Mesoscale modelling of polyelectrolyte electrophoresis
MMesoscale modelling of polyelectrolyte electrophoresis
Kai Grass a and Christian Holm a,b,ca Frankfurt Institute for Advanced Studies, Goethe University,Ruth-Moufang-Str. 1, 60438 Frankfurt/Main, Germany.E-mail: grass@fias.uni-frankfurt.de b Max-Planck-Institut f¨ur Polymerforschung,Ackermannweg 10, 55128 Mainz, Germany. c Institute for Computational Physics, University of Stuttgart,Pfaffenwaldring 27, 70569 Stuttgart,Germany.E-mail: [email protected] 28, 2018
The electrophoretic behaviour of flexible polyelectrolyte chains ranging from singlemonomers up to long fragments of hundred repeat units is studied by a mesoscopicsimulation approach. Abstracting from the atomistic details of the polyelectrolyte andthe fluid, a coarse-grained molecular dynamics model connected to a mesoscopic fluiddescribed by the Lattice Boltzmann approach is used to investigate free-solution elec-trophoresis. Our study demonstrates the importance of hydrodynamic interactions forthe electrophoretic motion of polyelectrolytes and quantifies the influence of surround-ing ions. The length-dependence of the electrophoretic mobility can be understood byevaluating the scaling behavior of the effective charge and the effective friction. Theperfect agreement of our results with experimental measurements shows that all chem-ical details and fluid structure can be safely neglected, and a suitable coarse-grainedapproach can yield an accurate description of the physics of the problem, provided thatelectrostatic and hydrodynamic interactions between all entities in the system, i.e. , thepolyelectrolyte, dissociated counterions, additional salt and the solvent, are properlyaccounted for. Our model is able to bridge the single molecule regime of a few nm upto macromolecules with contour lengths of more than 100 nm, a length scale that iscurrently not accessible to atomistic simulations.
Nowadays, electrophoresis methods are widely used to separate biomolecules [1, 2] suchas peptides, proteins, DNA, as well as synthetic polymers [3, 4]. In order to be ableto improve the processes involved in current electrophoretic separation methods it isa prerequisite to gain a thorough understanding of the behaviour of polyelectrolytes1 a r X i v : . [ c ond - m a t . s o f t ] F e b PEs) in an externally applied electric field. Several theories [5, 6, 7, 8] have beendeveloped to describe PE electrophoresis and successfully described qualitatively theexperimentally observed behaviour of various PEs under bulk conditions. However, themobility of small oligomeric PEs showed under low salt conditions a non-monotonicbehaviour that current theories had not been able to explain.In a recent publication [9], we employed a mesoscopic coarse-grained model usingmolecular dynamics simulations in connection with a Lattice-Boltzmann (LB) algo-rithm to extend the theoretical understanding on a more detailed level, and in particu-lar, we intended to investigate the role of hydrodynamic interactions in these systems.Our results were able to match the free-solution electrophoretic mobility µ of shortpolyelectrolyte chains, here polystyrene sulfonate (PSS), as a function of the numberof repeat units N with quantitative agreement to experiments as shown in Figure 4.Since the three data sets have different solvent viscosities the mobility is normalized bythe corresponding constant mobility for long chains, the so-called free-draining mobility, µ FD . The electrophoretic mobility increases for short oligomers, reaches a maximum forintermediate degrees of polymerization, and slowly decreases towards a plateau valuefor long chains. To understand this observation, the hydrodynamic interactions wereinvestigated in detail and we found that they are actually the major driving force forthe length dependent mobility for short and intermediate chain lengths. The constantmobility for long chains can be attributed to an effective screening of hydrodynamicinteractions, which leads to the so-called free-draining behavior. The inset in Figure 4shows a comparison to a coarse-grained simulation that neglects hydrodynamic inter-actions. This leads to a qualitatively completely different behavior, showing a mono-tonically decreasing mobility. Agreement to the experimentally observed behaviour isonly achieved as long as hydrodynamic interactions are included correctly as has beenshown in detail in our previous investigations [10, 11].In this article, we will extend our work by studying the electrophoresis of genericflexible polyelectrolyte chains ranging from single monomers to long fragments of hun-dred repeat units. Abstracting from the atomistic details of the polyelectrolyte andthe fluid, a coarse-grained molecular dynamics model connected to a mesoscopic fluiddescribed by the Lattice Boltzmann approach is used to investigate the free-solutionbehavior under varying salt concentration.In the next section we will introduce the employed simulation model. In Section 3,the main results of this study are presented and discussed. We conclude with finalremarks in Section 4. We employ molecular dynamics (MD) simulations using the ESPResSo package [12] tostudy the behaviour of linear polyelectrolytes (PE) of different lengths. The PEs aremodelled by a totally flexible bead-spring model. The monomers are connected to eachother by finitely extensible nonlinear elastic (FENE) bonds [13] U FENE ( r ) = 12 kR ln (cid:18) − (cid:16) rR (cid:17) (cid:19) , with stiffness k = 30 (cid:15) , and maximum extension R = 1 . σ , where r is the distancebetween the interacting monomers. Additionally, a truncated Lennard-Jones or WCA2otential [14] U LJ ( r < r c ) = (cid:15) (cid:18)(cid:16) σ r (cid:17) − (cid:16) σ r (cid:17) + 14 (cid:19) , is used for excluded volume interactions between all monomers. A cutoff value of r c = √ σ ensures a purely repulsive potential. All dissociated counterions and additionalsalt ions are modelled by appropriately charged spheres using the same WCA potential.Here, (cid:15) and σ define the energy and length scale of the simulations. We use (cid:15) = k B T , i.e. the energy of the system is expressed in terms of the thermal energy.The length scale σ defines the size of the monomers and the dimension of the system.For this study, σ is chosen to be 4 ˚A. Different polyelectrolytes can be mapped bychanging σ . Unless mentioned otherwise, all observables are expressed in reducedsimulation units, and we will not use σ and (cid:15) explicitly from now on.The chain length is varied from N = 1 to N = 128 and all chain monomers carrya negative electric charge q = − e , where e is the elementary charge. For chargeneutrality, N monovalent counterions of charge +1 e are added. Additional monovalentsalt is added to the simulation, corresponding to concentrations between c s = 0 mMand c s = 160 mM. The later concentration being equivalent to a particle density ofthe salt ions of ρ s = 0 . E = 0 . F E = qE on all charged particles, and thus inducing anelectrophoretic mobility. It has been carefully checked that the field strength is withinthe linear response regime, i.e. , it does not influence the chain conformation or thedistribution of the surrounding ions [11].Full electrostatic interactions are calculated with the P3M algorithm using theimplementation of Reference 15. The Bjerrum length l B = e / (4 π(cid:15) (cid:15) r k B T ) = 1 . (cid:15) r ≈ L of the box is varied to realize a constant monomer concentrationof c PE = 16 mM independent of chain length. This is equivalent to a monomer density ρ PE = 0 . ν = 1 .
0, and a fluid density ρ = 1 .
0. The resulting fluid has a dynamic vis-cosity η = ρν = 1 .
0. The simulation box is discretised by a grid with spacing a = 1 . v and the fluid velocity at the particle position u : F R = − Γ bare ( v − u ) . bare = 20 .
0. Additional random fluc-tuations introduced to the particles and fluid act as a thermostat. The interactionbetween particles and fluid conserve total momentum, and this algorithm has beenshown to yield correct long-range hydrodynamic interaction between individual parti-cles [17].Additionally, a second type of MD simulation is used which is based on the Langevinequations of motions with a velocity dependent dissipative and a random term inaddition to the inter particle forces. Together, both additional terms implicitly modelthe effects of a solvent surrounding the particles: the dissipative force, F D = − Γ v , with Γ = 1 . F R = ξ ( t ) , mimic thermal kicks (Brownian motion). In order to fulfill the fluctuation-dissipationtheorem, dissipative and random force have to be coupled together: (cid:104) ξ i ( t ) · ξ j ( t (cid:48) ) (cid:105) =6Γ k B T δ ij δ ( t − t (cid:48) ). This approach only offers local particle-fluid interactions, andtherefore destroys long-range hydrodynamic interactions. Nevertheless one can use itto compute the effective charge as has been presented in [10, 11]. This effective chargeis used to obtain the effect friction in the presence of hydrodynamic interactions andillustrates their importance for the electrophoretic mobility.All simulations are carried out with a MD time step τ MD = 0 .
01 and LB time step τ LB = 0 .
05. After an equilibration time of 10 steps, 10 steps are used for generatingthe data. The time-series of four independent simulations are analyzed using auto-correlation functions to estimate the statistical errors as detailed in Reference 18. Errorbars of the order of the symbol size or smaller are omitted in the figures. We determine the electrophoretic mobility µ as the ratio between the measured centerof mass velocity v PE and the magnitude of the electric field E : µ = vE . For comparison, the results are normalized by the monomer mobility µ .Figure 2 displays the characteristic behaviour of flexible polyelectrolytes for vanish-ing salt concentration c s = 0 mM: initially, the electrophoretic mobility increases with N to reach a maximum at intermediate chain lengths and then slowly decays towardsa constant value for long chains. This constant value, often called the free-draininglimit µ FD , can be explained by the length independence of the ratio between effectivecharge and effective friction for long chains as we will show in this article.In the presence of added salt, the long chain mobility is reduced, which is consistentwith the experimentally observed [19] behavior. Furthermore, the shape of the curveis influenced, and the maximum at intermediate chains is suppressed for increased saltconcentration. At c s = 160 mM the maximum disappears and the measured mobility4ecomes length independent within the resolution of the simulation. A further increaseof the added salt concentration leads to a further reduction of the limiting mobility µ FD , not shown here, while the monomer mobility µ remains almost unchanged. Thisleads eventually to an inverted length-dependence with a monotonic decrease of themobility towards the limiting value.In the simple local force picture, the constant center of mass velocity v PE thatdetermines the electrophoretic mobility is a direct result of the cancellation of twoacting forces: the electric driving force F E = Q eff E is canceled by the solvent frictionor drag force F D = Γ eff v PE . Here, Q eff is the effective charge of the polyelectrolyte,which can be thought of as the bare charge of the polyelectrolyte reduced by oppositelycharged ions in solution that associate to the polyelectrolyte chain. The association ofcounterions to a PE chain is known as counterion condensation [20, 21]. The compoundformed by the polyelectrolyte and the associated ions is moved through the solventunder the influence of the external field and experiences a Stokesian drag force withan effective friction coefficient Γ eff that is a priori unknown. In the steady state bothforces balance and the mobility is given by µ = vE = Q eff Γ eff . Next, let us compare the results of Figure 2 to the case when long-range hydro-dynamic interactions between the particles are neglected in simulations, i.e. , by usinga standard Langevin thermostat. The results are shown in Figure 3. One immedi-ately notices that the observed electrophoretic mobility differs significantly from thebehaviour observed in Figure 2. Independent of the salt concentration, the mobilitydecreases monotonically with chain length and slowly approaches a constant value forlong chains which is independent of the salt concentration. This difference to the ex-perimental observations and to the LB simulation including hydrodynamics will beanalyzed in detail in the following sections.
To analyze the observed influence of the added salt on the polyelectrolyte mobility, wewill determine the effective charge, and can then calculate Γ eff = Q eff /µ to obtain anestimate for the effective friction of the polyelectrolyte-ion compound. A word of carehas to be taken here, since the value of the effective charge depends on definition. Qual-itatively one can differentiate between a static definition and a dynamic definition [22].In our case it is obviously a dynamic definition. In [10, 11], we introduced severalstatic and dynamic estimators for the effective charge and showed their equivalence atvanishing salt concentration. Here, three of them will be reviewed and applied to thecase of added salt.The local force picture described above can be used to estimate the effective chargeof the polyelectrolyte based on the measurement of the electrophoretic mobility in theabsence of hydrodynamic interactions. Let N CI be the number of associated counterionsreducing the bare charge of the polyelectrolyte which is equal to N . The effective chargeis then given by Q eff = N − N CI . Without long-rang hydrodynamic interactions the interaction of each particle with thesolvent is purely local and directly given by the friction constant Γ of the Langevin5lgorithm. The total effective friction of the polyelectrolyte and the ions is then:Γ eff = Γ ( N + N CI ) . This results in an expression for the electrophoretic mobility µ = N − N CI Γ ( N + N CI )from which an expression for N CI is obtained. Therefore we can express the effectivecharge purely as a function of the mobility measurements shown in Figure 3 and on ourinput value for Γ , independent of the knowledge of the value of N c l by the followingexpression: Q (1)eff = N (cid:18) − − µ Γ µ Γ (cid:19) . An alternative way of characterizing the associated ions is presented in Ref. 23who suggested to determine the ion velocity with respect to the distance to the centerof mass of the polyelectrolyte. For this method we use the LB algorithm to includehydrodynamical interactions, and the result for a chain of N = 64 at c s = 16 mM canbe inspected in Figure 4. The average ion velocity in the direction of the electric field v CI is a function of the distance d to the center of mass of the polyelectrolyte chain andin general depends on the chain length N and the salt concentration c s . As shown, ionsclose to the center move with the chain at negative speed, whereas ions far away fromthe center move with the single particle velocity v = µ E into the opposite direction.The association of ions to the chain is strong enough to move them against the electricfield. For every chain length and every salt concentration, the distance d at which v CI ( d ) = 0 is used to separate co-moving, associated ions from non-associated ones.We use this distance d to define the effective charge by summing up the total chargein the system found within this distance to the center of mass of the polyelectrolyte: Q (2)eff = N (1 − I ( d )) , where I ( d ) is the integrated fraction of neutralizing charges found by adding the num-ber of counterions and positively charged salt ions reduced by the number of negativelyor like-charged salt ions. Far away from the center of mass of the chain, the total barecharge of the polyelectrolyte is neutralized and I = 1.The effective charge Q eff as obtained from both estimators is presented in Figure 5a.Initially, Q eff is close to the bare charge N , but as ion condensation sets in, the effectivecharge is reduced. Longer chains show a linear increase of their charge close to theManning prediction for counterion condensation in the salt free case Q eff = (1 /ξ ) N ,where Manning parameter ξ = l B /b is the ratio between the Bjerrum length and thecharge spacing along the polyelectrolyte backbone. For the model used here b = 0 . ξ = 2 .
0. We note that there is no apparent dependence of the effectivecharge for long polyelectrolyte chains on the salt concentrations when measured by thedynamic effective charge estimators presented here.Figure 5b plots the effective charge per monomer, Q eff /N . Here, the influence ofthe salt concentration for short and intermediate chain length can be seen. The higherthe concentration of the added salt, the faster the electric charge of the polyelectrolyteis reduced by condensed counter ions. For long chains, the charge per monomer is againindependent of the salt concentration and comparable to the Manning prediction 1 /ξ .6he difference for short and intermediate chains at different salt concentrations can beattributed to stronger association of counterions with increasing salt concentrations.For short chains, smaller than the Debye length, effects due to the finite size play aleading role in the ability to condense counterions [24, 25, 26].Additionally, Figure 5 shows the equivalence of the two dynamic estimators Q (1)eff and Q (2)eff independently of the presence or absence of hydrodynamic interactions alsoin the presence of additional salt. This new observation supports the applicabilityand importance of these charge estimators for the study of polyelectrolytes duringelectrophoresis.The charge estimators Q (1)eff and Q (2)eff measure the effective charge of the movingpolyelectrolyte and its surrounding counterions. Therefore, they measure the effectivedynamic charge of the polyelectrolyte. Similarly, it is possible to define a static estimateof the effective charge using the following simple method Q (3)eff = N PE − N CI ( d < d ) , where N CI ( d < d ) is the average number of counterions that can be found within adistance d to the closest monomer. Here, the threshold d chosen to be d = 2 σ .The second static charge estimator used in Ref. 11 based on the inflection criterionto estimate the threshold of counterion condensation [27, 28, 29] breaks down in thepresence of high salt concentrations and therefore can not be applied here.In Figure 6, we compare the static charge estimate Q (3)eff for varying salt concentra-tions to the dynamic charge estimate obtained by Q (1)eff . Unlike the dynamic estimatethe static charge estimate shows a strong dependence on the salt concentration. Whileboth estimators agree for vanishing salt concentration as previously shown in Ref. 11,the static charge estimate shows a decreases with higher salt concentrations, hence anincrease in counterion condensation, as could be expected from a mean-field compari-son [29], and eventually falls below the Manning prediction.The higher salt concentrations increase the number of counterions in the close vicin-ity of the chain as measured by Q (3)eff . At the same time, the electrostatic interactions inthe system are reduced due to electrostatic screening, which also reduces the strengthof the coupling between the polyelectrolyte and the counterions. The independenceof the dynamic effective charge on the salt concentration for long chains as shown inFigure 5 has to be understood as the cancellation of both effects: with increasing saltconcentration more counterions in close vicinity to the polyelectrolyte are influencedby the chain but the strength of the interactions is reduced in such a way that thecombined action remains unchanged and yields a concentration independent dynamiceffective charge.In the following, we will use the dynamic effective charge to calculate the effectivefriction of the polyelectrolyte-ion compound. When long-range hydrodynamic interactions are present, the effective friction of thepolyelectrolyte and the associated counterions cannot be given in a simple analyticform. We therefore obtain it from the measurements of the mobility and the effectivecharge presented above: Γ eff = Q eff µ .
7n Figure 7a, the effective charge Γ eff is displayed as a function of chain length N for different salt concentrations c s . The friction increases monotonically with chainlength.Neglecting the contribution of the counterions the effective friction can be obtainedfrom the hydrodynamic radius R h of the polyelectrolyte defined by: (cid:28) R h (cid:29) = 1 N (cid:88) i (cid:54) = j (cid:28) (cid:107) (cid:126)r i − (cid:126)r j (cid:107) (cid:29) . Here, (cid:126)r i is the position of the i -th chain monomer, and (cid:126)r cm the center of mass of thepolyelectrolyte chain. The angular brackets (cid:104) . . . (cid:105) indicate an ensemble average. Thehydrodynamic radius is expected to exhibit a power law scaling R h ∼ ( N − ν , wherethe scaling exponent ν depends on the system. For an uncharged polymer with idealchain behaviour one should get ν ≈ .
588 (Flory exponent) [30], whereas for a fullycharged polyelectrolyte without electrostatic screening one expects ν = 1. Dependingon the salt concentration, we obtain values between ν ≈ .
66 for c s = 0 mM and ν ≈ .
59 for c s = 160 mM (not shown here).Initially, the friction increases with N as given by the hydrodynamic size of thepolyelectrolyte Γ = 6 πηR h ∝ N . . With the onset of counterion condensation the friction exceeds the value of the barepolyelectrolyte and for long chains becomes linear in N . The higher the concentrationof the additional salt, the earlier the transition between the two regimes is observed.We furthermore note that the absolute friction value is increased with the addition ofexternal salt.The role of the additional salt can be best understood when looking at the effectivefriction per monomer as presented in Figure 7b. Γ eff /N shows an initial decrease withchain length which can be understood by hydrodynamic shielding: the monomers ofshort polyelectrolyte chains are hydrodynamically coupled and shield each other fromthe effect of the solvent. This reduces the friction per monomer below the value of asingle particle. The decrease in friction due to the hydrodynamic shielding is strongerthe lower the salt concentration is. The presence of ions in the vicinity of the chainmonomers reduces the hydrodynamic coupling. For longer length scale, i.e. , for longerchains, the ions effectively decouple different parts of the polyelectrolyte chain suchthat the friction per monomer becomes a constant value. The chain length N FD forwhich this transition occurs is depending on the salt concentration. The higher the saltconcentration, i.e. , the shorter the Debye length in the system, the more confined is thehydrodynamic shielding effect and the earlier the effective friction becomes constant.The role of hydrodynamic interactions for the effective friction of the polyelectrolytecan be seen by comparing Figure 7 to Figure 8 that shows the effective friction obtainedwith the Langevin algorithm neglecting long-range hydrodynamic interactions. In Fig-ure 8a, the initial increase of the effective friction is super linear, but linear scaling isreached for longer chains. The absolute friction value for long chains is independentof the salt concentration. This can be understood by realizing that the total effectivefriction of a polyelectrolyte in the Langevin algorithm is only based on the local frictionparameter Γ and the number of co-moving particles, i.e. , the sum of N monomers and N CI condensed counterions: Γ eff = Γ ( N + N CI ). As shown in Figure 5 the effectivecharge, and therefore also N CI and Γ eff , is only influenced by the salt concentration for8hort and intermediate chains, but not for long chains. Figure 8b shows the increaseof the effective friction per monomer from Γ eff (1) = 1 /µ ≈ Γ to a constant valuefor long chains, which is comparable to the plateau value predicted using counterioncondensation theory: Γ eff /N = Γ (2 − /ξ ).Figure 9 schematically illustrates how counterions and salt in the vicinity of poly-electrolyte chains influence the hydrodynamic interactions during electrophoresis. Fig-ure 9a indicates the regime, where all parts of the chain can interact via hydrodynamicinteractions. The individual chain segments provide hydrodynamic shielding to eachother. During electrophoresis, see Figure 9, the counterions within the polyelectrolytelimit the range of the hydrodynamic interaction. The hydrodynamic screening lengthdepends on the ion concentration in the vicinity of the chain. This relation betweenthe ion density and the hydrodynamic screening length was previously suggested bydifferent authors [31, 32].The connection between electrostatic screening and hydrodynamic screening canbe easily motivated by the following reasoning: the Debye length is the length-scaleon which the charge of the polyelectrolyte is screened by the surrounding ions. Whenlooking at this object from the outside, the total force exerted by the applied electricfield is zero, i.e. , no momentum is transferred to the polyelectrolyte-ion complex. Dueto momentum conservation, the interaction with the fluid has to result in a vanishingtotal force.The counterions that associate with the polyelectrolyte influence the solvent flowaround it, effectively canceling the beneficial shielding effects. When additional saltis added to the system, the like charged salt ions likewise contribute to this effect asshown in Figure 9c. The higher the ion concentration is, i.e. , the shorter the Debyelength is, the shorter is the length scale on which different polyelectrolyte monomerscan interact hydrodynamically. On a length scale comparable to the Debye length inthe system, different parts of the polyelectrolyte become decoupled. For longer chains,the effective friction per segment does not depend on the length of the polyelectrolyteanymore. Consequently, the effective friction per monomer becomes independent ofthe length of the polyelectrolyte chain, as seen in Figure 7b. We presented a detailed study of the electrophoretic behaviour of flexible polyelec-trolyte chains by means of a mesoscopic coarse-grained molecular dynamics modelincluding full hydrodynamic and electrostatic interactions.The electrophoretic mobility exhibits a characteristic length dependence for shortpolyelectrolyte chains and a constant length independent value for long chains. Weshowed that both, the shape and the constant long chain value, depend on the saltconcentration of the solution if hydrodynamical interactions were properly accountedfor. The long chain mobility was then found to be decreasing with increasing saltconcentration, in agreement with experimental observations.Direct measurements of the effective charge by two independent estimators showedthat the dynamic effective charge for long chains is independent of the salt concentra-tion. We therefore conclude, that the dependence of the long chain mobility on thesalt concentration is not due a reduced effective charge but has to be attributed to achange in the effective friction. On the other hand, the effective charge for short andintermediate chains is influenced by the salt concentration which explains the different9ehaviour of the electrophoretic mobility in this regime.We note that a static estimate of the effective charge shows a dependence on thesalt concentration leading to to different charge estimates for finite salt concentrations:a static and a dynamic effective charge.We showed that the effective friction of the polyelectrolyte is strongly influencedby the presence of ions in the solution. For short chains and low salt concentrationsno counterions are associated with the polyelectrolyte chain and the effective frictionis given by the hydrodynamic radius. The presence of ions in the vicinity of thechain reduces the hydrodynamic shielding between the chain monomers and leads toan increased friction. The longer the chains and the higher the salt concentration, themore the shielding is reduced, until for chains longer than a specific length N FD thefriction becomes linear with chain length. In this regime, different parts of the chainare effectively decoupled.From this, the specific behaviour of the electrophoretic mobility as observed inexperiments can be understood: the hydrodynamic shielding between the monomersallows for an initial increase in the mobility. The onset of counterion condensationcounteracts this increase as it reduces the effective charge and at the same time in-creases the effective friction. For long chains, charge and friction both become linearlydependent on chain length which therefore results in the well-known constant mobility,or free-draining limit.The presence of salt reduces the length scale on which the chain monomers caninteract hydrodynamically. This reduces the initial hydrodynamic shielding and there-fore suppresses the mobility maximum. At the same time, the total friction is increasedleading to a reduced long-chain mobility. Salt concentrations exceeding the ones in thissimulation can cause a total decoupling of the individual chain monomers, which canthen be simulated without hydrodynamic interactions. We expect a length-dependenceof the mobility as shown in Figure 3 and an effective friction per monomer, cf. Fig-ure 7b, that does not depend on N .The study shows that chemical details and fluid structure can be neglected, and ahigher level of abstraction yields an accurate description of the physics of the problem,as long as electrostatic and hydrodynamic interactions between all entities in the sys-tem, i.e., the polyelectrolyte, dissociated counterions, additional salt and the solvent,are properly accounted for. In this way we were able to model a process bridging thesingle molecule regime of a few nm up to macromolecules with contour lengths of morethan 100 nm, a regime currently not accessible to atomistic simulations. Acknowledgements
Funds from the the Volkswagen foundation, the DAAD, and DFG under the TR6 aregratefully acknowledged. All simulations were carried out on the compute cluster ofthe Center for Scientific Computing (CSC) at Goethe University Frankfurt/Main.10 eferences [1] ed. P. G. Righetti,
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The normalized electrophoretic mobility µ/µ FD as a function of the num-ber of repeat units N for simulation data including hydrodynamic inter-actions (HI), and experimental data coming from capillary electrophoresis(CE) and from electrophoretic NMR. The inset compares to simulationdata obtained with a model neglecting hydrodynamic interactions. Fig. 2
The normalized electrophoretic mobility µ/µ of polyelectrolyte chains oflength N for three different salt concentrations using the LB algorithm.The added salt not only influences the absolute mobility, but likewisechanges the characteristic shape of the mobility with respect to chainlength N . Fig. 3
The normalized electrophoretic mobility µ/µ for different chain length N at varying salt concentrations c s without hydrodynamic interactionsdiffers significantly from the behaviour observed in Figure 2. The mobilityshows a salt-dependent monotonic decrease for short chains and a salt-independent constant value for long chains. Fig. 4
The average ion velocity in the direction of the electric field v CI (herefor a chain with N = 64 monomers, salt concentration c s = 16 mM, andhydrodynamics included) depends on the distance d to the center of massof the polyelectrolyte. Ions close to the center co-move with the chain’svelocity (dashed line), whereas ions far away from the center move with thesingle particle velocity v = µ E into the opposite direction. The distance d at which v CI ( d ) = 0 is used to separate co-moving, associated ionsfrom non-associated ones. The solid line shows the integrated fraction ofcharges I that is found up to the distance d of the center of mass. Fig. 5 (a) The effective charge Q eff as a function of chain length N (symbolsfor Q (1)eff , lines for Q (2)eff ). Both charge estimators show good agreement.Initially, Q eff is close to the bare charge N (dotted line), but as ion con-densation sets in, the effective charge is reduced. Longer chains show alinear increase of their charge close to the Manning prediction (1 /ξ ) N (dashed line). (b) The effective charge per monomer Q eff / N is influencedby the salt concentration for short and intermediate chains. The higherthe concentration of the added salt, the faster the electric charge of thepolyelectrolyte is reduced by condensed ions. For long chains, the chargeper monomer is again independent of the salt concentration and compa-rable to the Manning prediction 1 /ξ (dashed line). Fig. 6
The effective charge as measured by the static estimator Q (3)eff shows astrong dependence on the the salt concentration. At c S = 0 mM the staticestimate agrees with the dynamic estimates (solid line). The higher thesalt concentration, the lower is the static charge estimate. For comparisonthe bare charge N (dotted line) and the Manning prediction (dashed-line)are plotted. 13 ig. 7 (a) The normalized effective friction Γ eff µ as a function of chain length N for different salt concentrations c s using the LB algorithm. Initially, thefriction increases as given by the hydrodynamic size of the polyelectrolyteΓ = ∼ N . (dotted line). With the onset of counterion condensationthe friction exceeds the value of the bare polyelectrolyte and for longchains becomes linear in N (dashed line). The absolute friction value isincreased with the addition of external salt. (b) The normalized effectivefriction per monomer Γ eff µ /N shows an initial decrease with chain lengththat is stronger the lower the salt concentration is. From a concentrationdependent value of N FD onwards, the friction per monomer becomes aconstant value that increases with increasing salt concentration (indicatedby dashed lines). Fig. 8 (a) The normalized effective friction Γ eff µ as a function of chain length N for different salt concentrations c s without long-range hydrodynamicinteractions. Initially, the increase of the effective friction is super linear,but linear scaling (dotted line) is reached for longer chains. For thesechains, the absolute friction value is independent of the addition of ex-ternal salt. (b) The normalized effective friction per monomer Γ eff µ /N shows an initial increase with chain length that is stronger the higherthe salt concentration is. For longer chains, a plateau value is reachedwhich is independent of the salt concentration and can be compared tothe predicted value based on the counterion condensation theory (dashedline). Fig. 9
Illustration of the influence of surrounding ions on to the long-range hy-drodynamic interactions between different parts of a polyelectrolyte chain.(a) For an uncharged polymer, the hydrodynamic interactions are un-screened and all chain monomers can interact with each other. (b) Thepresence of counterions during electrophoresis of polyelectrolytes limitsthe hydrodynamic interaction. (c) The more salt is added to the system,the higher is the ion density in the vicinity of the chain, which reducesthe hydrodynamic interaction range even further, so that most parts ofthe chain appear to be hydrodynamically decoupled.14
10 20 30 40 N μ / μ F D Simulation (HI)Experiment (NMR)Experiment (CE) N μ / μ F D HIno HI
Figure 1: The normalized electrophoretic mobility µ/µ FD as a function of the numberof repeat units N for simulation data including hydrodynamic interactions (HI), andexperimental data coming from capillary electrophoresis (CE) and from electrophoreticNMR. The inset compares to simulation data obtained with a model neglecting hydro-dynamic interactions. 15
10 100N012 m / m c s = 0 mMc s = 16 mMc s = 160 mM Figure 2: The normalized electrophoretic mobility µ/µ of polyelectrolyte chains oflength N for three different salt concentrations using the LB algorithm. The addedsalt not only influences the absolute mobility, but likewise changes the characteristicshape of the mobility with respect to chain length N .16
10 100N00.51 m / m c s = 0 mMc s = 16 mMc s = 160 mM Figure 3: The normalized electrophoretic mobility µ/µ for different chain length N atvarying salt concentrations c s without hydrodynamic interactions differs significantlyfrom the behaviour observed in Figure 2. The mobility shows a salt-dependent mono-tonic decrease for short chains and a salt-independent constant value for long chains.17
10 20 30d-2-1012 Iv CI /v Figure 4: The average ion velocity in the direction of the electric field v CI (here fora chain with N = 64 monomers, salt concentration c s = 16 mM, and hydrodynamicsincluded) depends on the distance d to the center of mass of the polyelectrolyte. Ionsclose to the center co-move with the chain’s velocity (dashed line), whereas ions faraway from the center move with the single particle velocity v = µ E into the oppo-site direction. The distance d at which v CI ( d ) = 0 is used to separate co-moving,associated ions from non-associated ones. The solid line shows the integrated fractionof charges I that is found up to the distance d of the center of mass.18
10 100N110100 Q e ff c s = 0 mMc s = 16 mMc s = 160 mMN(1/ x) N (a) Q e ff / N c s = 0 mMc s = 16 mMc s = 160 mM1/ x (b) Figure 5: (a) The effective charge Q eff as a function of chain length N (symbols for Q (1)eff , lines for Q (2)eff ). Both charge estimators show good agreement. Initially, Q eff isclose to the bare charge N (dotted line), but as ion condensation sets in, the effec-tive charge is reduced. Longer chains show a linear increase of their charge close tothe Manning prediction (1 /ξ ) N (dashed line). (b) The effective charge per monomer Q eff / N is influenced by the salt concentration for short and intermediate chains. Thehigher the concentration of the added salt, the faster the electric charge of the poly-electrolyte is reduced by condensed ions. For long chains, the charge per monomer isagain independent of the salt concentration and comparable to the Manning prediction1 /ξ (dashed line). 19
10 100N110100 Q e ff c s = 0 mMc s = 16 mMc s = 160 mMN(1/ x) N Figure 6: The effective charge as measured by the static estimator Q (3)eff shows a strongdependence on the the salt concentration. At c S = 0 mM the static estimate agreeswith the dynamic estimates (solid line). The higher the salt concentration, the loweris the static charge estimate. For comparison the bare charge N (dotted line) and theManning prediction (dashed-line) are plotted.20
10 100N110100 G e ff m c s = 0 mMc s = 16 mMc s = 160 mMNN (a) G e ff m / N c s = 0 mMc s = 16 mMc s = 160 mM (b) Figure 7: (a) The normalized effective friction Γ eff µ as a function of chain length N fordifferent salt concentrations c s using the LB algorithm. Initially, the friction increasesas given by the hydrodynamic size of the polyelectrolyte Γ = ∼ N . (dotted line).With the onset of counterion condensation the friction exceeds the value of the barepolyelectrolyte and for long chains becomes linear in N (dashed line). The absolutefriction value is increased with the addition of external salt. (b) The normalized effec-tive friction per monomer Γ eff µ /N shows an initial decrease with chain length that isstronger the lower the salt concentration is. From a concentration dependent value of N FD onwards, the friction per monomer becomes a constant value that increases withincreasing salt concentration (indicated by dashed lines).21
10 100N110100 G e ff m c s = 0 mMc s = 16 mMc s = 160 mMN (a) G e ff m / N c s = 0 mMc s = 16 mMc s = 160 mM2-1/ x (b) Figure 8: (a) The normalized effective friction Γ eff µ as a function of chain length N fordifferent salt concentrations c s without long-range hydrodynamic interactions. Initially,the increase of the effective friction is super linear, but linear scaling (dotted line) isreached for longer chains. For these chains, the absolute friction value is independentof the addition of external salt. (b) The normalized effective friction per monomerΓ eff µ /N shows an initial increase with chain length that is stronger the higher the saltconcentration is. For longer chains, a plateau value is reached which is independentof the salt concentration and can be compared to the predicted value based on thecounterion condensation theory (dashed line).22 IHI ++ + ++ + + + ++ +++ + +++ +++
HIHI + ++ + + +++ ++ + + ++ ++ +++++ + +++ ++ + + + ++++ - ++ - ++ - - ---- - --- - - ---- -- + + HIHI a) b) c)a) b) c)