Self-Assembly of Lysine-Based Dendritic Surfactants Modeled by the

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Self-assembly of lysine-based dendritic surfactants modeled by the self-consistent field approach. Oleg Shavykin, Frans A.M. Leermakers, Igor Neelov, and Anatoly Darinskii Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03825 • Publication Date (Web): 29 Dec 2017 Downloaded from http://pubs.acs.org on December 31, 2017

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Self-assembly of lysine-based dendritic surfactants modeled by the self-consistent field approach. O. V. Shavykin,∗,† F. A. M. Leermakers,‡ I.M. Neelov,†,¶ and A.A. Darinskii†,¶ St. Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy pr. 49, St. Petersburg, 197101, Russia, Physical Chemistry and Soft Matter, Wageningen University, 6703 HB Wageningen, the Netherlands, and Institute of Macromolecular Compounds, Russian Academy of Sciences, Bolshoi Prospect 31, V.O., St. Petersburg, 199004, Russia E-mail: [email protected]

Abstract Implementing a united atom model, we apply self-consistent field theory to study structure and thermodynamic properties of spherical micelles composed of surfactants that combine an alkyl tail with a charged lysine-based dendritic head group. Following experiments, the focus was on dendron surfactants with varying tail length and dendron generations G0, G1, G2. The heads are subject to acetylation modification which reduces the charge and hydrophilicity. We establish a reasonable parameter set which ∗

To whom correspondence should be addressed St. Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy pr. 49, St. Petersburg, 197101, Russia ‡ Physical Chemistry and Soft Matter, Wageningen University, 6703 HB Wageningen, the Netherlands ¶ Institute of Macromolecular Compounds, Russian Academy of Sciences, Bolshoi Prospect 31, V.O., St. Petersburg, 199004, Russia †

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results in semi-quantitative agreement with the available experiments. The critical micellization concentration, the aggregation number and micelle size are discussed. The strongly charged dendronic surfactants micelles are stable for generation numbers G0 and G1, for progressively higher ionic strengths. Associates of G2 surfactants are very small and can only be found at extreme surfactant concentration and salt strengths. Micelles of corresponding weaker charged acetylated variants exist upto G2, tolerate significantly lower salt concentrations, but loose the spherical micelle topology for G0 at high ionic strengths.

Introduction Surfactant self-assembly in aqueous solutions is a classical field of research important for lubrication, wetting, cleaning and formulation sciences and is the workhorse in nano-technology. 1–3 Surfactants exist of two complementary molecular constituents which have opposite affinities for water. The apolar part, which typically is referred to as the tail, gives the driving force for the assembly. The tails aggregate into dense cores of the assemblies and thereby largely avoid contact with water. The polar entities, with good water solubility, remain hydrated and give the stopping force that prevents the assemblies from growing to macroscopic sizes. The resulting objects typically have either a spherical, cylindrical or lamellar topology. Hence at least one of the dimensions of the assemblies remains comparable with the size of the surfactants. Generically these assemblies may be referred to as micelles, but here we reserve this term for spherical objects. It is clear that upon self-assembly individual surfactants give up translational entropy in favor of reduction of solvatation energy. These ingredients may lead to the assumption that micellization is a first-order phase transition. However, the micellar aggregates in turn have translational degrees of freedom, and therefore micelles obtain a concentration dependent chemical potential. As a consequence micellization-transition is only a first-orderlike process. The larger is the aggregation number of the micelles the sharper is the transition. 2


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Nevertheless, experimentally the appearance of micelles is rather sharp and the transition concentration is known as the critical micellization concentration or cmc. 4–6 The cmc (in dimensionless units represented by ϕbcmc ) is known to be a strongly decreasing function of the tail length Nt , i.e. ϕbcmc ∝ exp −Nt . Also the head group properties contribute to the cmc. For ionic surfactants the cmc increases with the number of charges per surfactant and decreases with salt concentration. For nonionic surfactants the cmc usually decreases weakly with an increase in size or length of the head group. There are many salient features and intricacies of self-assembly. In first order we only have freely dispersed surfactants in solution below the cmc (dimers and trimers etc. have exponentially low probability) and well-defined micelles with a given number of surfactants in it (the aggregation number) above the cmc. Hence in the size distribution of aggregates has a characteristic gap which may be seen as the hallmark for self-assembly. The micelles are in equilibrium with freely dispersed surfactants with a concentration close to the cmc. Upon closer inspection we know that above the cmc the (average) micelles size and the freely dispersed surfactant concentration are weakly increasing functions of the overall surfactant concentration. 7 The structure of classical as well as polymeric micelles has been investigated and early reports date back to last decades of the previous century. 8,9 From experiments we know that the inner region of the micelles, i.e. the core, is basically dry and thus virtually free of water molecules. 10,11 From theoretical considerations we understand that the build-up of an osmotic pressure, when solvated head groups are forced to be close to each other in the outer-part (corona) of the micelles, provides the stopping mechanism of the growth of the assemblies. 7 Furthermore, it is clear that in many cases there is a rather sharp boundary between the core and corona. 12 These packing characteristics are behind the success of the surfactant packing parameter p = v/(al) that is used to rationalize micellization. 13 Here v is the volume of the alkyl tail, l the length of the tail and a the «surface» area each surfactant occupies at the core-corona interface. The volume v and the length l are simple

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functions of number of carbons Nt in the tail. The area per molecule a is, however, a more complex entity, which might be obtained from, e.g., Langmuir adsorption isotherms 4 and its estimate is one of the challenges for analytical modeling of micelles. Roughly speaking one may expect spherical, cylindrical micelles or lamellar (bilayer) aggregates when the packing parameter is around 1/3, 1/2 or unity, respectively. 14,15 In the case of polymeric micelles one can judge transitions from one micelle geometry to the other by considering the size of the core compared to the size of the corona. In general when the corona size of a spherical micelle is less than the core, one should expect a transition towards cylindrical micelles. Similarly in case the corona of cylindrical micelles is small compared to the (radial) size of the core one should expect bilayers. 16 As mentioned already for charged surfactants the critical micellization concentration, above which the micelles are formed, depends on the ion concentration. 17 The concentration dependence is semi-quantitatively explained by the Gouy Chapman (GC) theory, as the periphery of the micelle can be seen as a classical electric double layer. 4 From this we know √ that the Debye length lD ∝ 1/ cs , where cs is the concentration of salt in the solution, governs the range of electrostatic interactions. It is understood that for such charged micelles the «surface» charge density is not one-to-one related to the head group area a because small counterions can penetrate (condense) into the head region (a Donnan layer). When they do so they effectively reduce the surface charge density. Nevertheless it is understood that, e.g., by increasing the salt concentration the charges in the head groups are better screened and therefore the area a is expected to go down. This renders the packing parameter p to go up and in turn this may trigger growth of the micellar size and possible changes in micelle geometry. 15,16,18–20 Indeed, the surfactant packing parameter is a suitable guiding principle to rationalize self-assembly from a qualitative point of view. It is less effective to rationalize concentration dependences, or provides other quantitative guidance for experiments. For prediction of quantitative measures it is viable to consider more detailed molecular level theory, 14–16,19–24

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computer simulations, 25–32 or numerical self-consistent field models, 33–45 where it is understood that the provided references to the literature is just a small (biased) selection. Typically the theoretical models give dependences with respect to the molecular weight, give trends with respect of some tuning parameter (ionic strength) and work for the polymer cases best in the long polymer limits, and in general work best when the micelles have a clear (pre-assumed) core-corona structure. Full-atomic MD simulations can give in principle realistic results for a given model and predict new morphologies, 46 but require a long computer time. One of the ways to reduce this time is to use DPD method based on the coarse-grained model. 30–32,46 Self-consistent field (SCF) theory is positioned in between DPD simulations and analytical approaches. SCF is based on a free energy functional and focuses on most-likely micelles with a well-defined core-corona interface. It presumes a given micellar topology, applies a mean field approximation accordingly, and then produces a wide range of structural and thermodynamic properties at relatively low CPU cost. The dense packing of surfactants in the aggregates leads to relatively few adverse effects of the mean-field approximation and this contributes favorably to the applicability of the method. In this paper we use this approach to consider the micellization of dendritic lysine surfactants. Lysine dendrimers (dendrons) were first synthesized in early 80th of last century. 47 They were used in many biomedical applications, for example, as carriers for drug and gene delivery as well as core for multiple antigen peptides (MAPs), as antibacterial, antiviral and antiamyloid agents. 48–51 Molecular dynamics simulation 52–56 and numerical self-consistent field calculations 57–59 and Brownian dynamics (BD) simulations 60 of these dendrimers were performed by us recently. Computer programs for these MD simulations were elaborated and applied by us in previous papers 61–67 and for BD simulations in these papers. 68–75 In recent years C14 and C16 surfactants with dendritic lysine (lysine aminoacid residue) based head groups have been synthesized and their scattering properties in water were studied. 76 Referring to figure 1 for an illustration, in the zeroth generation dendrimer (G0) there

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Figure 1: Schematic illustration of the (default) dendritic surfactants used. The arrows point to the «acetylation» points. For the first generation dendrimer we have shown the effect after acetylation (indicated by circled fragments). The coloring refers to the type of segment used as indicated. The corresponding Flory-Huggins interaction parameters are specified in table 1. Below a shorthand notation is used to refer to the dendrimers. The tail length information is typically removed, that is, we will use G0 for G0-C16. The acetylated variants are referred to by G0a = G0(acetyl)-C16 is just one lysine in the head group with two positive charges (here and below we will assume that the pH is near neutral where all the lysine charges are permanent). In the first generation dendrimer G1, the surfactant has already three lysine residues two of which are terminal with 4 positive charges. In the second generation G2 there are seven lysine residues four of which are terminal with 8 charges, etcetera. In fact for the micellization properties of single tail surfactants only G0, G1 and G2 variants are relevant. Experimentally it proved possible to reduce the charge in the dendrimers by an acetylation reaction. In figure 1 the sites of the acetylation are pointed at by arrows and the implementation on the united atom level of the reaction is illustrated for the G1-C16 case. The characterization of both types of micelles (dendron as well as acetylated variants) as a function of the surfactant concentration and the surfactant tail length were recently reported for a given value of the ionic strength. 76 Dendron micelles are potentially interesting objects. They can be used as carriers for both polar as well as apolar compounds. The application for hydrophobic material is classical and needs no further explanation. The individual dendron heads also may be seen as a molecular «containers», and therefore such micelles can transport different components that are mutually incompatible or should be separated from each other. Micelles composed of dendrons

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with higher generations have many terminal groups that can be further functionalised. This feature can be used to target the micelles in biomedical applications. 77–79 Dendritic micelles may have a similar size than some higher generation classical dendrimer. Higher generation dendrimers may feature a lack of space for their «arms» and then loose some of their carrier function. The corresponding micelles do not suffer from the same problem. The SCF modeling of such micelles may guide the experimental micellization efforts and help develop applications. The method may also be used to underpin strategies to change the dendronic surfactants such that the corresponding micelles may contain, e.g., the proper number of «containers». Here we will use a molecularly detailed self-consistent field theory to model the selfassembly of lysine-based dendritical surfactants and the acetylated variants. The theory features a freely-jointed chain model to deal with the conformational degrees of freedom for the (surfactant) molecules. It implements the full Poisson-Boltzmann (PB) formalism to account for the electrostatics. In the classical PB formalism, ions are modeled as point charges. Here we take also the volume of ions into account, albeit that this is done on a «generic» level, that is without having a specific type of ion into account. The approach deals automatically both with the Gouy-Chapman (electric double layer is outside the micelle corona) as well as the Donnan limits (the charge in the corona is locally compensated by the ions). It employs Flory-Huggins interaction parameters to account of the solvency effects, necessary, e.g. to account for the the driving force for self-assembly, the degree of hydration of units in the dendron heads. From fig. 1 it is clear that the model features many different segment types. As a result there is a rather large parameter set which influences the self-assembly of these molecules. As in the mean field approximation the probability of a contact between two types of segments is proportional to the product of the (local) volume fractions, we realize that the most important interactions are those that involve either water or hydrocarbon tails. In line with this we have tuned these to find a reasonable correlation between theoretical predictions and experimental data. We haste to mention that still the

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resulting set of parameters should be considered as preliminary. In mean-field theory and thus in SCF, there is an issue that contacts between units that have a low density are marginally accounted for. In principle for such interactions one should implement, e.g. quasi chemical models. 80 Here we have not implemented this. As a result non-electrostatic interactions, e.g., between ions and charged groups in the dendron head are basically ignored. We therefore understand that we can not correctly catch the true partitioning of ions between bulk and the corona. As a result we can only qualitatively predict the screening of the charges in the head group region. To correct for the neglect of these ion correlation effects, we accept the use of an adjustable parameter needed to translate the volume fraction of salt (in the model) to the molar concentration of salt (in experiment). This adjustable parameter will be discussed below in more detail. The thermodynamics of small systems 81 is used to judge the relevance of micelle structures that present themselves once the SCF equations are solved. In previous publications of SCF methods for self-assembly 38,82–88 this method has been thoroughly discussed. Below we give a brief outline. It is important to mention that it is due to this thermodynamic framework that with SCF one can focus on a most-likely micelle as these thermodynamic features point to stable, meta-stable or unstable micelles and allows a prediction of experimental observables such as (i) the critical micellization concentration, (ii) the most likely micelle size (aggregation number) and micelle size distribution e.g. as a function of the surfactant concentration, ionic strength and degree of acetylation. Above it was mentioned that the SCF theory is inbetween formally exact computer simulations and analytical theory. The comparison of the numerical SCF predictions with analytical theory is helpful, e.g., to rationalize the numerical results. In turn the comparison may also give guidance to the analytical theory by checking the validity of implemented approximations. The remainder of the paper is the following. In the beginning we will overview the main characteristic features of the modeling while introducing the molecular model and its

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parameters and the definition of the most important characteristics of the micelles. More detailed theoretical background is presented in a Support Information. In the results section we focus on surfactants with lysine based dendritic heads and acetylated variants. Special emphasis is on the comparison with experimental data.

Model The theoretical framework outlined in a Support Information is constructed around five assumptions. (i) The volume is discretized in the form of a lattice wherein lattice sites are organized in layers numbered r = 1, 2, · · · , M . Here r is used as the coordinate, while the number of lattice site L(r) for given r depends on the geometry (it is a constant for the planar, it is proportional to r (L(r) ∝ r) for cylindrical-, and is proportional to r2 (L(r) ∝ r2 ) for spherical geometry). (ii) Molecules are assumed to be composed of (united) atoms which we call segments. The segments are taken to be of equal size fitting the lattice site. A given molecule i has Ni segments. (iii) A mean-field approximation is applied. Instead of using discrete positions of the segments, the theory implements local volume fractions at specified locations r, that is, ϕ(r). As a consequence we lost information about the exact location of the segments inside a given layer r as these are averaged out. What is left are radial distributions, which can be used to obtain the aggregation number, the sizes of the core and corona, the ζ-potential, etcetera. The volume fractions are also used to estimate the interactions between segments. These interactions are parameterized with Flory-Huggins interaction parameters. Estimates of various FH parameters are known from previous works, 41,89 but not for all. (iv) The conformational degrees of freedom are approximated by a freely jointed chain model. This model allows for direct backfolding of chain-conformations. We understand that this is not realistic, but we accept this approximation because it allows for an extremely efficient scheme to compute the statistical weight of all possible conformations, and hence the computation of the (mean field) partition function. The latter quantity leads

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to the free energy of the micelle formation from the surfactant solution. (v) The system is assumed to be incompressible. This constrained corrects in part for the backfolding problem mentioned at (iv). The natural concentration units in the lattice model are volume fractions ϕ and the incompressibility condition specifies that the sum over all species of the volume fractions add up to unity at each «coordinate» r. Our interest is in surfactant self-assembly in selective solvents. Typically the solvent is a small molecule which occupies just one or a few lattice sites. Here and below we will suggest that the solvent is water, albeit that in a mean field model the true properties of water are only poorly represented. The idea is to focus on the (most-likely) micelle that is effectively restricted to the region of small values of r, while its constituents are free to partition between the micelle and the bulk. The optimization of the mean-field free energy (see a Support Information) leads to a set of equations that are solved numerically. The fixed point is known as the self-consistent field solution. At such solution both structural as well as thermodynamic information of the system is available. The latter is essential because the micelle in the simulated volume is not automatically physically relevant. For example, its aggregation number may be too high and the splitting up into two micelles is suppressed by the mean field approximation. Similarly, the micelle in the simulated volume may have a too low aggregation number. Then it would have liked to go together with another one. Again this process is suppressed by the mean field approximations. These physically unimportant micelles, but also the stable micelles are recognized from studying the work of formation of micelles Ωm of specified size g, i.e., from the function Ωm (g). From the thermodynamics of small systems (see also a Support Information) we know that we should focus on systems for which the work of formation of the micelles is positive and for which this work is a decreasing function of the aggregation number g. In principle one further needs to consider various lattice geometries to compare spherical, cylindrical or lamellar micelles thermodynamically to know the best structure for the system at interest. Such information is judged from the value of the chemical potentials µi of the molecular constituent i, where it is understood that

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the system which features the lowest chemical potentials (for a specified overall composition of the system) is the preferred one. For the modeling of micelles the critical micellization concentration (cmc) is always an issue. The larger the micelles the more abrupt is their first appearance. It implies that we can have different measures to locate the cmc. This is also true in the SCF model for selfassembly. Within the SCF framework we can point to the first appearance of the micelles. This occurs when the work of formation has its maximum Ωm (g) value, or equivalently when the chemical potential for the surfactants µi (g) has a minimum. At this point the micelles have their smallest size and their concentration is often extremely low so that this point is experimentally not easily recognized. We may therefore choose to analyze micelles that are located at other points on the stability curve which are experimentally more relevant. As the micelle concentration is an exponential function of the work of formation (see a Support Information), i.e., ϕm = exp −Ωm (g) (with Ωm in units of the thermal energy kB T ), we choose to consider micelles for which the micelle concentration is (experimentally) noticeable. Rather arbitrarily we to focus on the case Ωm = 5 (energy units are kB T ). Such micelles are usually significantly larger than the smallest ones referred to above. Taking spherical micelles as an example, the modeling results in structural information in terms of radial density profiles ϕi (r) for the volume fraction of molecule i at coordinate r (where r = 0 is at the center of the micelle). From this we may compute the so-called aggregation number gi in the micelle as the excess number with respect to the bulk solution (far away from the micelle; indicted by the super-index b):

gi =

X

L(r)

r

ϕi (r) − ϕbi Ni

(1)

As it is clear from the context that the aggregation number refers to the surfactant and when there is only one of these in the system we often omit the index i. For more detailed structural information we may turn to density profiles per segment

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type (see below), i.e., ϕX (r) which is the volume fraction of segment type X at coordinate r. From this we can compute the second moment of the distribution of a specified segment (or segment type) X by v uP uM u (ϕX (r) − ϕbX )L(r)r2 u RX = u r=1 t nlayer P (ϕX (r) − ϕbX )L(r)

(2)

r=1

to locate the average position of particular segments. Alternatively, we can do this for a specified chain fragment, e.g., the tail segments leading to Rtail , or the head group segments leading to Rhead . Below we will discuss the size of the core Rcore ≡ Rtail and that of the corona Hshell ≡ Rhead − Rcore . In these equations the volume fraction in the bulk of the corresponding segments or chain fragments are given by the super index b, is the volume fraction that is found far outside the micelle where all inhomogeneities have died out. When our interest turns to the electrostatic forces in the micelle we may consider the radial charge distribution q(r) or the corresponding electrostatic potential ψ(r) profile, both of these feature in the Poisson Equation:

∇2 ψ = −

q(r)

(3)

This equation is correct when the dielectric permitivity is constant in space. We therefore P implement a density weighted refractive permittivity ε(r) = 0 X ϕX (r)εX where εX is the relative dielectric permittivity of a phase composed of units of type X and 0 is the dielectric permittivity of vacuum. The Poisson Eqn 3 is modified accordingly (see a Support P Information). Similarly q(r) = e X ϕX (r)vX , wherein e is the elementary charge and vX the valence of segments of type X. We may now turn to the molecular architectures that are implemented, c.f. fig. 1 and the parameters collected in table 1. Similarly as used in recent SCF modeling of self-assembly, 89 we consider the major component water as a cluster that occupies 5 lattice sites (a fourarmed star). Obviously this is a poor model for water and the simple reason for using such 12


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a model is to implement somehow the known feature of water that it associates with other water molecules. For the current choice to put N = 5 water is unable to go as a single monomer into the micellar core. Their «connectivity» to four others therefore results in a core which is starved in water which is consistent with the literature data 4 (albeit that in reality it is expected that the cores are even more dry). The alternative of having monomeric species N = 1 would lead to micelles with a noticeable amount of water (several percent) in the core which is less acceptable, e.g., when parameters are chosen to represent correct cmc values. The default surfactants in our study have a single hydrocarbon tail of 16 segments, i.e. C16 and a dendritic head groups. Referring again to fig. 1 we have G0 which has one lysine-like unit and two positive charges, G1 which has two lysines and four charges, while G2 combines 4 lysines and 8 charges. The structure of the head group is dendritic with branches subsequent branches depending on the generation 0, 1, 2. Experimentally it is possible to reduce the charge density in the dendrimers by acetylation. In Figure 1 we show how the result of acetylation implemented on the surfactant G1 and we refer to this variant to G1(acetyl) or in short G1a. Arrows point in the other structures to similar acetylation sites. We carefully constructed the molecules by choosing as few as possible number of different segment types. The ones that are used in the figure are listed in the square box and are distinguished by color. The same segment types are used in table 1 where the parameters of the modeling are collected. For the majority of most important parameters we adopted the values used in previous SCF work for lipid bilayers. 89 We will briefly address the essential ones. It must be mentioned that for a truly accurate comparison with experiments one should fine tune the parameters. This has not been attempted yet, and therefore the set should be considered as preliminary or approximate. First and fore all there is the repulsion between C and W, implemented by χC,W = 1.2. 89 This parameter provides the driving force for assembly. 90 The numerical value is consistent

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Table 1: Top: Set of Flory-Huggins interaction parameters χ between two unlike segments. The segment types W (water), C (CH, CH2 , CH3 ), charged units is surfactant NH+ 3 , neutral NH and O, the generic co-ion Na+ and counter-ion Cl− . Bottom: valency vX of the segments X, and relative dielectric permittivity εrX . χ W C NH+ 3 NH O Na Cl v εr

W C NH+ 3 0 1.2 0 1.2 0 3 0 3 0 −0.6 2 0 −0.6 2 0 0 2 0 0 2 0 0 0 1 80 2 5

NH −0.6 2 0 0 0 0 0 0 5

O Na −0.6 0 2 2 0 0 0 0 0 0 0 0 0 0 0 1 5 10

Cl 0 2 0 0 0 0 0 −1 10

with the known trend of the log(ϕbcmc ) with an increase of the tail length (roughly an increase in tail length by 3 segments decreases the cmc by a factor of 10). It must be understood that χC,W = 1.2 is very rough attempt to account for the hydrophobic effect. In line with repulsion between hydrocarbon segments and water we have implemented repulsion between hydrocarbon and charged species. We understand that it is unfavorable for ions to be in the hydrocarbon phase and therefore we put a high price to this. Even though in practice we do not expect to find counter ions and co-ions to have exactly similar affinities for the hydrocarbon phase we do not make a difference, that is, χC,N a = χC,Cl = χC,N H3+ = 2 in the default case. 89 Some disparities in these interactions may contribute to a charge mechanism in the micellization. The interaction of NH and O with the hydrocarbons is also taken to be repulsive: χC,N H = χC,O = 2. 89 It was noticed that such repulsion between these units does help the formation of stable micelles as it correlates with a reduction between overlap of heads and tails. When lysine dendrons are acetylated a negative charge is replace by a small hydrophobic fragment surround by an NH and and O (cf Fig.1). These groups are hydrophilic as one is H-bond donating and the other is H-bond acceptor. We have set χW,N H = χW,O = −0.6. 91 This negative value compensates effectively the drop of the polarity of the head group upon 14


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acetylation. Alternatively we could have chosen for the two C-segments that are added to the head group upon acetylation to be (slightly) less hydrophobic than the C-segments in the tails. This was judged less realistic than the negative values for χW,N H and χW,O . The remainder of the contacts are between hydrophilic segments. In the default parameter setting we choose for simplicity to set these to the athermal value. This means that without mentioning otherwise we take only the volume into account and ignore the specific affinities between the segments. It is of considerable interest to know how the micellization takes place in the absence of such interactions, or in other words to know the generic effects. Molecular specific interactions, which are harder to predict especially using a mean field model, will modulate the self-assemble and thus limit the capability of the theory to follow experimental data. In table 1 also the segment valencies and the relative dielectric permittivities are specified. In this work we assume that the lysines are all fully charged, that is that the pH is sufficiently low. In the Poisson equation (eq. S15 in a Supporting information) we need a value for the local dielectric permittivity. Here we follow the usual approximation to compute these by a volume fraction weighted average. This approach may not necessarily be fully accurate however it implements the physical intuition that the local relative dielectric constant is approximately 80 in bulk water and very low (approximately 2) in the hydrocarbon phase. The value of 10 for the co- and counterion as well as the value of 5 for the remainder of the segments has no strong background, but by the density averaging we will have a gradient in the corona going from the value close to 2 in the core to a value of somewhat less than 80 in the bulk (depending on the salts used). To implement the Poisson equation (eq. S15 in a Supporting information), which relates the charge density profile (as well as the profile for the dielectric permittivity) to the electrostatic potential, it is necessary to have a value for the lattice cell size. We have used a united atom description to specify the molecules and consistent with this we have implemented a cell-size of b = 3 × 10−10 m. In this paper we will use dimensionless (in lattice units) sizes and

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distances. These values can be multiplied by b to convert to values in meters. The conversion of dimensionless volume fractions ϕ to molar concentrations c requires a conversion factor k, i.e., c = k × ϕ. It is easy to see that k =

10 , NA 1000b3 N

were NA is Avogadro’s number. For

monomeric species N = 1 the conversion factor is k ≈ 60. We argued above that due to the mean field approximation we basically ignore the specific affinities of ions for the hydrophilic groups in the dendron micelles. We suggest a pragmatic solution to this problem and take k as an extra tuning parameter. The default parameter setting discussed above has been established by assuming k = 10. Hence when a bulk volume fraction of salt is specified by ϕbN a = 0.01 we advice that this corresponds to 0.1M salt instead of 0.6M. Again this «engineering» solution should be understood as a correction for the failure of mean field theory to account for specific interactions of the ions and is implemented for the ions only.

Results

Figure 2: (a) The grand potential Ω ≡ Ωm as a function of the aggregation number g and (b) the corresponding chemical potential of the surfactant as a function of the aggregation number g. The stable branches of the three generations of LYS-dendrons G0, G1 and G2 are indicated by solid lines. The meta-stable parts are dotted. On the G1 case the symbols mark the maximum of the grand potential (triangle point), the case for which Ω = 5 kB T (round point), and the case for which work of formation is zero (square point). Interaction parameters given in table 1 and volume fraction of the co-ion ϕbN a = 0.01. Both co- and counter ions are surrounded by four W segments.

The self-consistent field theory has been used for self-assembly of ionic surfactants, 92 non16


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ionic surfactants, 93,94 lipids, 83–85 copolymers. 86,88 The self-assembly of charged lysine-based dendrons and the corresponding acetylated variants closely follows the protocols established in these works. We will first consider a case study exemplifying the structure and thermodynamics of the default dendron micelles. We then focus on the aggregation number, the cmc, the core and corona sizes, the zeta potential and the fluctuations of the micelle size as a function of the salt concentration. A study about the effects of the surfactant tail lengths and that of acetylation is presented at the end of the results section. There exists experimental estimates for the quantities that are predicted by the model. In several figures discussed below experimental data points 76 are added as dots. The first steps of any SCF analysis of self-assembly centers around the work of formation of micelles, or Ωm (g). To generate this dependence we need to make a choice for the micelle morphology. Following experiments 76 we focus on spherical micelles. In fig. 2a three curves for the work of formation Ωm (g), that is, for G0, G1 and G2 Lysine-based surfactants are presented for which the default parameter setting has been applied. As outlined in the previous section and more systematically elaborated on in Support Information we may distinguish meta-stable (dotted parts) from the stable branch (solid line) and negative values for Ωm should be and have been discarded. The corresponding curves for the chemical potential of the surfactants which also are functions of the aggregation number g are plotted in fig. 2b. Also in this graph the stable «branch» is given by the solid lines and the metastable parts are dotted (for G2 we not found meta-stable solutions). The ionic strength is fixed by setting ϕbN a = 0.01 which should correspond to an experimental ionic strength of approximately 0.1 M. It is instructive to elaborate on the interpretation of these graphs. As mentioned in a Support Information the work of formation curves and the chemical potential curves are interrelated by a Gibbs-Duhem equation (see, eq. S6 in a Supporting Information). This implies that when Ωm goes through a maximum, the chemical potential has a minimum and vice versa as well as that the slopes are related. This implies that many remarks made about the grand potential can be restated in terms of the surfactant chemical

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potentials. We will not pursue this and use either one or the other quantity when this is easiest. (i) Obviously, the minimum in the chemical potential curve corresponds to the lowest possible chemical potential consistent with the presence of micelles. This point, marked by a dot on the G1 curve, signals the first stable micelle. Clearly with increasing size of the surfactant head the concentration at which the first micelles are found increases (recall that there is a one-to-one relation between bulk volume fraction and chemical potential). (ii) The aggregation number for the first stable micelle is very small for G2, is just below 20 for G1 and is close to 40 for G0. We can understand this trend from the action of the surfactant packing parameter; a larger head group leads to smaller values for p and related to this we expect smaller aggregation numbers. When we follow the equilibrium branch of the grand potential curve starting at the maximum we notice that this corresponds to the lowest overall surfactant concentration (cf eqn S5 in a Supporting Information). For the G0 surfactant at this point the micellar concentration ϕm = exp −33 is much lower than, e.g. for G2 for which ϕm = exp −6.7. Following the grand potential curves to larger aggregation numbers we see that the grand potential (work of formation) decreases. This means that the micelle volume fraction increases. The system for which the grand potential is 5 kB T is indicated for the G1 surfactant and occurs in this case at g ≈ 30 which is about 50% larger than the first stable micelles. It is important to mention that the grand potential curve should not be over-interpreted. For example one can follow the curve to even larger aggregation numbers and find the system for which the work of formation vanishes, i.e. Ωm = 0. Application of Eqn. S3 in a Supporting Information would then imply that ϕm ≈ 1. However, the curve presented in fig. 2 was computed for isolated micelles and we understand that the high surfactant concentration limit needs special attention. For example to model surfactant system at high concentration requires the evaluation of interacting micelles and possibly the transition from spherical to cylindrical or lamellar objects. In generating Fig. 2 these effects were not taken into account.

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Apart from the region very close to the first appearance of the micelles, where the slope ∂Ωm /∂g ≈ 0, we notice that over a wide region near a practical cmc, i.e. near Ωm = 5, the slope ∂Ωm /∂g is approximately constant. The precise value Ωm is not very important, because the change of g in this area is less than fluctuations of g.

Figure 3: Radial profiles for spherical micelles made of G1 Lysine-dendron surfactants with Ωm ≈ 5 kB T (green dot in fig. 2). a) the distribution of tail (red), head groups (green) and overall (yellow). The distributions of the charges in the surfactant, i.e. the NH+ 3 groups (black) and the two ions (pink and gray) are also given in the inset in an enlarged scale. The vertical dotted lines represent measures for the core and the overall size of the micelle, respectively. b) the corresponding electrostatic potential (in Volt) in log-lin coordinates (left ordinate; black line) and the dimensionless charge q(r) ≡ q(r)/e (right ordinate; red line); interpreted as the number of charges at a distance r. The electrostatic potential at the dashed line is argued to be an estimate for the ζ-potential that can be measured by electrokinetic experiments.

Let us next consider the radial density profiles of the micelles. In figure 3a we show such profiles for the G1 Lysine-dendron surfactant system at the default parameter setting for the micelle with work of formation Ωm = 5 kB T . In this figure we see the distribution of the tails, the head group segments and the solvent. In the inset the distribution of the charged species is enlarged. In this figure we also show by vertical dotted lines a measure for the core size Rcore (left dotted line) and the overall micelle size (right dotted line). Note that r = 0 is the center of the micelle and with increasing r we go in the radial direction. The core of the micelle extends to about r = 5. For r < 5 the volume fraction of tails is close to unity. Water and ions avoid this region. The dendritic head group have

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some overlap with the core, but this overlap essentially should be interpreted as the corecorona interface. We also see that the overall density of segments in the corona tops at about 0.4. This unusually high value is attributed to the rather compact dendritic structure of the surfactant heads. The maximum of the charge density is of course a factor of 10 lower following the fraction of charged groups in the dendron head. Most of the dendron charges are situated slightly towards the periphery of the corona (see inset). It is of interest to mention that the counterions more or less copy the distribution of the dendron charge and that most of the difference can be attributed to a depletion of the co-ions. In figure 3b we show the radial electrostatic potential ψ in log-lin coordinates (left ordinate) together with the corresponding radial (dimensionless) charge q/e distribution (right ordinate). The dotted line in this graph is the same as the right dotted line of panel a. The electrostatic potential at the position that corresponds to the size of the micelle is identified pragmatically as the ζ-potential, which is understood as a potential that can be measured by electrokinetic experiments. In line with the distribution of charges discussed above, the radial charge distribution only marginally deviates from zero. There is a minor positive excursion at the location of the position of the dendron charges and there are negative values outside the micelle. The latter is attributed to the diffuse part of the electric double layer. Even though the estimated number of charges due to the dendron heads is approximately 120, there are only about 10 uncompensated charges. This low number is typically pointing to a Donnan layer and the electrostatic potential in the head group region may be interpreted as the Donnan potential. At the relatively high ionic strength conditions selected for this example, the charge density outside the micelle is small. The described charge separation gives causes (via the Poisson eqn 3) a radial electrostatic potential profile (fig.3b, left ordinate) which is approximately constant in the head group region and decays exponentially outside the micelle as known from the Gouy Chapman theory. This decay length is exactly given by the Debye length in the system. Inspection of the electrostatic potential profile (fig.3b gives that the

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estimate for the electrokinetic potential ζ ≈ 50 mV. When we would have positioned the plane of shear to be somewhat larger than the micelle size, we will find correspondingly lower values. Hence we should interpret the value of the ζ-potential as «indicative». By dynamic light scattering one can find estimates for the hydrodynamic size of the micelle, and we here assume that the we may use the same measure of the micelle size based on the second moment of the head group segments as a measure for the hydrodynamic shear plane.

Figure 4: (a) The aggregation number nagg of Lysine-dendron surfactants as a function of the volume fraction of salt ϕbs . The square dots refer to experimental estimates. 76 b) left ordinate (solid lines), the volume fraction of freely dispersed surfactants at the cmc as a function of the volume fraction of salt for G0, G1 and G2 cases as indicated. Right ordinate (dotted lines) we plot ln ϕm or equivalently −Ωcmc (where cmc refers to the maximum of the work of formation curve) as a function of the volume of salt. Other parameters have the default value.

An important measure of self-assembly is the mass of the associates which, e.g., may be found by static light scattering. When we divide this by the mass of the surfactant one gets the micellar aggregation number g. Our theoretical estimate of this aggregation number is based on the excess amount of surfactants in the most-likely micelle which has a grand potential Ωm = 5 kB T and we use the variable nagg for this case. In fig. 4a we present the aggregation numbers for the three Lysine-dendron surfactants G0, G1 and G2 as a function of the dimensionless salt concentration. The solid points are corresponding data points from experiments 76 which will be discussed at the end of the paper. As anticipated above, the micelles for the G2-surfactants exist only at relatively high

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ionic strengths (it vanishes below ϕbs ≈ 0.005). The G1-surfactant only forms micelles above 0.002, and the G1-micelles can tolerate ionics strengths below ϕbs < 0.001. Also anticipated from general considerations we notice that the aggregation number is an increasing function of the ionic strength. In more detail, the aggregation numbers for the G2-micelles is less than 10 even when the ionic strength is of order 1 M. The G1 system has significantly larger aggregation numbers unless the system is close to the disintegration at low ionic strengths. Finally, the G0 case is characterized by typical micelle aggregation numbers of order 50 to 100 depending on the ionic strength. Again at very low ionic strength the aggregation number drops and when the micelle falls below a critical aggregation number the micelle stability vanishes. We may compare the growth of the micelles for the various generations. In absolute sense the growth is largest for the G0 and smallest for the G2 micelles. However, the relative growth, that is, when the aggregation number is scaled to the value at ϕbs = 0.02 we find the opposite to be true. Hence to double the size of the micelles for G2 needs less of a change in ionic strength than for G0. SCF calculations allow for the evaluation of the volume fraction of the freely dispersed surfactants at the cmc, which we refer to as ϕbcmc , and the (dimensionless) concentration of surfactants in micelles ϕm = − ln Ωcmc . The sum of these two contributions give the overall surfactant concentration. These two quantities are plotted in fig. 4b as a function of the logarithm of the ionic strength for the G0, G1 and G2 surfactants. For both contributions to the total surfactant concentration (at the cmc) a significant increase is predicted with decreasing ionic strengths. We may understand that by decreasing the salt concentration the charges in the corona are less effectively screened and therefore the chemical potential of the surfactants (at the cmc) must increase. Interestingly the slope of the log of the free surfactant concentration with salt is strongest for the highest generation surfactants which correlates with the largest number of charges per molecule. The volume fraction of micelles at the cmc also increases with decreasing ionic strength and the curves for G0, G1 and

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G2 are following similar trends. It is noticed that for the G0-case at the cmc the micellar contribution is much lower than the freely dispersed surfactants, whereas this no longer is the case for G2. Experimentally we can only measure the total, that is the sum of ϕm and ϕb . This implies that for the G2 surfactant the cmc is relatively high due to the many surfactants needed to make the micelles, while the cmc for the G0 case is fully determined by the equilibrium concentration of freely dispersed surfactants.

Figure 5: (a) The size of micelle core Rcore and (b) that Hshell shell of micelle corona, both in lattice units, as functions of the bulk volume fraction ϕbs of salt. Micelles with Ω = 5 kB T were selected for dendrimer surfactants as indicated. Parameters have the default value. Square dots refer to results from experiments. 76

Closely linked to the aggregation number of a micelle are the measures for the size of the core and that of the shell. The sum of these two obviously give the overall size of the micelle. In figure 5 we show both quantities for the G0, G1 and G2 surfactants, again as a function of the ionic strength. As before micelles were selected for which the work of formation Ωm = 5 kB T (that is for micelles which exist at a reasonably high micelle volume fraction of ϕm ≈ 0.007). As the density of tails in the core is close to unity, it is obviously the 1/3

core size is strongly linked to the aggregation number (Rcore ∝ nagg ). Hence, an increase of the radius by two requires an increase in the aggregation number by 23 = 8 and this explains the trends found in figure 5a. Note that for the G2 case the size of the core becomes of order unity and this also explains why the micelles loose their stability. Even though a similar scaling is strictly not expected for the overall size, because the density in the corona is not 23


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unity, the trend (not shown) remain the same. This is because the corresponding corona size presented in figure 5b, is only a very weak function of the salt concentration. It is not hard to understand that the corona size is larger for the G2 surfactant than for the G0 variant. It is noticed that the overall size of G0-micelles is larger than that of G1-micelles, which in turn are larger than G2-ones, even though the size of the head groups varies oppositely. The relatively large head group size (for G2) explain why the overall size of the micelles is not a strong function of the generation number.

Figure 6: (a) The zeta-potential ζ (in Volt) and (b) fluctuations in micelle sizes in system with Ω ≈ 5kB T as functions of the bulk volume fraction ϕbs of salt for G0, G1 and G2 micelles as indicated. Interaction parameters taken from the table 1.

It is not immediately clear what to expect for the ζ-potential as a function of the ionic strength. With decreasing salt concentration in general the potentials grow to higher values, but as we have seen the aggregation number and hence the number of charges in the micelle decreases with salt. Inspection of fig. 6a proves that the first effect is the strongest. The ζ-potential increases with decreasing salt concentration. It is of some interest to note that the surfactant with the fewest charges in the head group (G0) forms micelles with the largest ζ-potential. This must be attributed to the much larger aggregation number. In our case total charge of a micelle consisting of surfactant molecules with smaller charge occurs to be much larger than that of a micelle consisting of more charged surfactants. Therefore zeta-potential in the first case will be higher, than in the second one. For example at a salt concentration of ϕbs = 0.01: for the surfactant G0 with 2 charges per molecule there are about 24


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100 surfactants, i.e. 200 charges in micelle. From another side for the surfactant G2 with 8 charges per molecule, there are only 10 surfactants, i.e. 80 charges in micelle. Unfortunately we do not have experimental data that verifies this prediction. The other noteworthy feature is that the ζ-potential only appears a strong function of the ionic strength at low values of the ionic strength. The work of formation of micelles as a function of the aggregation number was already presented in fig. 2a. The relative fluctuations δg/g (see Support Information) of the micelle size, or equivalently the width of the micelle size distribution can be found from −∂Ωm /∂g. Inspection of fig. 2a reveals that the fluctuations in micelle size is non-monotonic with the dendrimer generation number: it is highest for the G2-micelles and lowest for the G1micelle and intermediate for G0. In figure 6 we have collected the fluctuations for the three surfactants as a function of the volume fraction of salt. From the presented result we see that the trends persist in the whole range of ionic strengths. In fact it is found that the salt concentration has little influence of the fluctuations. The variance of the size distribution √ δg for the G2 is about 0.3nagg ≈ 0.5nagg for the G2 and correspondingly less for the other two. Currently we have no experimental data that support this prediction. We may speculate why the G1 has the sharpest micelle size distribution. One may argue that the lowest fluctuations occur for systems that are far from the first appearance of the micelles and far from the second cmc, that is the concentration for which worm-like micelles appear in the system. Because of the large head group size the G2 micelles are typically close to the smallest possible micelles, the G0 micelles on the other hand may be closer to the second cmc. We judge the morphology by comparison of the value of the grand potential for spherical micelle in an equilibrium state and that of the spherical micelle with the same chemical potential as for cylindrical micelles in equilibrium state. 4 Following the experimental data by Misharghi et al, 76 it is of interest to discuss how micelle characteristics are expected to change as a function of the tail length Nt . For convenience we focus on the micelles with work of formation Ωm = 5 kB T . It was shown in, 90

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Figure 7: The aggregation number nagg for micelles with grand potential Ω = 5kB T and salt volume fraction ϕbs = 0.01 (0.1 M) as functions of the tail length Nt in log-log coordinates. Interaction parameters taken from the table 1. The dots refer to experiments. 76

that for longer tails the driving force is larger. Therefore it is natural to expect the resulting micelles to be larger (both in size as well as in aggregation number). In figure 7 we see these trends for all three types of surfactants in double logarithmic coordinates. Even though we can vary the tail length of the surfactants only in a narrow interval (for large Nt G0-surfactants do not form spherical micelles), we can try to interpret the slope of the dependences. Naively one could think that the size of the core should scale proportional to the tail length. In this case the aggregation number should scale as the third power with Nt . In this argument one fully ignores the stopping force of the head groups and as a result we expect weaker power-law coefficients. Inspection of the result of fig. 7 shows that for the dendron micelles we find the highest slope for the G0- and the lowest slope for the G1- while G2 has an intermediate slope albeit that it is close to that of G1. The Figures 8a and b present the sizes of core and shell, respectively, as functions of tail length, for the corresponding micelles presented in fig. 7. In this case we use linlin coordinates. Again, the cores size is directly linked to the aggregation number of the micelles because the core density is close to unity. The size of the corona is a very weakly increasing function of the tail length. It is natural to expect that when the strength of the driving force increases (increase of Nt ) that also an increase in the stopping force is necessary. 26


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Figure 8: (a) The size of micelle core Rcore and (b) Hshell of the shell of corona of micelle consists of Lysine-dendron surfactants, both in lattice units, as functions of the tail length Nt . Micelles with Ω = 5 kB T were selected and under ϕsb = 0.01. Square dots refer to results from experiments. 76

This increased force results in a stronger overlap of the dendron heads. This overlap leads to a stretching of the corona in the radial direction. As the variation in the tail length is modest the increase in corona size is correspondingly modest.

Figure 9: Aggregation number for acetylated Lysine-dendron surfactants in system with Ω = 5kB T as functions of the salt concentration ϕbs (a) and tail length Nt (b) for three types of surfactants G0a, G1a and G2a. Interaction parameters taken from the table 1. The dot is an experimental data point. 76

A special feature of the lysine-based dendrons is the option to perform an acetylation reaction. The most dramatic effect of this reaction is the reduction of the number of charges. This renders the head groups to be less hydrophilic. On top of this a few hydrocarbons are added to the head group. The appearance of and «O» and an «NH» group compensate 27


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somehow the drop in hydrophilicity of the head groups. Experimentally it is found that acetylated variants, which we here will denote by G0a, G1a and G2a have noticeably changed self-assembly characteristics. Lets start with acetylated variants that have the default tail length Nt = 16 and discuss the ionic strength dependence of the aggregation numbers. As usual we focus on micelles with grand potential 5 kB T . We limit our results to cases for which the spherical micelles are still stable. This means that for the G0a-case no results are given for higher ionic strengths. From figure 9a we find that similarly as the lysine variants (cf fig. 4) the aggregation number increases with salt concentration. The numerical values for the aggregations numbers significantly increased due to the acetylation reaction (compare figs 9a with 4a). This trend is in line with the expected reduced hydrophilicity of the acetylated variants. It is of interest to mention that unlike for G2 the acetylated variant G2a also has stable micelles for very low ionic strengths. Hence the window for micellization increased for the second generation surfactants and decreased (for spherical micelles) for the zeroth generation surfactants. In figure 9b we show the results for the aggregation numbers for the micelles formed by acetylated surfactants as a function of the tail length at a fixed ionic strength of ϕbs = 0.01. Again it is found that the micelles aggregation number strongly increases with tail length and the strongest dependence is found for the G0a case. This result can directly be compared to the corresponding figure for the lys-micelles given in fig. 7. In particular for G0 the slope is the same on both graphs. For both the first and second generation dendrimer micelles the power-law coefficient is slightly reduced (difference in slope is equal to 0.4) by acetylation. It means, that the effect of the acetylation on the growth of the aggregation number by the increase of the hydrophobic tail is weak. It is of considerable interest to present also the size characteristics of the micelles formed by the acetylated surfactants. In figure 10a the dependence of the corona sizes is shown as a function of the salt concentration for the C16 tail length, while in figure 10b the same quantities are plotted as a function of the tail length for given ionic strength ϕbs = 0.01. It

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Figure 10: The size of micelle corona Hshell in lattice units (a) as functions of the bulk volume fraction ϕbs of salt and (b) as functions of the tail length Nt of micelle consists of acetylated Lys-dendron surfactants. Micelles with Ω = 5 were selected and under ϕsb = 0.01. Square dots refer to results from experiments. 76

is seen that the corona size for high generation surfactant micelles is larger than that of the low generation micelles. Quantitatively the dependences with salt concentration and chain length are different from the corresponding results for the lysine-base surfactants, but the qualitative features are identical. It therefore suffices to point to a few noteworthy points. In general we find that the corona sizes for the acetylated systems are less strongly a function of the generation number than in the lysine case. We may attribute this to the reduced number of charges in the corona. Together with the increased hydrohobicity of the heads the average density in the corona layer increased and we therefore expect at ternary interactions have contributed more to the stopping forces in the acetylated case than for the lysine ones. Next, it is found that the corona size Hshell is a weakly decreasing function of the salt concentration. This result is found for the lysine case but is more clearly visible for the acetylated variants. This size decrease is due to the screening of the charges which reduces the repulsion between dendron heads, which apparently is not overcompensated by the increase in the aggregation number. Finally, we point to the observation that the core size of G0a tends to be significantly larger than the corona (see, a Supporting Information). As explained above this is a indication that spherical micelles might become unstable against the formation of worm-like micelles.

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Discussion Molecularly detailed modeling of the self-assembly of lysine-based surfactants and the acetylated variants by self-consistent field theory reveals a rather complete picture with many interesting features. Many of these are at least qualitatively in line with the surfactant literature. It is of more than average interest to compare such predictions more quantitatively with experimental data. In several figures discussed above we have added experimental data points found by careful experiments reported by Misharghi et al. 76 We may now go back to the various quantities and find out that also quantitatively the performance of the theory is rather good. This is, admittedly, also due to the small adjustments of the parameter set given in table 1 compared to previous estimates (see e.g. ref 89 ). Of course one may always argue that by specific set of parameters one can hide away the mean-field errors. Of course this can be the case but we believe that the parameters given in table 1 are reasonable. The value for χW,O = χW,N H = −0.6 more specifically was use to make agreement with both sets (lysinebased and acetylated variants) more perfect. We realize of course that water is a complex solvent and the use of such a value might still be acceptable. We are particularly satisfied with the agreement for the prediction of the aggregation numbers (cf fig. 4 and 9). In this case the numbers almost quantitatively agree. As it comes to the measures of the sizes of the core and the corona the agreement is a bit poorer but still may be qualified as good (cf. fig. 5 for the lysine micelles and 10 for the acetylated variants). We note that the overall size of the micelle (sum of core and corona) is even more quantitatively in agreement with experiments, because SCF typically slightly overpredicted the core size and underpredicted the dimensions of the corona. The latter deviations were a bit larger for the acetylated case than for the lysine based micelles. This might indicate that the assumption of spherical symmetry of the core and corona might be a bit worse for the acetylated case than for the lysine based micelles. Finally, the SCF theory also reasonable accurately predicted the tail length dependences (figs 7), but we admit that there are not 30


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many data points. It is important to mention here as well that we have used a conversion for the dimensionless volume fraction of salt to the molar concentration a factor of k = 10 whereas the lattice approximations of SCF call for a factor that is 6 times higher. It is possible to avoid this ad hoc choice, but for this we need to introduce salt ions with different interactions with the solvent (water), apolar phase (C) and importantly with the hydrophilic groups in the corona. Here we did not attempt to change the parameters in this direction and opted for a pragmatic (engineering) solution. In future however we may return to this problem and consider non-ideal properties of the ions, including the effects of size and hydrophobicity of the ions. In this work we have focused on spherical micelles only and omitted the regimes for which the worm-like micelles become the dominant species. More specifically we found worm-like micelles to occur for the G0a system especially for high ionic strengths and for large chain lengths Nt . This prediction is worth exploring in experimental systems.

Conclusions We have presented a detailed SCF analysis for the formation of spherical micelles of lysinebased dendritic surfactant and acetylated variants and compared these to experimental data. The theory relies on mean field and lattice approximations, implements a freely jointed chain model for the chain statistics and accounts for solvency effects on a Flory-Huggins- and electrostatic interactions on a Poisson Boltzmann level. A relatively simple set of parameters was used to find semi-quantitative agreement between the SCF results and experiments. In a wide range of ionic strengths, that are experimentally accessible, we find spherical micelles for dendritic micelles of generation G0, G1 and also G2. The latter micelles only exist at high ionic strengths but the acetylated variants exists also at lower salt concentrations. At the same time the acetylated variant for the lowest generation G0a is vulnerable to

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suffer a transition to worm-like micelles especially at high ionic strengths. There are several predictions of the modeling which still await experimental verification. For example, we have presented a non-trivial result for the ζ-potential which was shown to be a decreasing function of the generation number even though a dendrimer with higher generation carries more charges per molecule. Another interesting result was found for the width of the micelle size distribution. This width goes was lowest for G1 and higher for G0 and G2 dendritic micelles.

Acknowledgement The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University. 95 This work was supported by grant of the Government of Russian Federation 074-U01 and by RFBR grant 16-03-00775.

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Self-Assembly of Lysine-Based Dendritic Surfactants Modeled by the

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