density of the hydrophobic core could be even higher than the density of amorphous PE. The densities of semi-crystalline PE lies in the range of 0.92-0.97 gm/sm3 and the density of crystalline PE is close to 1 g/cm3. As we can see the density of the hydrophobic core is indeed a little higher than the density of amorphous PE in the region with high order and low concentration of terminal methyl groups.

The standard picture of the ionic micelle includes a homogeneous density of the hydrophobic micelle core. It should be noticed that this means obviously a homogeneous density of electron clouds. However the mass density of the hydrophobic core is not necessarily homogeneous. It was established17 for spherical micelles that most of the terminal methyl groups are concentrated in the centre of the micelle. It is reasonable to expect that most of the terminal methyl groups are also concentrated in the centre of a cylindrical micelle. The volume of the terminal methyl groups is roughly twice the volume of a CH2 group and that is why a high concentration of terminal methyl groups results in low mass density. The scattering density of the hydrophobic micelle core is also not necessarily homogeneous. As it was already mentioned in the results section, the averaged neutron scattering density of CH3 group is several times higher than the averaged scattering density of a CH2 group. The averaged scattering density of an ionic head group of a TAB molecule is almost zero. That is why it is reasonable to expect that the scattering density of a micelle formed by TAB molecules should grow from the edge of the micelle to the micelle centre. In general, the scattering density of a micelle would depend on the chemical formula of the head group; but the scattering density of the hydrophobic core should dominate at the micelle centre. It is accepted that head groups sit outside the hydrophobic core, hence the scattering density of the hydrophobic core should not mix with the scattering density of the head groups. That is why it does not look reasonable to use a model of a homogeneous micelle to fit neutron scattering data from protonated micelles.


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Despite of the large number of experimental studies it appears that it was not possible to establish experimentally the existence of the hole. SANS which is the most reliable experimental technique does not provide enough resolution to determine a hole with a radius of several angstroms in the centre of a protonated micelle13. On the other hand it was established that micelles formed by TAB and similar amphiphilic molecules may have a hydrophobic core with a diameter larger than the length of fully extended alkyl tails. The suggested explanation of this phenomena was a non spherical form of the micelles. However the experimental results can be explained as well by the presence of the hole in the micelle centre. Interestingly, it should be possible to determine the inhomogeneous density in the deiteurated micelle centre, but no experimental studies on neutron scattering were found in the literature, where the scattering cross section of a completely deteriorated micelle was determined.

On the other hand it was predicted theoretically that ampiphilic molecules may tend to form micelles with a hole in the centre15,16. The highest possible cost of the hole formation with a volume of 54 Å3 was estimated to be 109 cal/mol of chains for a spherical micelle with a hydrophobic core radius of 20.4 Å. A more realistic estimate gives only 45 cal/mol of chains. The energy cost of hole formation was estimated as work against internal pressure in the micelle to form the hole and loss of Van-der-Waals energy by chains surrounding the hole. Different estimates of internal pressure also lead to different estimates for energy cost of hole formation. The energy of the hole formation is very small indeed: for comparison, the estimated entropic cost of micelle formation is roughly 0.6 kcal/mol of chains and the experimentally measured free energy, which C12 chain gain when being transferred from water to bulk n-alkane is approximately 12 kcal/mol. The formation of the hole allows to increase the aggregation number. For example, the formation of a hole with a volume of 54 Å3 in a micelle with a radius of 20.4 Å allows to increase the aggregation number by 30%. On the other hand it was also argued that even small deviations away from a perfect sphere would eliminate a small hole.

The deviation from the spherical shape increases the surface area of the micelle which is energetically unfavourable. On the other hand a deviation from the spherical shape eliminates the hole in the centre of the micelle and hence gains energy equal to the cost of hole formation. As the aggregation number of the micelle does not change when it changes its form the energy cost of deviation from the spherical form is equal to the surface change multiplied by the tension of the water/n-alkane interface, i.e. 50 mN·m-1. A spherical micelle with a radius of 22 Å and a hole in its centre with a radius of 2 Å should converge to an ellipsoid with main axes equal to 20, 23.1 and 23.1 Å in order to eliminate the hole. The energy cost of the increased surface area is 23 cal/mol of


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chains. Taking into account the estimated energy of the hole formation, the elliptical micelle is energetically more favourable by 2-8 kcal/mol or 3.4-13.6 kT. The estimated energy difference between a spherical micelle with the hole in its centre and an elliptical micelle is very small indeed. For example the difference between gauche and trans conformations in alkyl chains is of the order of 1 kT. Another example: during 1 ps of MD simulation of a cylindrical micelle in water the potential energy varied more than by 100 kcal/mol and the total potential energy of the system cylindrical micelle of TAB molecules in water environment was minus 2x105 kcal/mol.

It should be emphasised again that estimated energy difference between a micelle with a hole and an elliptical micelle is very low. It was assumed that the elimination of the hole does not change the entropy of the micelle, which is obviously wrong. When the micelle adopts an elliptical form the alkyl chains should also adopt different conformations and this should lead to additional free energy cost. This means that a micelle with a hole could be even energetically more favourable. Let us assume that the estimated energies are close to reality. Then amphiphilic aggregates would adopt different shapes between spherical with hole and elliptical with probabilities given by the Boltzman distribution:

where lambda denotes the path between spherical and elliptical forms. The estimate of energy cost of hole formation was performed for the spherical micelle, but one would expect that the mechanism of the hole formation in a cylindrical micelle is the same.

It is interesting to notice that when a micelles adopts different shapes from elliptical to spherical with a hole in the micelle centre and the probabilities are given by a Boltzman distribution. On average a micelle would have an inhomogeneous density in its centre.

The simulations of a cylindrical micelle and a half cylindrical micelle on gold suffers from the fact that the metallic properties of gold are not taken into account. The TAB molecules interact with the gold atoms only via Lennard-Jones potentials since the gold atoms are completely uncharged. In classical physics it is well known that charge close to the metal surface would cause the redistribution of the free electrons which would lead to the appearance of an effective mirror charge. If we leave this uncertainty it is interesting to compare how substrates with different Lennard-Jones interaction with TAB molecules


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would affect the adsorbed structures. For instance the Lennard-Jones interaction between gold and carbon atoms is much higher than between carbon and carbon.

The structure of a half cylindrical micelle on gold after MD simulation still exhibit a very well ordered first layer (Fig 28-29). The layer of TAB molecules close to the substrate is separated from the rest of the micelle by several angstroms with relatively low density, which causes high fluctuation of the order parameter in this part. In a half cylindrical micelle on paraffin (Fig 33-34) it is also clearly possible to define a layer of TAB molecules oriented by the paraffin substrate. But in comparison to the gold substrate the order fluctuations are higher and the first layer is smeared out indicating that the half cylindrical micelle is better bound to the gold substrate. It is interesting to notice that the order for molecules close to the substrate surface is the same for paraffin and gold substrates.

The order parameter Sm for a cylindrical micelle on a gold substrate (Fig. 29) clearly indicates that these structure is not yet equilibrated. The deviation from the mean of the order parameter for molecules close to the surface is very high indicating that these molecules still have to adopt their equilibrium conformations. Because of the high fluctuations it is questionable to conclude a general trend in the order parameter. A lower order parameter for the molecules close to the substrate surface indicates that the molecules close to the surface tend to orient themselves perpendicular to the surface, which is consistent with the flattening of the cylindrical micelle toward the surface and with neutron reflection experiments10, where it was shown that aggregates of TAB molecules on the substrate silica are in good agreement with the model of a bilayer.


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4.3. Conclusions

MD simulations of a cylindrical micelle formed by TAB molecules in water environment, a half cylindrical micelle of TAB molecules on gold and paraffin substrates and a cylindrical micelle formed by TAB molecules on a gold substrate were performed. A good qualitative and quantitative agreement between the simulated cylindrical micelle in water and corresponding experimental results was found, except for the non-homogeneous density of the hydrophobic core. The simulated cylindrical micelle was qualitatively in good agreement with the standard picture of an ionic spherical micelle. Good quantitative agreement was found between experimentally established and MD simulated structure regarding order parameter and distribution of terminal methyl groups. It was difficult to compare relaxation times given by NMR experiments and calculated from MD trajectories. As it was argued it is not possible to give just a single relaxation time for a given carbon in the alkyl chain, rather spectra of relaxation times.

The MD simulated cylindrical micelle formes a hole (vacuum) in its centre and in addition the density of the hydrophobic core composed of alkyl chains was not homogeneous. The proposed explanation of the inhomogeneous density of the micelle core was related to the distribution of the terminal methyl groups in the micelle centre. The explanation for the hole formation in the centre of the micelle was based on the work of Gruen15,16. It was argued that the formation of a small hole in the micelle centre lead to sufficient increasing of micelle aggregation number, which could be energetically favourable The small deviation from the spherical form that may eliminate the hole also increases the micelle surface, hence costs additional energy.

The influence of gold and paraffin substrates on the structure of half cylindrical micelles was characterised. The ability of metals to form mirror charges was not included in the model of gold. The gold was simulated just a crystalline material formed by gold uncharged atoms. Since the model of the gold substrate used in this investigation does not include free electrons, gold and paraffin substrates were compared as two substrates with different Lennard-Jones interaction with adsorbed structures. It was found that the gold substrate induces higher order in adsorbed half cylindrical micelle than a paraffin substrate.

The time evolution of a cylindrical micelle on a gold substrate was simulated. It was found that the cylindrical micelle started to flatten on the gold substrate and the alkyl chains close to the substrate tend to orient perpendicular to the substrate plane. The flattening and the orientation of alkyl chains


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perpendicular to the substrate was compared favourably to the experimental established flattening of the structures formed by TAB molecules on silica.


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