Advertisement

From the Atomistic to the Macromolecular Scale: Distinct Simulation Approaches for Polyelectrolyte Solutions

  • Jens Smiatek
  • Christian HolmEmail author
Living reference work entry

Abstract

Polyelectrolytes reveal interesting properties in solution. At short length scales, the dissociation of counterions is heavily affected by the chemical structure of the polyelectrolyte, the properties of the solution, and specific ion effects. At larger length scales, the structure of polyelectrolyte solutions is dominated by long-range interactions. In the special case of dissolved polyanions and polycations, polyelectrolyte complexes or multilayers can form. In this review we present distinct simulation approaches to study the corresponding effects at different length scales in more detail. Whereas at short length scales, atomistic molecular dynamics simulation is often the method of choice, semi-coarse-grained and coarse-grained models with a lower level of details reveal their benefits at larger length scales.

1 Introduction

In accordance with the IUPAC definition, polyelectrolytes are charged macromolecules, in which a substantial portion of the constitutional units contains ionic or ionizable groups, or both (McNaught and Wilkinson 1997). Strong polyelectrolytes like DNA are completely ionized, whereas weak polyelectrolytes like polyacrylic acid show an incomplete and pH-dependent dissociation behavior. Consequently, the properties of polyelectrolytes are dominated by a combination of electrostatic and molecular interactions, which heavily affect the occurring conformations and the resulting ion dissociation behavior. In contrast to uncharged polymers, the long-range decay of electrostatic interactions between the ionic groups impedes the derivation of scaling relations in terms of simple mean-field approaches (de Gennes 1979; Doi and Edwards 1988; Dobrynin and Rubinstein 2005; Dobrynin 2008). Thus, the complex interplay between the polyelectrolyte, the ions in the solution, and the solvent reveals many interesting phenomena at different length scales, which can be studied in more detail with the help of distinct molecular dynamics (MD) simulation approaches in combination with appropriate models for the considered species.

At short length scales, previous atomistic MD simulations demonstrated that the molecular solvation behavior significantly influences the amount of ionized groups and thus the corresponding conformation with regard to counterion condensation effects and repulsive electrostatic interactions along the polyelectrolyte backbone (Smiatek et al. 2014). Vice versa, the behavior of the surrounding solvent molecules is also modified by the presence of ionic groups, which is reflected by local variations concerning the dielectric permittivity and the solvation of ions (Fahrenberger et al. 2015a; Mukhopadhyay et al. 2012). In contrast to these local interactions, the aggregation of polyelectrolytes becomes important at larger length scales, such that polycations and polyanions form polyelectrolyte complexes and multilayers, whose stability significantly depends on enthalpic and entropic contributions and the properties of the embedded solvent molecules (Cerdà et al. 2009; Qiao et al. 2011, 2012). Also in bulk solution, molecular properties determine the configurational behavior of the polyelectrolyte, as can be seen by the formation of polyelectrolyte micelles, pearl-necklace structures, or the onset of microphase separation processes between polar and apolar regions (Limbach and Holm 2003; Limbach et al. 2004; Dormidontova et al. 1994). Thus, appropriate models with a sufficient degree of detail are needed for reliable simulations of polyelectrolyte solutions in order to elucidate the properties of interest.

In this chapter, we introduce several well-established approaches for the simulation of polyelectrolyte systems at different length and time scales. The benefits and drawbacks of atomistic, semi-coarse-grained and simple coarse-grained models are discussed, and we will present representative examples for various applications. In the last section, we summarize the main points and address current limitations concerning the general applicability of the methods.

2 Simulation Approaches for Polyelectrolyte Solutions at Different Length Scales

The discussion of methods in this section follows a bottom-up approach, which means that we start with atomistic models and the description of molecular force fields. Hereafter, we decrease the level of resolution in terms of semi- and simple coarse-grained approaches, which are best suited for the simulation of processes at large length and long time scales. All models rely on the consideration of explicit particles in combination with molecular dynamics or Monte Carlo time integration schemes. It has to be noted that other continuum or self-consistent field approaches can also be used, for instance, as described in Schmid (1998). Over the last years, multiscale simulation approaches were also developed, which rely on a combined consideration of the following methods in terms of matching and adaptive resolution schemes. The reader is referred to Fritz et al. (2011) and Praprotnik et al. (2008) for more details and for other contributions to this volume.

2.1 Atomistic Models: Importance of Chemical and Molecular Details

Due to the massive increase of computational power over the last years, it is nowadays possible to study the properties of short polyelectrolyte chains, the so-called oligoelectrolytes, for hundreds of nanoseconds via atomistic MD simulations in combination with accurate molecular force fields. In terms of modern force fields, molecular properties are dictated by bonded and nonbonded interactions, which are represented by classical potential functions in order to avoid an extremely expensive evaluation of the electronic behavior. In more detail, standard atomistic force fields like OPLS/AA (Jorgensen et al. 1996) include parameters for bonded, angular, and dihedral potentials in addition to nonbonded interactions, which are usually represented by Coulomb and Lennard-Jones contributions. In combination with advanced electrostatics algorithms (Deserno and Holm 1998; Arnold et al. 2013) and the sophisticated use of graphics processing units (GPUs), the properties of polyelectrolyte solutions with a dimension up to several nanometers can be studied for hundreds of nanoseconds by standard atomistic MD approaches. Hence, detailed information on the molecular interactions, the solvent behavior, and the corresponding distribution functions are accessible.

As an illustrative example, the stable conformations for a sulfonated oligosulfonic acid with sodium counterions in water and chloroform are displayed in Fig. 1, where more details of this system can be found in Smiatek et al. (2014). It can be clearly seen that the oligoelectrolyte reveals a swollen conformation in polar solvents like water when compared with chloroform. This finding can be related to molecular solvation effects and standard polymer theories, which distinguish between good, poor, and theta solvents (de Gennes 1979). Thus, the average size of the polyelectrolyte with Np monomers can be described by a power-law behavior according to \(R\sim N_p^{\nu }\) with the excluded-volume parameter ν, which is usually ν = 0.588 for uncharged polymers in good solvents (Doi and Edwards 1988). Depending on the actual value of the excluded-volume parameter, good and poor solvents are defined by ν > 1∕2 and ν < 1∕2, respectively, whereas a theta solvent is characterized by ν = 1∕2. A polyelectrolyte chain at infinite dilution should scale with ν = 1. Hence, the molecular properties of the polyelectrolyte adapt directly to the polarity of the solvent, which is reflected by an increase or a decrease of the solvent-accessible surface area and the size.
Fig. 1

Sulfonated oligosulfonic acid with nine monomers according to Np = 9 and an equivalent number of sodium counterions (blue spheres) in water (left side) and chloroform (right side). The solvent molecules are not shown for reasons of clarity, and the chemical structure of one monomer is depicted in the middle

Interestingly, further analysis concerning atomistic models for polyelectrolytes revealed that the counterion behavior differs significantly from standard theoretical descriptions (Manning 1969, 1996; Deserno et al. 2000; Deserno and Holm 2001), which highlights the benefits of atomistic MD approaches in order to verify molecular theories and to study the corresponding deviations. As a result of these simulations (Lund et al. 2008; Heyda and Dzubiella 2012; Smiatek et al. 2014; Batys et al. 2017), it was found that the counterion distribution around the polyelectrolyte can deviate from standard mean-field predictions in terms of the Poisson-Boltzmann theory (Andelman 1995), which can be attributed to specific solvation effects and molecular interactions (Smiatek et al. 2014). Furthermore, it was also observed that specific ion effects (Marcus 2009; Kunz 2010; Lo Nostro and Ninham 2012) and conformational properties of polyelectrolytes (Wohlfarth et al. 2015) influence the corresponding counterion distribution and yield results in good agreement with experimental data. To summarize, also atomistic simulations reveal short-range deviations from simple mean-field theories; however, Poisson-Boltzmann theory and its generalizations often agree surprisingly well for global observables like the osmotic pressure or effects that are dominated by the Coulomb interactions (Deserno et al. 2001; Antypov et al. 2005; Antypov and Holm 2006; Lu et al. 2008).

2.2 Semi-Coarse-Grained Approaches: MARTINI Force Fields and Refined Models

Atomistic MD simulations mainly intend to study the properties of short oligoelectrolyte chains and their molecular interactions with counterions and other components of the solution. Although also larger systems were studied, for instance, polyelectrolyte complexes and multilayers (Farhat et al. 1999; Micciulla et al. 2014), it has to be mentioned that these simulations are very CPU-time demanding and usually restricted to several tens of nanoseconds, such that even a reasonable equilibration of these complex solutions is hard to achieve. In order to introduce efficient but still accurate models for these larger systems, the use of semi-coarse-grained approaches is highly beneficial. In general, coarse-grained approaches intend to reduce the degrees of freedom when compared with atomistic simulations, which results in a significant acceleration of the computation times and the possibility of using larger time steps (Winger et al. 2009; Marrink et al. 2010). Standard coarse-grained frameworks for polyelectrolytes and polymers include the introduction of simple bead-spring models (Doi and Edwards 1988). Hence, several atoms of a monomer are represented by one interaction site, the so-called bead, while the individual beads are connected with springs, which are usually modeled by classic harmonic or modified Finitely Extensible Nonlinear Elastic (FENE) potentials. In contrast to the most simple coarse-grained methods, semi-coarse-grained approaches like the MARTINI force field (Marrink and Tieleman 2013) refine these very generic models by consideration of important chemical details. For instance, the MARTINI force field usually relies on a 4:1 mapping scheme, such that four heavy atoms are combined into one interaction site (CG bead) with parameterized polarity values and hydrogen bond acceptor and donor abilities (Marrink et al. 2007). In more detail, the MARTINI CG beads can be divided into different particle-type classes (polar (P), nonpolar (N), apolar (C), and charged (Q) species). The subtypes within these classes are categorized due to their ability to form hydrogen bonds (donor (d), acceptor (a), both donor and acceptor (da), and none of them (0)) and with regard to their polarity (from 1 = low polarity to 5 = high polarity) (Marrink et al. 2007; Marrink and Tieleman 2013). Further subclasses were also defined to increase the local resolution (Marrink et al. 2007). All CG beads reveal different Lennard-Jones parameters and partial charges, which were parameterized according to partitioning coefficients for similar atomic groups in oil/water mixtures (Marrink et al. 2007; Marrink and Tieleman 2013).

Over the last years, several MARTINI models for polymers and polyelectrolytes were developed. Specific examples are DNA (Uusitalo et al. 2015), poly(styrene sulfonate) (PSS) and poly(diallyldimethylammonium) (PDADMA), which were both used for the study of polyelectrolyte complexes (Vögele et al. 2015b). The molecular topologies of PSS and PDADMA in terms of the MARTINI force field and the corresponding particle types according to Marrink et al. (2007) are shown in Fig. 2. In combination with refined polarizable water models (Yesylevskyy et al. 2010; Michalowsky et al. 2017), it was shown that MARTINI simulations are well suited to study the properties of highly charged systems at large length and time scales (Vögele et al. 2015b,a; Uusitalo et al. 2015).
Fig. 2

Chemical structure of PSS (left side) and PDADMA (right side) with three monomers. The differently shaded regions denote spherical MARTINI CG beads with particle-type classes as defined in Marrink et al. (2007)

The advantages of MARTINI models are mostly given by their flexibility and their transferability, but it has to be noted that solvent particles indeed have to be considered explicitly. In order to circumvent time-consuming calculations, the matching of potential of mean forces between atomistic and coarse-grained simulations provides a computationally efficient route in terms of tabulated potentials and thus an implicit solvent approach (Brini et al. 2013; Reith et al. 2002; Li et al. 2012; Lyubartsev and Laaksonen 1999; Savelyev and Papoian 2010; Hsu et al. 2012). Although this method, which is also often called iterative Boltzmann inversion technique (Reith et al. 2003), avoids time-consuming calculation of interactions between solvent particles and between solvent and polyelectrolyte groups, it has to be mentioned that the force-matching method is mostly applicable for homogeneous solutions without interfaces. More refined coarse-grained models for DNA and ionomer systems, based on the matching of ion mobilities, were published in Lu et al. (2014), Weik et al. (2016), and Rau et al. (2017). A semi-coarse-grained approach for the simulation of weak polyelectrolytes was recently introduced in Landsgesell et al. (2017b,a). Furthermore, the well-known decrease of the dielectric permittivity around charged objects and the corresponding consequences were recently studied in coarse-grained polyelectrolyte solution via a modification of the Maxwell equation molecular dynamics algorithm (Fahrenberger and Holm 2014; Fahrenberger et al. 2015a,b).

In summary, refined or semi-coarse-grained models can be used for the study of systems at intermediate length and time scales. Nevertheless, for the study of long time-scale processes like transport behavior or the influence of hydrodynamics on polyelectrolyte motion, the use of simple coarse-grained methods remains the most suitable choice.

2.3 Simple Coarse-Graining: Generic Bead-Spring Models with Explicit Charges

In general, all simple coarse-grained models are composed of single interaction sites, which have a lower resolution when compared with semi-coarse-grained approaches and thus usually correspond to individual monomers or the number of monomers within the corresponding persistence length (Doi and Edwards 1988). All adjacent beads are connected by springs in terms of simple harmonic or FENE potentials, which restrict the length of the bonds to the equilibrium distance in order to avoid entanglement effects in polymer melts (Kremer and Grest 1990; Stevens and Kremer 1993a,b). In contrast to semi-coarse-grained approaches, simple bead-spring models do not include angular or dihedral potentials per definition, and the corresponding nonbonded interactions are represented by simple Lennard-Jones (LJ) and Coulomb interactions. Most often, the WCA potential, a purely repulsive shifted and truncated version of the Lennard-Jones potential, is used to mimic hard spheres for beads and ions (Weeks et al. 1971). In contrast to MARTINI models, the solvent is often modeled implicitly by consideration of a global dielectric constant, which is thus inserted into the Coulomb potential. Furthermore, the model can be even more simplified by using a screened electrostatic potential or neglecting Coulomb interactions all together (Hickey et al. 2012; Szuttor et al. 2017; Roy et al. 2017). Specifically in solvents with high values of the dielectric constant, for instance, water, electrostatic interactions between charged groups and ions dominate only at short distances (Collins 2004). This can be mostly attributed to the low value of the Bjerrum length at room temperature λB = e2∕4π𝜖0𝜖rkBT with the elementary charge e, the vacuum permittivity 𝜖0, the dielectric constant 𝜖r, the temperature T, and the Boltzmann constant kB. The Bjerrum length estimates the distance where the thermal energy dominates over the electrostatic energy and which is for water at 300 K around λB ≈ 0.7 nm corresponding to two hydration shells (Collins 2004; Marcus and Hefter 2006). For larger and highly charged objects like polyelectrolytes or colloidal particles, also the salt concentration plays a significant role in order to induce a fast decay of electrostatic interactions. Hence, by a simple linearization of the Poisson-Boltzmann equation in terms of the Debye-Hückel approximation, the corresponding electrostatic screening length reads \(\lambda _{D}=(\epsilon _r\epsilon _0k_BT/(\sum _i 2z_i^2e^2\rho _i))^{1/2}\) with the valency zi and the ion density ρi (Andelman 1995). With regard to this relation, one usually obtains a screening length of λD ≈ 1 nm for water at room temperature with a salt concentration of 0.1 mol/L, which implies that the monomers are only weakly affected by the electric field of the surrounding polyelectrolyte groups (Szuttor et al. 2017). Hence, for large objects like λ-DNA, it is often sufficient to neglect electrostatic interactions between the monomers, if the Debye and the Bjerrum length are significantly smaller than the size of the polyelectrolyte in accordance with λD ≈ λB ≪ R (Szuttor et al. 2017; Roy et al. 2017).

Most of these simple coarse-grained models are used to study the dynamics of polyelectrolytes and other components in solution, often under the influence of external forces in order to induce transport processes. Recent reviews highlighted the benefits of these models in combination with sophisticated mesoscopic simulation techniques for the study of transport processes and electrokinetic effects in microchannels (Slater et al. 2009; Pagonabarraga et al. 2010; Smiatek et al. 2012). In more detail, mesoscopic simulation approaches induce a stochastic motion of the solute species in the system, which is related to the behavior at long time scales. A simple but effective and thermodynamically consistent approach is represented by Langevin or Brownian dynamics (Kremer and Grest 1990). However, as it has been pointed out in Dünweg (1993), momentum is not conserved in Langevin dynamics, such that this approach cannot capture any hydrodynamical effects (Doi and Edwards 1988; Ober and Thomas 1997; Grass et al. 2008; Frank and Winkler 2009). Most often, one is indeed specifically interested in hydrodynamic effects, such that either the Langevin dynamics approach can be modified by introducing the Oseen tensor (Ermak and McCammon 1978) or efficient Navier-Stokes solvers have to be used. In terms of mesoscopic approaches, the most common techniques are dissipative particle dynamics (DPD) (Groot and Warren 1997; Smiatek et al. 2012), coupled lattice Boltzmann/molecular dynamics (LBMD) (Dünweg and Ladd 2009) or multiparticle collision dynamics (MPCD) (Gompper et al. 2008). Previous articles highlighted the good quantitative agreement between DPD and LBMD simulations (Smiatek et al. 2009) and their applicability to study transport processes in confined geometries (Smiatek and Schmid 2010, 2011; Smiatek et al. 2012; Weik et al. 2016). In general, the use of mesoscopic simulation methods can be nowadays regarded as a standard approach in order to study the influence of hydrodynamics in many different research fields.

Furthermore, simple coarse-grained models were often used to validate analytical mean-field approaches like counterion condensation theories (Deserno et al. 2000), stretching forces on tethered polymers (Szuttor et al. 2017), electrohydrodynamic screening effects (Grass et al. 2008), or the combined influence of electroosmotic and electrophoretic motion (Smiatek and Schmid 2010). Interestingly, also the influence of the solvent quality on the resulting conformations can be studied via the use of simple approaches. In terms of experimental and atomistic simulation results, it is known that poor solvents imply collapsed conformations, whereas good solvents lead to a swelling of the polyelectrolyte. Hence, these findings can be transferred to tunable attractive or repulsive interactions between the beads in order to correct for the presence of an implicit solvent. Hence, in combination with electrostatic interactions and attractive bead potentials, it was even possible to enforce the occurrence of pearl-necklace polyelectrolyte structures (Dobrynin et al. 1996; Micka et al. 1999; Limbach et al. 2002; Limbach and Holm 2003). Furthermore, also more complicated topologies like in polymeric ionic liquids (PILs) (Mecerreyes 2011; Yuan et al. 2013) can be modeled via simple coarse-grained approaches (Weyman et al. 2018). A simple example for a common alkylimidazolium-based PIL and a snapshot of a system conformation are shown in Fig. 3. As can be seen in the bottom, a simulation of 30 PIL chains with Np = 30 reveals the occurrence of a microphase separation between polar and apolar beads. The aggregation of apolar beads was initiated by attractive LJ interactions, which initiate the formation of apolar and polar microphases for larger side chain lengths. In summary, simple coarse-grained polyelectrolyte models are computationally efficient and can be used for the study of different systems at larger scales. The consideration of further details can be simply achieved via effective generic or tunable potentials and provides reasonable results, if the specific molecular details and processes are of minor importance.
Fig. 3

Chemical structure of a typical alkylimidazolium-based PIL with counterions X (top left). The corresponding simple bead-spring model with Np = 4 is shown at the upper right part. The charged terminal groups are shown as red spheres and the counterions as blue spheres. All neutral beads have a gray color. A snapshot of the simulation with 30 polymers with Np = 30 in terms of the coarse-grained model is shown at the bottom. Neutral beads are colored in red, while charged beads and counterions are colored in green

3 Summary and Conclusion

In this chapter, we described distinct approaches for the simulation of polyelectrolyte solutions at different length and time scales. For the study of fast processes and molecular interactions at short length scales, the use of atomistic models with regard to appropriate force fields is advised. In terms of larger length and longer time scales, semi- and simple coarse-grained models with different levels of detail can often be considered as the method of choice. A promising approach is the MARTINI force field, which provides a beneficial transferability between different systems without the need of a proper reparameterization for distinct models. Furthermore, also tabulated potential methods as well as refined coarse-grained models were also developed that can be interpreted as coarse-grained approaches with basic molecular properties. For the simulation of transport processes and hydrodynamic effects, simple coarse-grained models in combination with mesoscopic simulation methods are highly beneficial, which can be rationalized by the fact that polymers show a universal scaling behavior at large scales, such that specific molecular details are of minor importance (de Gennes 1979; Doi and Edwards 1988).

In summary, the presented methods can be used for a broad range of systems at different time and length scales. However, which method is most appropriate will depend crucially on the question to be answered, and one cannot give any general advice. However, it has to be emphasized that all methods rely on crucial approximations, and thus, if possible, one should always verify the simulation results with experimental findings or theoretical predictions. A prominent example are specific ion effects, which are modeled explicitly only in few atomistic MD force fields (Fyta and Netz 2012). It is thus a challenging task to transfer this information to coarse-grained models. Moreover, the study of apolar organic solvents is significantly more complicated in comparison to polar solvents, due to the fact that the Debye and the Bjerrum lengths can easily exceed the simulation box size, which induces electrostatic correlation effects between the polyelectrolyte and its periodic images. These finite-size effects modify structural and dynamic properties of the solution and are thus a crucial problem in order to bring simulation outcomes in quantitative agreement with experimental results. Moreover, also non-ideal effects, as they are well known for higher component solutions (Krishnamoorthy et al. 2016), are often not correctly reproduced. In general, all considered methods rely on potential functions to mimic the electronic behavior. Hence, an accurate study of bond formation and cleavage processes in combination with varying pH values of the solution is often impossible. A promising new route is the introduction of reactive force fields (Senftle et al. 2016), which are, although time consuming, less computationally expensive than ab initio simulations. In accordance with the simplification of the electronic behavior, atomic polarization effects are also usually neglected, which have nowadays become an active field of research and model improvement (Lemkul et al. 2016; Bordin et al. 2016).

Despite their limitations, the presented methods and models are the most promising approaches for the reliable study of effects and processes in polyelectrolyte solutions. It can be expected that new and refined approaches in combination with longer simulation times will allow a more accurate study of these systems in the coming years.

Notes

Acknowledgements

We thank Alexander Weyman, Martin Vögele, Anand Narayanan Krishnamoorthy, Florian Fahrenberger, Jonas Landsgesell, Kai Szuttor, Owen A. Hickey, Florian Weik, Tobias Rau, Stefan Kesselheim, Steffen Hardt, Tamal Roy, Andreas Wohlfarth, Klaus-Dieter Kreuer, Lars V. Schäfer, Paulo Telles de Souza, Johannes Zeman, Axel Arnold, Baofu Qiao, Juan J. Cerd\({\grave {\textrm a}}\), Rafael Bordin, Rudi Podgornik, Burkhard D\(\ddot {\textrm u}\)nweg, and Siewert-Jan Marrink for valuable discussions. We thank the Deutsche Forschungsgemeinschaft for funding through AR593/7-1, HO/1108-22-1, HO/1108 26-1, and the Cluster of Excellence Simulation Technology (EXC 310) and the collaborative research center 716 (SFB 716).

References

  1. Andelman D (1995) Handbook of biological physics. School of Physics and Astronomy, Tel Aviv University, p 603, chap 12Google Scholar
  2. Antypov D, Holm C (2006) Optimal cell approach to osmotic properties of finite stiff-chain polyelectrolytes. Phys Rev Lett 96:088302Google Scholar
  3. Antypov D, Barbosa MC, Holm C (2005) A simple non-local approach to treat size correlations within poisson-boltzmann theory. Phys Rev E 71:061106Google Scholar
  4. Arnold A, Fahrenberger F, Holm C, Lenz O, Bolten M, Dachsel H, Halver R, Kabadshow I, Gähler F, Heber F, Iseringhausen J, Hofmann M, Pippig M, Potts D, Sutmann G (2013) Comparison of scalable fast methods for long-range interactions. Phys Rev E 88:063308.  https://doi.org/10.1103/PhysRevE.88.063308
  5. Batys P, Luukkonen S, Sammalkorpi M (2017) Ability of Poisson–Boltzmann equation to capture molecular dynamics predicted ion distribution around polyelectrolytes. Phys Chem Chem Phys 19:24583–24593CrossRefGoogle Scholar
  6. Bordin JR, Podgornik R, Holm C (2016) Static polarizability effects on counterion distributions near charged dielectric surfaces: A coarse-grained molecular dynamics study employing the drude model. Eur Phys J Special Top 225(8):1693–1705.  https://doi.org/10.1140/epjst/e2016-60150-1ADSCrossRefGoogle Scholar
  7. Brini E, Algaer EA, Ganguly P, Li C, Rodriguez-Ropero F, van der Vegt NFA (2013) Systematic coarse-graining methods for soft matter simulations – a review. Soft Matter 9:2108–2119. https://doi.org/10.1039/C2SM27201FADSCrossRefGoogle Scholar
  8. Cerdà JJ, Qiao B, Holm C (2009) Understanding polyelectrolyte multilayers: an open challenge for simulations. Soft Matter 5:4412–4425. https://doi.org/10.1039/b912800jADSCrossRefGoogle Scholar
  9. Collins KD (2004) Ions from the Hofmeister series and osmolytes: effects on proteins in solution and in the crystallization process. Methods 34(3):300–311. https://doi.org/10.1016/j.ymeth.2004.03.021CrossRefGoogle Scholar
  10. de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca. http://books.google.com/books?id=ApzfJ2LYwGUC&lpg=PP1&num=15&pg=PP1#v=onepage&q&f=false
  11. Deserno M, Holm C (1998) How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines. J Chem Phys 109:7678ADSCrossRefGoogle Scholar
  12. Deserno M, Holm C (2001) Cell-model and poisson-boltzmann-theory: a brief introduction. In: Holm C, Kékicheff P, Podgornik R (eds) Electrostatic effects in soft matter and biophysics, NATO science series II – mathematics, physics and chemistry, vol 46. Kluwer Academic Publishers, Dordrecht, pp 27–50CrossRefGoogle Scholar
  13. Deserno M, Holm C, May S (2000) Fraction of condensed counterions around a charged rod: comparison of Poisson-Boltzmann theory and computer simulations. Macromolecules 33:199–206. https://doi.org/10.1021/ma990897oADSCrossRefGoogle Scholar
  14. Deserno M, Holm C, Blaul J, Ballauff M, Rehahn M (2001) The osmotic coefficient of rod-like polyelectrolytes: Computer simulation, analytical theory, and experiment. Eur Phys J E 5:97–103ADSCrossRefGoogle Scholar
  15. Dobrynin AV (2008) Theory and simulations of charged polymers: from solution properties to polymer nanomaterials. Curr Opin Colloid Interface Sci 13:376–388. https://doi.org/10.1016/j.cocis.2008.03.006, http://www.sciencedirect.com/science/article/pii/S1359029408000411CrossRefGoogle Scholar
  16. Dobrynin AV, Rubinstein M (2005) Theory of polyelectrolytes in solutions and at surfaces. Prog Polym Sci 30(11):1049–1118. https://doi.org/10.1016/j.progpolymsci.2005.07.006, http://www.sciencedirect.com/science/article/B6TX2-4H2G8WN-1/2/4b378f3016fcf641ad1821b4ded9d389CrossRefGoogle Scholar
  17. Dobrynin AV, Rubinstein M, Obukhov SP (1996) Cascade of transitions of polyelectrolytes in poor solvents. Macromolecules 29(8):2974ADSCrossRefGoogle Scholar
  18. Doi M, Edwards SF (1988) The theory of polymer dynamics, vol 73. Oxford University Press, New YorkGoogle Scholar
  19. Dormidontova EE, Erukhimovich IY, Khokhlov AR (1994) Microphase separation in poor-solvent polyelectrolyte solutions: phase diagram. Macromol Theory Simul 3(4):661–675CrossRefGoogle Scholar
  20. Dünweg B (1993) Molecular dynamics algorithms and hydrodynamic screening. J Chem Phys 99(9):6977–82ADSCrossRefGoogle Scholar
  21. Dünweg B, Ladd AJC (2009) Lattice boltzmann simulations of soft matter systems. In: Advanced computer simulation approaches for soft matter sciences III. Advances in polymer science, vol 221. Springer, Berlin, pp 89–166. https://doi.org/10.1007/12_2008_4
  22. Ermak DL, McCammon J (1978) Brownian dynamics with hydrodynamic interactions. J Chem Phys 69:1352ADSCrossRefGoogle Scholar
  23. Fahrenberger F, Holm C (2014) Computing the coulomb interaction in inhomogeneous dielectric media via a local electrostatics lattice algorithm. Phys Rev E 90:063304.  https://doi.org/10.1103/PhysRevE.90.063304
  24. Fahrenberger F, Hickey OA, Smiatek J, Holm C (2015a) Importance of varying permittivity on the conductivity of polyelectrolyte solutions. Phys Rev Lett 115:118301. http://link.aps.org/doi/10.1103/PhysRevLett.115.118301
  25. Fahrenberger F, Hickey OA, Smiatek J, Holm C (2015b) The influence of charged-induced variations in the local permittivity on the static and dynamic properties of polyelectrolyte solutions. J Chem Phys 143:243140. http://scitation.aip.org/content/aip/journal/jcp/143/24/10.1063/1.4936666ADSCrossRefGoogle Scholar
  26. Farhat T, Yassin G, Dubas ST, Schlenoff JB (1999) Water and ion pairing in polyelectrolyte multilayers. Langmuir 15(20):6621–6623CrossRefGoogle Scholar
  27. Frank S, Winkler RG (2009) Mesoscale hydrodynamic simulation of short polyelectrolytes in electric fields. J Chem Phys 131(23):234905. https://doi.org/10.1063/1.3274681ADSCrossRefGoogle Scholar
  28. Fritz D, Koschke K, Harmandaris VA, van der Vegt NF, Kremer K (2011) Multiscale modeling of soft matter: scaling of dynamics. Phys Chem Chem Phys 13(22):10412–10420CrossRefGoogle Scholar
  29. Fyta M, Netz RR (2012) Ionic force field optimization based on single-ion and ion-pair solvation properties: going beyond standard mixing rules. J Chem Phys 136(12). https://doi.org/10.1063/1.3693330ADSCrossRefGoogle Scholar
  30. Gompper G, Ihle T, Kroll DM, Winkler RG (2008) Multi-particle collision dynamics: a particle-based mesoscale simulation approach to the hydrodynamics of complex fluids. Adv Polym Sci 221:1–87Google Scholar
  31. Grass K, Böhme U, Scheler U, Cottet H, Holm C (2008) Importance of hydrodynamic shielding for the dynamic behavior of short polyelectrolyte chains. Phys Rev Lett 100:096104Google Scholar
  32. Groot RD, Warren PB (1997) Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107(11):4423–4435ADSCrossRefGoogle Scholar
  33. Heyda J, Dzubiella J (2012) Ion-specific counterion condensation on charged peptides: Poisson–boltzmann vs. atomistic simulations. Soft Matter 8(36):9338–9344ADSCrossRefGoogle Scholar
  34. Hickey OA, Shendruk TN, Harden JL, Slater GW (2012) Simulations of free-solution electrophoresis of polyelectrolytes with a finite debye length using the debye-hückel approximation. Phys Rev Lett 109:098302.  https://doi.org/10.1103/PhysRevLett.109.098302
  35. Hsu CW, Fyta M, Lakatos G, Melchionna S, Kaxiras E (2012) Ab initio determination of coarse-grained interactions in double-stranded DNA. J Chem Phys 137(10):105102ADSCrossRefGoogle Scholar
  36. Jorgensen WL, Maxwell DS, Tirado-Rives J (1996) Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J Am Chem Soc 118(45):11225–11236CrossRefGoogle Scholar
  37. Kremer K, Grest GS (1990) Dynamics of entangled linear polymer melts: a molecular-dynamics simulation. J Chem Phys 92(8):5057–5086ADSCrossRefGoogle Scholar
  38. Krishnamoorthy AN, Zeman J, Holm C, Smiatek J (2016) Preferential solvation and ion association properties in aqueous dimethyl sulfoxide solutions. Phys Chem Chem Phys 18:31312–31322. https://doi.org/10.1039/C6CP05909KCrossRefGoogle Scholar
  39. Kunz W (ed) (2010) Specific ion effects. World Scientific, SingaporeGoogle Scholar
  40. Landsgesell J, Holm C, Smiatek J (2017a) Simulation of weak polyelectrolytes: a comparison between the constant ph and the reaction ensemble method. Eur Phys J Special Top 226:725–736.  https://doi.org/10.1140/epjst/e2016-60324-3ADSCrossRefGoogle Scholar
  41. Landsgesell J, Holm C, Smiatek J (2017b) Wang-landau reaction ensemble method: simulation of weak polyelectrolytes and general acid-base reactions. J Chem Theory Comput 13(2):852–862.  https://doi.org/10.1021/acs.jctc.6b00791CrossRefGoogle Scholar
  42. Lemkul JA, Huang J, Roux B, MacKerell Jr AD (2016) An empirical polarizable force field based on the classical drude oscillator model: development history and recent applications. Chem Rev 116(9):4983–5013.  https://doi.org/10.1021/acs.chemrev.5b00505CrossRefGoogle Scholar
  43. Li YC, Wen TC, Wei HH (2012) Electrophoretic stretching of tethered polymer chains by travelling-wave electric fields: tunable stretching, expedited coil–stretch transition, and a new paradigm of dynamic molecular probing. Soft Matter 8(6):1977–1990ADSCrossRefGoogle Scholar
  44. Limbach HJ, Holm C (2003) Single-chain properties of polyelectrolytes in poor solvent. J Phys Chem B 107(32):8041–8055CrossRefGoogle Scholar
  45. Limbach HJ, Holm C, Kremer K (2002) Structure of polyelectrolytes in poor solvent. Europhys Lett 60(4):566–572ADSCrossRefGoogle Scholar
  46. Limbach HJ, Sayar M, Holm C (2004) Polyelectrolyte bundles. J Phys Condens Matter 16(22):2135–2144ADSGoogle Scholar
  47. Lo Nostro P, Ninham BW (2012) Hofmeister phenomena: an update on ion specificity in biology. Chem Rev 112(4):2286–2322. https://doi.org/10.1021/cr200271jCrossRefGoogle Scholar
  48. Lu BZ, Zhou YC, Holst MJ, McCammon JA (2008) Recent progress in numerical methods for the poisson-boltzmann equation in biophysical applications. Commun Comput Phys 3(5):973–1009Google Scholar
  49. Lu K, Rudzinski JF, Noid W, Milner ST, Maranas JK (2014) Scaling behavior and local structure of ion aggregates in single-ion conductors. Soft Matter 10(7):978–989ADSCrossRefGoogle Scholar
  50. Lund M, Vácha R, Jungwirth P (2008) Specific ion binding to macromolecules: effects of hydrophobicity and ion pairing. Langmuir 24(7):3387–3391CrossRefGoogle Scholar
  51. Lyubartsev A, Laaksonen A (1999) Effective potentials for ion–DNA interactions. J Chem Phys 111(24):11,207–11,215ADSCrossRefGoogle Scholar
  52. Manning GS (1969) Limiting laws and counterion condensation in polyelectrolyte solutions I. colligative properties. J Chem Phys 51:924–933ADSCrossRefGoogle Scholar
  53. Manning GS (1996) Counterion condensation theory constructed from different models. Physica A 231(1–3):236–253ADSCrossRefGoogle Scholar
  54. Marcus Y (2009) Effect of ions on the structure of water: structure making and breaking. Chem Rev 109(3):1346–1370. https://doi.org/10.1021/cr8003828CrossRefGoogle Scholar
  55. Marcus Y, Hefter G (2006) Ion pairing. Chem Rev 106(11):4585–4621CrossRefGoogle Scholar
  56. Marrink SJ, Tieleman DP (2013) Perspective on the MARTINI model. Chem Soc Rev 42(16):6801–6822. https://doi.org/10.1039/C3CS60093ACrossRefGoogle Scholar
  57. Marrink SJ, Risselada HJ, Yefimov S, Tieleman DP, de Vries AH (2007) The MARTINI force field: coarse grained model for biomolecular simulations. J Phys Chem B 111(27):7812–7824. https://doi.org/10.1021/jp071097fCrossRefGoogle Scholar
  58. Marrink SJ, Periole X, Tieleman DP, de Vries AH (2010) Comment on using a too large integration time step in molecular dynamics simulations of coarse-grained molecular models by M. Winger, D. Trzesniak, R. Baron and WF van Gunsteren. Phys Chem Chem Phys 2009, 11, 1934. Phys Chem Chem Phys 12(9):2254–2256Google Scholar
  59. McNaught AD, Wilkinson A (1997) Compendium of chemical terminology, vol 1669. Blackwell Science OxfordGoogle Scholar
  60. Mecerreyes D (2011) Polymeric ionic liquids: broadening the properties and applications of polyelectrolytes. Prog Polym Sci 36(12):1629–1648. https://doi.org/10.1016/j.progpolymsci.2011.05.007CrossRefGoogle Scholar
  61. Micciulla S, Sanchez PA, Smiatek J, Qiao B, Sega M, Laschewsky A, Holm C, von Klitzing R (2014) Layer-by-layer formation of oligoelectrolyte multilayers: a combined experimental and computational study. Soft Mater 12:S14. https://doi.org/10.1080/1539445X.2014.930046, http://www.tandfonline.com/eprint/eCn9vDIc5aMbyBmB6DV5/fullCrossRefGoogle Scholar
  62. Michalowsky J, Schäfer LV, Holm C, Smiatek J (2017) A refined polarizable water model for the coarse-grained MARTINI force field with long-range electrostatic interactions. J Chem Phys 146(5):054501. https://doi.org/10.1063/1.4974833ADSCrossRefGoogle Scholar
  63. Micka U, Holm C, Kremer K (1999) Strongly charged, flexible polyelectrolytes in poor solvents – a molecular dynamics study. Langmuir 15:4033CrossRefGoogle Scholar
  64. Mukhopadhyay A, Fenley AT, Tolokh IS, Onufriev AV (2012) Charge hydration asymmetry: the basic principle and how to use it to test and improve water models. J Phys Chem B 116(32):9776–9783CrossRefGoogle Scholar
  65. Ober MMCK, Thomas EL (1997) Competing interactions and levels of ordering in self-organizing polymeric materials. Science 277:1225–1232CrossRefGoogle Scholar
  66. Pagonabarraga I, Rotenberg B, Frenkel D (2010) Recent advances in the modelling and simulation of electrokinetic effects: bridging the gap between atomistic and macroscopic descriptions. Phys Chem Chem Phys 12:9566–9580. https://doi.org/10.1039/C004012FCrossRefGoogle Scholar
  67. Praprotnik M, Junghans C, Site LD, Kremer K (2008) Simulation approaches to soft matter: generic statistical properties vs. chemical details. Comput Phys Commun 179(1–3):51ADSCrossRefGoogle Scholar
  68. Qiao B, Sega M, Holm C (2011) An atomistic study of a poly(styrene sulfonate)/poly(diallyldimethylammonium) bilayer: the role of surface properties and charge reversal. Phys Chem Chem Phys 13(36):16336–16342. https://doi.org/10.1039/C1CP21777ACrossRefGoogle Scholar
  69. Qiao B, Sega M, Holm C (2012) Properties of water in the interfacial region of a polyelectrolyte bilayer adsorbed onto a substrate studied by computer simulations. Phys Chem Chem Phys 14:11425–11432. https://doi.org/10.1039/C2CP41115FCrossRefGoogle Scholar
  70. Rau T, Weik F, Holm C (2017) A dsDNA model optimized for electrokinetic applications. Soft Matter 3918–3926. https://doi.org/10.1039/C7SM00270J, http://pubs.rsc.org/en/content/articlehtml/2017/sm/c7sm00270jADSCrossRefGoogle Scholar
  71. Reith D, Müller B, Müller-Plathe F, Wiegand S (2002) How does the chain extension of poly (acrylic acid) scale in aqueous solution? a combined study with light scattering and computer simulation. J Chem Phys 116(20):9100–9106ADSCrossRefGoogle Scholar
  72. Reith D, Pütz M, Müller-Plathe F (2003) Deriving effective mesoscale potentials from atomistic simulations. J Comput Chem 24(13):1624–1636CrossRefGoogle Scholar
  73. Roy T, Szuttor K, Smiatek J, Holm C, Hardt S (2017) Stretching of surface-tethered polymers in pressure-driven flow under confinement. Soft Matter 13:6189–6196. https://doi.org/10.1039/C7SM00306DADSCrossRefGoogle Scholar
  74. Savelyev A, Papoian GA (2010) Chemically accurate coarse graining of double-stranded DNA. Proceedings of the National Academy of Sciences of the United States of America 107(47):20340–20345.  https://doi.org/10.1073/pnas.1001163107ADSCrossRefGoogle Scholar
  75. Schmid F (1998) Self-consistent-field theories for complex fluids. J Phys Condens Matter 10(37):8105ADSGoogle Scholar
  76. Senftle TP, Hong S, Islam MM, Kylasa SB, Zheng Y, Shi YK, Junkermeier C, Engel-Herbert R, Janik MJ, Aktulga HM, Verstraelen T, Grama A, van Duin ACT (2016) The ReaxFF reactive force-field: development, applications and future directions. Comput Mater 2:15011.  https://doi.org/10.1038/npjcompumats.2015.11
  77. Slater GW, Holm C, Chubynsky MV, de Haan HW, Dubé A, Grass K, Hickey OA, Kingsburry C, Sean D, Shendruk TN, Zhan L (2009) Modeling the separation of macromolecules: a review of current computer simulation methods. Electrophoresis 30(5):792–818.  https://doi.org/10.1002/elps.200800673CrossRefGoogle Scholar
  78. Smiatek J, Schmid F (2010) Polyelectrolyte electrophoresis in nanochannels: a dissipative particle dynamics simulation. J Phys Chem B 114(19):6266–6272. https://doi.org/10.1021/jp100128p, http://pubs.acs.org/doi/abs/10.1021/jp100128pCrossRefGoogle Scholar
  79. Smiatek J, Schmid F (2011) Mesoscopic simulations of electroosmotic flow and electrophoresis in nanochannels. Comput Phys Commun 182(9):1941–1944. https://doi.org/10.1016/j.cpc.2010.11.021, http://www.sciencedirect.com/science/article/pii/S0010465510004674. Computer Physics Communications Special Edition for Conference on Computational Physics Trondheim, Norway, June 23–26, 2010ADSCrossRefGoogle Scholar
  80. Smiatek J, Sega M, Holm C, Schiller UD, Schmid F (2009) Mesoscopic simulations of the counterion-induced electro-osmotic flow: a comparative study. Journal of Chemical Physics 130(244702). https://doi.org/10.1063/1.3152844ADSCrossRefGoogle Scholar
  81. Smiatek J, Harishchandra RK, Rubner O, Galla HJ, Heuer A (2012) Properties of compatible solutes in aqueous solution. Biophys Chem 160(1):62–68. https://doi.org/10.1016/j.bpc.2011.09.007CrossRefGoogle Scholar
  82. Smiatek J, Wohlfarth A, Holm C (2014) The solvation and ion condensation properties for sulfonated polyelectrolytes in different solvents-a computational study. New J Phys 16(2):025001. http://stacks.iop.org/1367-2630/16/i=2/a=025001CrossRefGoogle Scholar
  83. Stevens MJ, Kremer K (1993a) Form factor of salt-free linear polyelectrolytes. Macromolecules 26:4717ADSCrossRefGoogle Scholar
  84. Stevens MJ, Kremer K (1993b) Structure of salt-free linear polyelectrolytes. Phys Rev Lett 71:2228ADSCrossRefGoogle Scholar
  85. Szuttor K, Roy T, Hardt S, Holm C, Smiatek J (2017) The stretching force on a tethered polymer in pressure-driven flow. J Chem Phys 147(3):034902. https://doi.org/10.1063/1.4993619ADSCrossRefGoogle Scholar
  86. Uusitalo JJ, Ingólfsson HI, Akhshi P, Tieleman DP, Marrink SJ (2015) MARTINI coarse-grained force field: extension to DNA. J Chem Theory Comput 11(8):3932–3945CrossRefGoogle Scholar
  87. Vögele M, Holm C, Smiatek J (2015a) Coarse-grained simulations of polyelectrolyte complexes: MARTINI models for poly(styrene sulfonate) and poly(diallyldimethylammonium). J Chem Phys 143:243151. https://doi.org/10.1063/1.4937805, http://scitation.aip.org/content/aip/journal/jcp/143/24/10.1063/1.4937805ADSCrossRefGoogle Scholar
  88. Vögele M, Holm C, Smiatek J (2015b) Properties of the polarizable MARTINI water model: a comparative study for aqueous electrolyte solutions. J Mol Liq 212:103–110. https://doi.org/10.1016/j.molliq.2015.08.062, http://www.sciencedirect.com/science/article/pii/S0167732215304657CrossRefGoogle Scholar
  89. Weeks JD, Chandler D, Andersen HC (1971) Role of repulsive forces in determining the equilibrium structure of simple liquids. J Chem Phys 54:5237ADSCrossRefGoogle Scholar
  90. Weik F, Kesselheim S, Holm C (2016) A coarse-grained DNA model for the prediction of current signals in DNA translocation experiments. J Chem Phys 145(19):194106. https://doi.org/10.1063/1.4967458ADSCrossRefGoogle Scholar
  91. Weyman A, Bier M, Holm C, Smiatek J (2018) Microphase separation and the formation of ion conductivity channels in poly (ionic liquid) s: A coarse-grained molecular dynamics study. J Chem Phys 148:193824ADSCrossRefGoogle Scholar
  92. Winger M, Trzesniak D, Baron R, van Gunsteren WF (2009) On using a too large integration time step in molecular dynamics simulations of coarse-grained molecular models. Phys Chem Chem Phys 11(12):1934–1941Google Scholar
  93. Wohlfarth A, Smiatek J, Kreuer KD, Takamuku S, Jannasch P, Maier J (2015) Proton dissociation of sulfonated polysulfones: influence of molecular structure and conformation. Macromolecules 48(4):1134–1143. https://doi.org/10.1021/ma502550fADSCrossRefGoogle Scholar
  94. Yesylevskyy SO, Schäfer LV, Sengupta D, Marrink SJ (2010) Polarizable water model for the coarse-grained MARTINI force field. PLoS Comput Biol 6(6):e1000810.  https://doi.org/10.1371/journal.pcbi.1000810ADSCrossRefGoogle Scholar
  95. Yuan J, Mecerreyes D, Antonietti M (2013) Poly(ionic liquid)s: an update. Prog Polym Sci 38:1009–1036. https://doi.org/10.1016/j.progpolymsci.2013.04.002CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Computational PhysicsUniversity of StuttgartStuttgartGermany
  2. 2.Helmholtz Institute Münster (HI MS), Ionics in Energy Storage, Forschungszentrum Jülich GmbHMünsterGermany
  3. 3.Institute for Computational PhysicsUniversity of StuttgartStuttgartGermany

Section editors and affiliations

  • Kurt Kremer
    • 1
  1. 1.MPI for Polymer ResearchMainzGermany

Personalised recommendations