of Charged Macromolecules

Christian Holm

Institut für Computerphysik, Universität Stuttgart Stuttgart, Germany

1 Charge stabilized

The analytical description of charged colloidal suspensions is problematic: n Long ranged interactions: electrostatics/ hydrodynamics n Inhomogeneous/asymmetrical systems n Many-body interactions

Alternative : the relevant microscopic degrees of freedom are simulated via Molecular Dynamics! ●Explicit particles (ions) with charges ε ●Implicit solvent approach, but hydrodynamic interactions of the solvent are included via a Lattice-Boltzmann algorithm Test of LB implementation for Poiseuille

Simulation box of size 80x40x10. Velocity profile for a Poiseuille flow in a channel, which is tilted by 45◦ relative to the Lattice-Boltzmann node mesh. Computed using ESPResSo.

Profile of the absolute fluid velocity of the Poiseuille flow in the 45◦ tilted channel. Red crosses represent simulation data, the blue line is the theoretical result and the dashed blue line represents the theoretical result, using the channel width as a fit parameter

3 EOF in a Slit Pore

Simulation results for a water system. Solid lines denote simulation results, the dotted lines show the analytical results for comparison. Red stands for ion density in particles per nm3, blue stands for the fluid velocity in x-direction, green denotes the particle velocity. All quantities in simulation units. 4 Colloidal Electrophoresis

local force balance FE = FDrag leads to stationary state ν FE F = Z E − Zeff E eff

Electric field E

v Zeff Observable: mobilty µ µ = = E γ eff

µred = 6πη Bµ

e2 viscosity η Bjerrum length  B = 4πεkBT SEM (O‘Brien, White 1978)

Dimensional Analysis

˜ Z = ZlB /R

Di: collective diffusion coefficient

2 2 κ = 4πlB ∑zi ci € # & 2 fi = ci /% ∑z jc j ( $ i ' €

€ View Online

1622 ELECTROPHORETIC MOBILITY OF A COLLOIDAL PARTICLE as a function of Ka for specific values of the dimensionless y=- 4 kT‘ (9.2) This is not the most convenient way of plotting p(r, h-a) if the graphs are to be used for converting experimental mobility measurements to zeta potentials, since these experiments are usually carried out at fixed electrolyte concentration, i.e., fixed Ka.

I

E

”E

E: reduced mobility µred 4

E

3

View Online 2 Downloaded by Universitat Stuttgart on 25 March 2011 Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 1 I I Electrophoretic Mobility of a Spherical Colloidal Particle

BY RICHARDw. O’BRIENt AND LEE R. WHITEf* Department of Applied Mathematics, Institute of Advanced Studies, I,, ll*,, 0 5 Research School of Physical Sciences, The Australian National University, y: reducedCanberra, zeta-potentoial A.C.T. 2600, Australia Y ζred FIG. 3.-Reduced mobility E of a spherical colloidal particle in a KCl solutionReceived as13th a February, function 1978 of reduced zeta potential y, for KU < 2.75. In this regime the mobility appears to increase monotoni- The equations which govern the ion distributions and velocities, the electrostatic potential and cally with zetathe potential. hydrodynamic flow field around a solid colloidal particle in an applied electric field are re- examined. By using the linearity of the equations which determine the electrophoretic mobility, We have adopted the alternative procedure forwe show plotting that for a Ecolloidal as a particle function of any shapeof y the for mobility various is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface. The mobility Ka values. This has the additional advantagedepends ofonly mimicking on the particle size the and formshape, the of properties a typical of the electrolyte in which it is suspended, and the charge inside, or electrostatic potential on, the hydrodynamic shear plane in the experiment where mobility is changed, atabsence fixed of anIC, applied by fieldsystematically or any macroscopic varyingmotion. the concentration of a potential determining ion. NewThis expressions is due for to the the forces monotonic acting on the particlevariation are derived and a novel substitution is developed which leads to a significant decoupling of the governing equations. These analytic of surface potential with the p.d.i. concentration.developments allowSince for the the construction chemist of a rapid,usually robust numericalhas scheme for the solution of the governing equations which we have applied to the case of a spherical colloidal particle in a general control over the value of Ica, the experiment electrolytecan be solution. performed We describe with a computer a Ka valueprogram corres-for the conversion of mobility measurements to zeta potential for a spherical colloidal particle which is far more flexible than the Wiersema graphs ponding to a computed curve. which have traditionally been used for the interpretation of mobility data. Furthermore it is free of the high zeta potential convergence difficulties which limited Wiersema’s calculations to moderate values of 5. Some sample computations in typical 1 : 1 and 2 : 1 electrolytes are exhibited which illustrate the existence of a maximum in the mobility at high zeta potentials. The physical explanation of this effect is given. The importance of the mobility maximum in testing the validity of the governing equations of electrophoresis and its implications for the chemist’s picture of the Stern layer are briefly discussed. Downloaded by Universitat Stuttgart on 25 March 2011

Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 1. INTRODUCTION The velocity at which a charged colloidal particle moves in the presence of a weak electric field is linearly related to E, the strength of the applied field. For a spherical particle this relation takes the form 5 U = pE, where U is the velocity, and p is termed the “electrophoretic mobility” of the particle. This paper deals with the problem of calculating the quantity p. The electrophoretic velocity U is determined by the balance between the electrical and viscous forces which act on the particle, a balance which is strongly affected by the concentrations of the electrolyte ions in the solvent. In order to understand the way in which the ions affect the forces which act on the particle, we must consider the motion of ions around a spherical particle moving under the action of a steady electric field. t Rothmans Fellow ; Queen Elizabeth I1 Fellow. 5 For a non-spherical particle the velocity U will depend on the particle orientation ; there is no reason to expect that the velocity and field strength E will in general be in the same direction, 1607 View Online

R. w. O’BRIEN AND L. R. WHITE 1623 The point is academic however since the method, presented in the previous sections of this paper, enables a rapid, exact numerical solution to the general mobility problem.* Thus the chemist, armed with this computer program can generate mobility as a function of zeta potential for the Ka value, the particular electrolyte solution, and temperature of interest to himself. In view of this, the utility of displaying graphs of p(5, K-a) for restricted electrolyte systems, as a conversion aid for the experimentalist, seems marginal. r 7-

6-

5-

4- E

3-

2- Downloaded by Universitat Stuttgart on 25 March 2011 Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607

Y FIG.4.-Variation of mobility in KCI with zeta potential for KLZ > 3. In this case the mobility curves have a maximum. However, we have solved the mobility equations for a typical 1 : 1 electrolyte, viz. KCl, in order to display some of the striking features of the function &, ica). In fig. 3, we have plotted the reduced mobility E in KCl solution as a function of reduced zeta potential y (on the positive side where Cl- is the counterion) for values of rca 5 2.75. The following important features of fig. 3 should be noted: (i) in * A copy of the program constructed as a self-contained Fortran subroutine is available from the authors. View Online

1624 ELECTROPHORETIC MOBILITY OF A COLLOIDAL PARTICLE the region where Wiersema could obtain to the equations [i.e., y < 5 ( N 125 mV)] our results are in agreement. (ii) For all rca values 52.75, the mobility is a monotonic increasing function of the zeta potential [at least in the experimentally accessible range y < lO(w-250 mV)]. (iii) At low values of zeta potential, all curves asymptote to the Huckel form E=y (9.3) irrespective of ica value. (iv) For values of Ka - 2.75 the mobility E plateaus at -3.17 For y 2 5. All these features are also present in Wiersema's work but are obscured by his method of displaying &, rca) and the restriction on the range of [ values that his method of solution imposes.

I

I I I *,I 21 ' ' """'5 10 50 100

ua FIG.5.-Maximum mobility of a spherical particle in KCl as a function of ua. a=R in lecture notes notation It is this restriction which prevented Wiersema from displaying, in a definitive way, Downloaded by Universitat Stuttgart on 25 March 2011 the most important feature of the p([, rca) function, viz. the maximum in the mobility as a function of zeta potential for all values of rca 2 2.75. In fig. 4, we display E Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 in KCI solution as a function of (positive) y for Ka 3 3. The following features should be noted: (i) a maximum in the mobility exists for all values of rca 2 3; (ii) the value of the maximum mobility increases with rca for rca > 4 (the very shallow maximum for rca = 3 is, however, slightly higher than the rca = 4 value); (iii) the reduced zeta potential corresponding to the maximum mobility is - 5 (125 mV) and increases very slowly as rca increases (the maximum for rca = 3 is at - 7, however) ; (iv) E(y) curves tend to the Smoluchowski form E = 3y (9.4) as rca is increased although, unlike the small Ka curves, there is no regime where all curves reduce to the Smoluchowski form and become independent of Ka. In fig. 5, we plot the maximum mobility as a function of rca for KCl electrolyte. The behaviour of the maximum around rca N 3 implies that the zeta potential corresponding to the maximum quickly moves to high values as Ka is reduced below -4. The maximum in the mobility presumably arises from the fact that the electro- phoretic retarding forces which act on a particle increase at a faster rate with c than does the driving force ; at low values of zeta, this is clearly the case, since the driving force, which is given by qE where q is the particle charge, is proportional to [, while View Online

R. w. O’BRIEN AND L. R. WHITE 1625 the electrical retarding force due to the distorted charge cloud is proportional to c2, for this term involves the product of the particle density and the field due to the charge cloud, and both these terms are proportional to [. The reason that no maximum is observed in the mobility curves for Ica < 2.75 in the range y < 10 is probably due to the fact that, for fixed zeta potential the distortion 6p(r) of the double layer and the body force on the fluid vanish as KU is decreased to zero. Therefore, although the retarding forces increase more rapidly with c than the driving force, they will have small magnitude for low Ica values and thus the maximum mobility will be shifted to high zeta potentials. (This shift is seen as KU is reduced from 4 to 3 as noted previously). Whether a maximum in p(c) exists for y > 10 and Ka 5 2.75 is of academic interest since this region corresponds to t: > 250 mV. Such values are not usually accessible to the experimentalist and would most likely invalidate the Poisson-Boltzmann equation on which the analysis is based.

E -/

I

Downloaded by Universitat Stuttgart on 25 March 2011 Y FIG. 6.-Effect of counterion valency on the form of the mobility against zeta potential curves,

Published on 01 January 1978 http://pubs.rsc.org | doi:10.1039/F29787401607 for Ka = 5. The marked effect of counter-ion valency on the mobility maximum is illustrated10 in fig. 6 where, for KU = 5, we plot p(5) for negative zeta potentials in 1 : 1 (KCl), 2 : 1 [Ba(NO,),] and 3 : 1 [La(CI),] electrolytes. The maximum in the mobility moves from y-5 in 1 : 1 electrolyte, to y-2.5 in 2 : 1 electrolyte and to y- 1.7 in 3 : 1 electrolyte. These values are in the ratio 1 : t : 3. The effect of increasing counter- ion valencies is explained by the observation that the distortion of the double layer by the applied field and the body force on the liquid increase as the valency of the counter-ion is increased. Thus the retarding forces will, for given [, increase with counter-ion valency with the result that the position of the mobility maximum will shift to lower values of t:. The shift would appear, from fig. 6, to be inversely proportional to the valency. It should be noted that the theories of Booth and Overbeek based on a perturba- tion expansion in powers of also reveal this mobility maximum. The numerical results of Wiersema strongly suggest the effect demonstrated here, but unfortunately the iteration scheme he used broke down at the value of 5 at which the maximum occurred (y-5 in 1 : I electrolyte, y-2.5 in 2 : 1 electrolyte). Indeed, the failure of the iterative scheme and the presence of the maximum are likely to be related. Electro-Hydrodynamical Model

charged colloid! ! n driven system ! (external constant E –field)! n 1 central Lennard-Jones ! (LJ) bead ! n 100 LJ monomers on the ! surface, connected to a network via FENE bonds! ncounterions: LJ beads!

simulation=> lattice (implicit) hydrodynamics Langevin MD + Lattice-Boltzmann algorithm periodic boundary conditions Ewald sum: P3M V. Lobaskin, B. Dünweg, CH J. Phys.: Condens. Matter 16, S4063-S4073 (2004) Ionic Distribution around the Colloid Ionic Distribution around the Colloid

the ions within the shear plane renormalize the charge

Z to Zeff Mobility as a Function of Charge Z counterion condensation influence of comoving counterions

Mobility is calculated at zero field with the Green- Kubo integral: linear regime Comparison to Experiments

ZEff=20-30, R = 2.2 nm 2 κ = 4π B (nsalt + nM Zeff + nH ,OH )

V. Lobaskin, B. Dünweg,M. Medebach,T. Palberg, CH, PRL 98, 176105 (2007) Salt and Concentration Effects

Salt-free Simulations at finite Φ can be mapped to simulations including salt Colloidal Electrophoresis

Comparison to experiments F. Kremer (Leipzig) I. Semenov, S. Raafatnia, M. Sega, V. Lobaskin, C. Holm, F. Kremer. Phys. Rev. E 87(022302), 2013.

AC E-Field outlet inlet A B

Fluidic Cell IR-blocking filter Sync. Output for LED CMOS

LED Indicator Function Generator

17 Simulation snapshot

18 0

2

-potential Planar ⇣ R=5 nm 4 R=20 nm R=100 nm reduced R=400 nm 6 5 4 3 2 1 10 10 10 10 10 , [M]

19 ILYA SEMENOV et al. PHYSICAL REVIEW E 87,022302(2013) potential drop between the bulk and the location of the experiments; see Fig. 5(a).Formonovalentsalt,amaximumin 4 shear plane, xsp 1.025 σlj .Ithasbeencheckedthatsmall the mobility is observed at low ionic strengths of 10− mol/l, variations in the= location of the shear plane do not alter the which is in agreement with the SEM and previous∼ findings results significantly. The electrophoretic mobility is eventually for similar monovalent ionic solutions [9,47–49]. A less pro- calculated using the SEM, taking as input the ζ potential nounced monotonic decrease of the mobility with increasing obtained from the simulation and the experimental κa value. ionic strength is observed for CaCl2 and LaCl3.Inthecaseof trivalent salt, the experimental results show a mobility reversal, IV. RESULTS as can be inferred from the 180 degrees phase offset change, i.e. the colloid oscillates in phase with the electric field at Experimental and simulation results for the electrophoretic 2 I>10− mol/l[Fig.5(b)]. mobility as well as the experimentally measured phase angle Using the experimentally determined colloidal surface are shown in Fig. 5 as a function of the ionic strength. As can charge density in the simulations, a quantitative agreement be seen in Fig. 5(b),atmostconditionsthecolloidalmotionis with the experiment is achieved for the monovalent salt in the out of phase (phase offset is 225 degrees) with respect to the ∼ peak area and the full curves are at least qualitatively repro- electric field due to its negative surface charge. The decrease duced without further amendments to the restricted primitive of the electrophoretic mobility with increasing ionic strength model in the monovalent and divalent cases [see Fig. 5(a)]. predicted by the SEM for all three valencies is observed in the However, no ζ potential (and hence mobility) reversal is observed in the trivalent case. This could have been expected, since in the strong coupling theory of Netz et al. [50]the 2 3 Coulomb coupling parameter $ 2πσlB z ,wherez denotes the counterion valency, needs to= be larger than 10 to produce charge inversion by pure electrostatic correlation effects, while for the investigated trivalent salt system we have only $ 1.7. Obviously, the colloidal surface charge density is too= low to bind a sufficient amount of trivalent ions electrostatically to produce the experimentally observed mobility reversal. This indicates the presence of other forces besides Coulombic attraction between the trivalent ions and the surface. In fact, 3 specific interactions between La + ions and Latex colloids have been reported by other authors [9,51–53]. Therefore, to model this ion-specific attraction the WCA interaction (a) between the colloidal surface and the counterions is replaced by a full LJ potential. The resulting reduced electric potential profiles is shown in Fig. 6 for different ionic strengths. An attractive well depth of εcoll 4 kB T is found to be necessary in orderDetermination to recover the mobility =of zeta-potential reversal in agreement with

(b)

FIG. 5. (Color online) The electrophoretic mobility (a) and phase angle (b) vs. salt ionic strength of aqueous solutions of varying valency (KCl, CaCl2,andLaCl3). Squares represent the measured data, with error bars indicating the standard deviation. 20 The measurements for each valency are carried out with the very FIG. 6. (Color online) Reduced electric potential ( eψ where ψ kB T same negatively charged PS colloid (diameter: 2.23 µm). The field is the electric potential) profiles at different ionic strengths obtained strength is varied in the range 1–18 V/cm. The laser power is 0.2W. from simulations of the trivalent case with full LJ interaction between The simulation results with and without LJ attraction are shown via the charged surface and the counterions. The position of the shear stars and triangles, respectively, connected by dotted lines for a guide plane is taken to be at xsp 1.025 σlj from the solid surface. It is = 2 to the eyes. In the monovalent case, the solid line represents SEM seen that for ionic strengths I>10− mol/l, the ζ potential changes calculations based on GC solutions, whereas the dashed and dotted sign and, consequently, the mobility is reversed. In the inset, the lines indicate SEM calculations using GC and spherical PB solutions, reduced potential profile is shown for the monovalent case at ionic 1 including the LJ attraction, respectively. strength 10− mol/l. No mobility reversal is observed in this case.

022302-4 Experiment vs. Simulations Monovalent Divalent Trivalent -6 MONOVALENT DIVALENT TRIVALENT -5 4 MD Sim Full LJ 4 -4 Planar PB MD Sim

/Vs] + 2+ Full LJ 3+ 2 Ka Ca La -3 Planar PB

[m 2 2 -8 -2

*10 e -1 Ion specific 0 0 interactions -0 Spherical PB 3+

Reduced Mobility for La SCE, ExperimentExperiment --1 MD Sim, full LJ 5 4 3 2 3 1 2 4 2 5 5 1 4 1 3 Mobility Mobility GC SCE, Experiment SCE, Experiment --2 GC+LJ MD Sim, full LJ MD Sim, full LJ 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Spherical PB+LJ MD Sim, WCA MD Sim, WCA --3 Salt Ionic Strength [M] -6 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 log Salt Ionic Strength [mol/l] Figure 2: Electrophoretic mobility as a function of ionic strength for monovalent (KCl), divalent (CaCl2)andtrivalent(LaCl3)saltsolutions.Experimentalresults,blacksquares, are from Semenov et al.10,31,32 for latex colloids of diameter 2.23 µm and surface charge density 0.31 µC/cm2 0.02 e/nm2. Simulation results are represented via blue circles (WCA interaction) and' red filled circles (full LJ interaction). The dashed and dotted lines represent calculations based on the numerical solution to the nonlinear planar and spherical PB, respectively. In the trivalent case, the red solid line represents the PB solution including 21 the full LJ interaction between the counterions and the surface. Data taken from. 10

13 mobilities of latex colloids in mono-, di- and trivalent salt (KCl, CaCl2 and LaCl3 respec- tively) as a function of ionic strength using a Mark II microelectrophoresis apparatus. We compare our simulation results to their experimental data for a colloid of diameter 0.753 µm and surface charge density 5.64 µC/cm2 0.35 e/nm2, see Fig. 4. Due to the large size of ' the colloid in comparison to the Debye length in the considered range of salt concentrations, 12 < R<3.91 102, the planar geometry inherent in our simulation technique is also Comparison⇥ to other Experimental Data appropriate for this system. MONOVALENT DIVALENT TRIVALENT MONOVALENT DIVALENT TRIVALENT

7 7 MD Sim Full LJ 5 5 Planar PB MD Sim 5 Full LJ 5 Planar PB 3 3 3 3 1 1 1 1 ExperimentExperiment reduced mobility

1Reduced Mobility 1 1 PlanarSpherical PB PB 1 MD Sim WCA MD Sim Full LJ 4 3 1 1 4 5 2 2 3 5 3 2 4 3 1 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 3 1 3 2 4 1 5 4 2 5 3 1 4 2 10 10 10 10 10 10 10 10 10 10 10 10 10 Salt Ionic Strength [M] 10 salt ionic strength [M] Figure 4: Electrophoretic mobility as a function of ionic strength of monovalent (KCl), divalent (CaCl2)andtrivalent(LaCl3)electrolyte.Simulationresults,representedviablue (WCA) and red filled circles (full LJ), are compared to the experimental results of Elimelech et.al indicated by black squares. Experiments were done by Elimelech et al.6 with latex colloids of diameter 0.753 µm and surface charge density 5.64 µC/cm2 0.35 e/nm2. The dashed and dotted lines represent calculations based on the numerical' solution to the nonlinear planar and spherical PB, respectively. In the trivalent case, the red solid line represents the PB solution including the full LJ interaction between the counterions and the 22 surface.

17 Conclusions

• Successful mapping of simulations of charged colloidal electrophoresis onto experimental values • Colloidal concentration can be mapped on salt concentration

• Intriguing minimum observed in µred as function of κR • SEM works well, even for multivalent salt solutions if combined with Simulations for determining the zeta potential • In specific effects (non-electrostatic attraction) might be important for some ion species (here La3+) Electrophoresis of

Christian Holm

Institut für Computerphysik, Universität Stuttgart Stuttgart, Germany

24 Outline n Coarse-grained simulation model for electrophoresis with and without HI n Results on diffusion & mobility compared to experiments n Effective charge estimators n Effective friction of polyelectrolytes in free- solution electrophoresis n ELFSE n Electrophoresis of charged colloids Why care about electrophoresis?

n Electrophoretic separation of DNA n Crucial step is gene analysis n Yields characteristic genetic finger prints n Today: Gel Electrophoresis based on entanglement n Widely applicable and reliable n Slowed down dynamics leads to long elution times n Future: Novel separation techniques based on hydrodynamic and chemical interactions n Micro-fluidic devices with structured surfaces n On-going design and development process First Goal: Understand free-flow Electrophoresis Free-Flow Electrophoresis n Charged polymers move in solution under the influence of an external electric field n Local force balance leads to constant velocity n Electrical driving force n Solvent friction force n Electrophoretic mobility µ n Size dependence of µ (N) determines separation process (N) (N) Short and long polyelectrolytes n Short PE chains: Experimental observations: n Extended rod-like conformation n Length dependent mobility µ(N) n Long PE chains: Free- draining n Random coil

conformations N NFD n Screening of long range hydrodynamic interactions Crossover:

n Length independent § DNA: NFD ~ 170 bp mobility (free-draining) § PSS: NFD ~ 100 units Short and long polyelectrolytes Not understood n Short PE chains: Experimental observations: n Extended rod-like conformation n Length dependent mobility µ(N) n Long PE chains: Free- draining n Random coil

conformations N NFD n Screening of long range hydrodynamic interactions Crossover:

n Length independent § DNA: NFD ~ 170 bp mobility (free-draining) § PSS: NFD ~ 100 units Model n No chemical details n Charged subgroups connected along a backbone by elastic springs n Ions modeled as free mobile charges n Implicit solvent model n continuous dielectric background n Explicit charges treated with full electrostatics via P3M in p.b.c. n Explicit hydrodynamics by frictionally coupling the beads to a Lattice-Boltzmann fluid n Can simulate with HI and without HI (Langevin) LB-Hydrodynamics

• Lattice Boltzmann equation • streaming and collision • discrete set of velocity populations • D3Q18 scheme • hydrodynamic fields from populations Hydrodynamics via LB n Frictional coupling of MD particles to Lattice Boltzmann fluid [1] n Modified Langevin equation:

n Momentum exchange between immersed particles and fluid n Total momentum conservation Hydrodynamic interactions

[1] P. Ahlrichs and B. Dünweg. International Journal of Modern Physics C, 9:1429-1438, 1998. Current D3Q19 Version with correct fluctuation spectrum due to Schiller, Duenweg implemented in ESPResSo 32 Observables

n (Self-)Diffusion D

n Electrophoretic mobility µ

n Zero-field mobility (Green-Kubo):

1 ∞ µ = q vr (0 )⋅ vr (τ) dτ 3k T ∑i i ∫ i PE B 0 33

€ Simulation parameters

n Mapping on sulfonated polystyrene (PSS) n All beads have diameter of 2.5 Ångström n Chain length N=1 …64 n lb = 7.1 Ångström (H2O at 20° C) n Monomer concentration 100 mMol n No added salt n Experimental conditions: n Böhme, Scheler, IPF Dresden, PFG-NMR n H. Cottet, CNRS Montpellier, Cap. Elec.

34 Results 1: Diffusion

K. Grass, U Böhme, U. Scheler, H. Cottet, and C. Holm, Phys. Rev. Lett. 100, 096104 (2008) Electrophoretic mobility Effect of HI

No HI!

K. Grass, U Böhme, U. Scheler, H. Cottet, and C. Holm, Phys. Rev. Lett. 100, 096104 (2008) Salt dependence cs of Mobility

Maximum reproduced No maximum

Independent. of cs!

with! without hydrodynamical interactions (HI)! -ion-complex

E Hydrodynamic Friction

n Important quantity to characterize PEs n Effective friction for polymers

n Obtainable from mobility measurements _ _

Q1: How to estimate the_ effective charge? Q2: Can we use Γeff = Γeff to measure Qeff 3 Estimators for Qeff (1) Ion velocity distribution to find co- moving counter ions

(2) Calculation from Langevin (no HD) mobility

(3) Counter ions within 2σ of PE chain Co-Moving Counter Ions n Co-moving counter ions reduce effective charge of PE-Ion complex n Look at the ion velocity distribution to find co-moving counterions Ion Velocity Distribution

Simulations with and without HD yield almost identical threshold => similar Qeff vIon: relative velocity to center of mass (PE + CI) frame Langevin Simulation (No HI) n The mobility measures the combined movement of PE and co-moving counter ions n Local friction of each particle is given in Langevin simulation as Estimator Comparison

(3) (2) (1) Qeff > Qeff ≈ Qeff

Manning fraction fM = 1-b/λB based on one infinite rod at infinite dilution dynamic Qeff= static Qeff Effective Friction

_ _ _

_

Debye length ≈ hydrodynamic screening length

PE effective friction deviates from single chain D for longer N! Influence of condensed counterions Effective Charge and Friction

_

Debye length ≈ hydrodynamic screening length

Constant friction and charge per monomer for long PEs. 3 Estimators for Qeff (1) Calculation from Langevin mobility (no HI)

(2) Ion velocity distribution to find co- moving counter ions

(3) Counter ions within 2σ of PE chain Ion velocity with respect to CM Effective charge vs. N

(1) Only the dynamical estimators Q eff and (2) Q eff are compared

(1) (2) Effective charge estimators Q eff and Q eff are numerically equivalent! No dependence on hydrodynamical interactions, weak salt dependence for small N, Qeff is independent of salt for large N, and Qeff /N = constant (3) Static effective charge Q eff

Contrary to the dynamical estimators we observe a strong salt dependence 50 Effective friction vs. N

versus

Zimm (?) scaling with! Rouse (?) scaling without hydrodynamical interactions! Effective friction vs. chain length

Hydrodynamic shielding Independant. of cs

With! Without hydrodynamical interactions.! For large N Γ/Ν = constant ! Hydrodynamic coupling scheme

Influence of surrounding ions on hydrodynamic screening length

unscreened medium screening by large screening by counterions counterions

Debye length ≈ hydrodynamic screening length Intermediate conclusion on FSE n Coarse-grained MD with full electrostatic and explicit hydrodynamic interactions reproduce experimental results on bulk electrophoresis of polyelectrolytes, with and without salt n Mobility maximum due to hydrodynamic shielding n Dynamic charge estimators do not depend on HI, and are independent of salt (the static one is, however, salt dependent!) n Effective friction is strongly influenced by co-moving ions and salt concentration: n Hydrodynamic shielding for short N, constant friction per monomer for long chains due to hydrodynamic screening

• K. Grass, U. Böhme, U. Scheler, H. Cottet, C. Holm, Phys. Rev. Lett., 100, 096104 (2008). • K. Grass, C. Holm, J. Phys.: Condens. Matter 20, 494217 (2008). • K. Grass, C. Holm, Soft Matter 5, 2079 (2009). How to separate long chains? n Attaching a label to the PE changes the scaling of effective friction („Molecular parachute“) ELFSE (End-Labeld Free Solution Electrophoresis) principle n Size-dependent free-solution mobility of long PE chains

• Mayer, P., et. al.; Anal. Chem. 1994, 66, 1777-1780

• Ren, H., et. al.; Electrophoresis 1999, 20, 2501-2509 End-labeles

• Linear tags • Branched tgs • Micellar tags Mobility influenced by label

nEnd-labeling restores size dependence of mobility n ⇒ separation possible Theoretically expected slowdown

§ Calculate persistence length

lL = 1.9, lPE = 4.7 --> αML ≈ 0.4*20 ≈ 8

§ Calculate α from simulation according to

• Desruisseaux, C. et. al.; Macromolecules 2001, 34, 44-52 Slow down as expected!

λD

Results for linear labels, no free fit parameters Conclusions on ELFSE

• ELFSE works according to the theoretical predictions • Size dependant sparation can be pushed to longer chain lengths • The insight gained by analyzing this process can lead to optimized experimental setups n For more details see:

K. Grass, C. Holm, G. Slater, Macromolecules, 42, 5352-5359 (2009) Governing Physics for Bare Colloids Stokes equation: ⌅⇧u N ⇥ = 2⇧u ⇧ P n z e⇧ ⇤ s ⌅t r r j j r j=1 X Poisson-Nernst-Planck equation: (~u ~v ) z e~ k T ~ log n = ~0 j j j r B r j Incompressibility: ~ ~u = ~0 r · ⇥s, ⇤u, : Solvent’s density, velocity and viscosity P : Pressure V : Particle’s velocity ~ : Electric field due to ions of charge z j and concentration n j r and the applied field j : ionic drag coefficient

Monday, July 7, 2014 Governing Physics for Soft Colloids The Darcy-Brinkman equation: ⌅⇧u N ⇥ = 2⇧u ⇧ P + (V⇧ ⇧u) n z e⇧ ⇤ s ⌅t r r l2 j j r j=1 drag force of the layer X Poisson-Nernst-Planck equation:| {z } (~u ~v ) z e~ k T ~ log n = ~0 j j j r B r j Incompressibility: ~ ~u = ~0 r ·

l2 : permeability of the grafted polymer layer l(r):Brinkman length related to the monomer distribution, decreases with increasing monomer density

Monday, July 7, 2014 Soft Colloid Electrophoresis

• Canonical ensemble (NVT) • L: Simulation box size (L=48σlj) • Raspberry-like colloid • Rcol: colloid radius (R=3σlj) • Qcol: colloid charge • N: number of grafted polymers (N=20) • M: degree of polymerization of each chain (M=20) • H: layer thickness ~ 4.5σlj • λ: fraction of charged monomers • cs: monovalent salt concentration • λD: Debye length • HI via LB fluid

3 ⌘e µred = µe 2 ✏rs✏kBT Hill’s numerical solver based on Darcy-Brinkman formalism* Monomer density parameters found from fit to simulation

* Hill et al., Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 31–49

Monday, July 7, 2014 Mobility of Neutral Soft Colloid

101 D/ 100 1 =0.1

Qcol = 40 e 0.5 Qnet =0e red µ 0 Simulation Theory 0.5 3 2 1 0 10 10 10 10 cs/M

Monday, July 7, 2014 Mobility of Neutral Soft Colloid

101 D/ 100 1 =0.1

Qcol = 40 e 706 Langm uir, Vol. 17, N o. 3, 2001 0.5 Harden et al. Qnet =0e2ModelandNotations

red We consider uniformly charged polyelectrolytes of

µ degree of polymerization N end-grafted at a number density σ 1/d 2 onto a flat charged surface, and focus on 0 the case of∼ high ionic strengths where the thickness of the D >H screening (Debye) layer κ-1 is much smaller than both the typicalsizeofa polymer andthedistancebetween graftingSimulation EOF 1I sites d.Insuchhighionicstrengthconditions,electrostatic interactions between monomers on a given polyelectrolyteTheory chain are screened, so that an unperturbed polymer EOF 1 0.5 behaves as a statistical coil of typical size R aNν,where a is the3 Kuhn statistical segment2 length (the∼ effective1 0 10persistence length) and10 the scaling exponent10νcharacter- 10 izes thesolvent quality(ν = 3/5 in good solvent conditions and ν = 1/2 in θ-solvent conditions).14 Two regimes are immediatelyapparent: the“mushroom”regime,cs/M σR 2 , 1, where grafted chains do not interact (Figure 1a), and the “br u sh ” r egim e, σR 2 .1, where the grafted polymers are strongly overlapping and stretch away from the grafting surface toavoid unfavorable monomer-monomer contacts Figure 1. A sketch of grafted polymers in (a) the “mushroom” (Figure 1b). Grafted polymer layers (see reviews in refs regime (d . R ), where grafted chains do not interact, and (b) 15 and 16) are soft structures that can stretch or tilt in the“brush”regime(d , R), where the grafted polymers stretch response to applied forces or flows.17-23 awayfromthegraftingsurfacetoavoidunfavorablemonomer- Monday, July 7, 2014 We want to address the situation where the surface monomer contacts. anditsgraftedpolymer layer aresubmittedtoa tangential electric field E.The problem at hand is,in principle,very a grafted wall and consequently the electrophoretic complex as it combines polymer dynamics and electro- mobilityofa particleonwhichpolymershavebeen grafted hydrodynamics.Therefore,wewilladopt a simplescheme (provided the particle is large compared to the thickness recently put forward24,25 to evaluate the resulting layer of the coating). Moreover, this would facilitate polymer deformation and flow perturbation. This simple picture coverage estimates from electrokinetic data. We will is strictly speaking valid in the limit ofvery thin polymers, consider separately the limits of low and high grafting small Debye lengths, and weak surface potentials, but density (Figure 1). permits a reasonably easy first-order description of the In section 2, we recall classical elements of electro- physics at work. This approach is based on the classical hydrodynamics and use recent progress in the under- strategy of replacing the explicit description of coupled standing of the behavior of charged polymers in electric electro-hydrodynamics by a problem of pure hydrody- fields to propose a rather simple model for the complex namics with slip boundary conditions.2 problem at hand. In particular, our model reduces the Let us start by considering the effect of an applied field general problem of electro-osmotic flow past a charged on the two components (surface and grafted polymers) surface with grafted polyelectrolyte chains to the case of taken separately, as sketched in Figure 2. Application of neutralchainsgraftedtoa surfacewith an effectivecharge the field along the bare charged surface results in a drag density. Section 3 describes our analysis of polymers on the nonneutralfluid in the thin Debye layer and thus grafted to a flat surface at low densities in which the solvent flow (Figure 2a). Neglecting the description of the polymers maybe treated on an individual(noninteracting) flow in this thin layer, the result is a macroscopic slip of basis. In particular, we derive the weak, intermediate, the fluid on the surface at a velocity V )-µsE,whereµs and strong deformation regimes for the isolated grafted is a mobility that is characteristic of the surface. In the chains. We show that chains grafted at low density give simplest picture, this parameter is typically proportional rise to screening of the electro-osmotic flow generated by tothesurfacechargedensityandtotheDebyelength and the surface. The magnitude of this screening exhibits inversely proportional to the fluid viscosity (see, e.g., refs nonmonotonic dependence on the applied electric field E, 1 and 2). As a result of this slip, the entire fluid flows in steady-state as a plug at a velocity V )-µ E everywhere, increasing rapidly with E near the onset of chain s which is the classical electro-osmotic flow. Consider next deformation, reaching a maximum value, and finally an unattached polyelectrolyte in the electric field E.This decreasing to a plateau value corresponding to fully chain drifts in a free-draining way at velocity µ E relative elongated grafted chains. In section 4, we generalize our p to the solvent. The polymer mobility µ is a local (mon- analysis to high grafting densities (the brush regime), p omeric) property that does thus not depend on the degree where the layer is found to be much less compliant. For weak electricfields, the surface-generated electro-osmotic (14) Barrat, J .-L.; J oanny, J .-F. Adv. Chem . Phys. 1996, 94,1. flow is exponentially screened by the polymer brush. For (15) Milner, S. T. S cience 1991, 251,905. sufficiently strong electric fields, however, the chains (16) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym . S ci. 1992, 100,33. become deformed near the grafting surface.In good solvent (17) Rabin, Y.; Alexander, S. Europhys. Lett. 1990, 13,49. conditions, this deformation leads to a slight swelling of (18) Barrat, J .-L. Macrom olecules 1992, 25,832. the polymer brush and significantly enhanced screening (19) Brochard-Wyart, F. Europhys. Lett. 1993, 23,105. of the surface-generated electro-osmotic flow as compared (20) Brochard-Wyart, F.; Hervet H.; Pincus P. Europhys. Lett. 1994, 26,511. with undeformed brush conditions. Finally, in section 5 (21) Brochard-Wyart, F. Europhys. Lett. 1995, 30,387. we summarize our results and conclude the paper with (22)Harden, J. L.; Cates, M. E. Physical R ev. E 1996, 53,3782. a discussion of the highlights and shortcomings of our (23) Harden, J . L.; Borisov, O. V.; Cates, M. E. Macrom olecules 1997, 30,1179. approach as well as a few suggestions for future experi- (24) Long, D.; Viovy, J . L.; Ajdari, A. Phys. R ev. Lett. 1996, 76,3858. mental and theoretical studies. (25) Long, D.; Ajdari A. Electrophoresis 1996, 17,1161. Mobility of Charged Soft Colloid

101 D/ 100

0 706 Langm uir, Vol. 17, N o. 3, 2001 Harden et al.

=02ModelandNotations.1 We consider uniformly charged polyelectrolytes of red degreeQcol of= polymerization90 e N end-grafted at a number 2 µ density σ 1/donto a flat charged surface, and focus on >H theQ casenet of∼ high= ionic50 strengths e where the thickness of the D 1 screening (Debye) layer κ-1 is much smaller than both the typicalsizeofa polymer andthedistancebetween grafting EOF 1I sites d.Insuchhighionicstrengthconditions,electrostatic interactions between monomers on a given polyelectrolyte Simulation chain are screened, so that an unperturbed polymer EOF 1 behaves as a statistical coil of typical size R aNν,where a is the Kuhn statistical segment length (the∼ effective Theory persistence length) and the scaling exponent ν character- 2 izes thesolvent quality(ν = 3/5 in good solvent conditions and ν = 1/2 in θ-solvent conditions).14 Two regimes are immediatelyapparent:3 the“mushroom”regime,2 σR 2 , 1, 1 0 where10 grafted chains do not interact10 (Figure 1a), and the10 10 “br u sh ” r egim e, σR 2 .1, where the grafted polymers are strongly overlapping and stretch away from the grafting surface toavoid unfavorable monomer-monomerc contactss/M Figure 1. A sketch of grafted polymers in (a) the “mushroom” (Figure 1b). Grafted polymer layers (see reviews in refs regime (d . R ), where grafted chains do not interact, and (b) 15 and 16) are soft structures that can stretch or tilt in the“brush”regime(d , R), where the grafted polymers stretch response to applied forces or flows.17-23 awayfromthegraftingsurfacetoavoidunfavorablemonomer- We want to address the situation where the surface monomer contacts.Monday, July 7, 2014 anditsgraftedpolymer layer aresubmittedtoa tangential electric field E.The problem at hand is,in principle,very a grafted wall and consequently the electrophoretic complex as it combines polymer dynamics and electro- mobilityofa particleonwhichpolymershavebeen grafted hydrodynamics.Therefore,wewilladopt a simplescheme (provided the particle is large compared to the thickness recently put forward24,25 to evaluate the resulting layer of the coating). Moreover, this would facilitate polymer deformation and flow perturbation. This simple picture coverage estimates from electrokinetic data. We will is strictly speaking valid in the limit ofvery thin polymers, consider separately the limits of low and high grafting small Debye lengths, and weak surface potentials, but density (Figure 1). permits a reasonably easy first-order description of the In section 2, we recall classical elements of electro- physics at work. This approach is based on the classical hydrodynamics and use recent progress in the under- strategy of replacing the explicit description of coupled standing of the behavior of charged polymers in electric electro-hydrodynamics by a problem of pure hydrody- fields to propose a rather simple model for the complex namics with slip boundary conditions.2 problem at hand. In particular, our model reduces the Let us start by considering the effect of an applied field general problem of electro-osmotic flow past a charged on the two components (surface and grafted polymers) surface with grafted polyelectrolyte chains to the case of taken separately, as sketched in Figure 2. Application of neutralchainsgraftedtoa surfacewith an effectivecharge the field along the bare charged surface results in a drag density. Section 3 describes our analysis of polymers on the nonneutralfluid in the thin Debye layer and thus grafted to a flat surface at low densities in which the solvent flow (Figure 2a). Neglecting the description of the polymers maybe treated on an individual(noninteracting) flow in this thin layer, the result is a macroscopic slip of basis. In particular, we derive the weak, intermediate, the fluid on the surface at a velocity V )-µsE,whereµs and strong deformation regimes for the isolated grafted is a mobility that is characteristic of the surface. In the chains. We show that chains grafted at low density give simplest picture, this parameter is typically proportional rise to screening of the electro-osmotic flow generated by tothesurfacechargedensityandtotheDebyelength and the surface. The magnitude of this screening exhibits inversely proportional to the fluid viscosity (see, e.g., refs nonmonotonic dependence on the applied electric field E, 1 and 2). As a result of this slip, the entire fluid flows in steady-state as a plug at a velocity V )-µ E everywhere, increasing rapidly with E near the onset of chain s which is the classical electro-osmotic flow. Consider next deformation, reaching a maximum value, and finally an unattached polyelectrolyte in the electric field E.This decreasing to a plateau value corresponding to fully chain drifts in a free-draining way at velocity µ E relative elongated grafted chains. In section 4, we generalize our p to the solvent. The polymer mobility µ is a local (mon- analysis to high grafting densities (the brush regime), p omeric) property that does thus not depend on the degree where the layer is found to be much less compliant. For weak electricfields, the surface-generated electro-osmotic (14) Barrat, J .-L.; J oanny, J .-F. Adv. Chem . Phys. 1996, 94,1. flow is exponentially screened by the polymer brush. For (15) Milner, S. T. S cience 1991, 251,905. sufficiently strong electric fields, however, the chains (16) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym . S ci. 1992, 100,33. become deformed near the grafting surface.In good solvent (17) Rabin, Y.; Alexander, S. Europhys. Lett. 1990, 13,49. conditions, this deformation leads to a slight swelling of (18) Barrat, J .-L. Macrom olecules 1992, 25,832. the polymer brush and significantly enhanced screening (19) Brochard-Wyart, F. Europhys. Lett. 1993, 23,105. of the surface-generated electro-osmotic flow as compared (20) Brochard-Wyart, F.; Hervet H.; Pincus P. Europhys. Lett. 1994, 26,511. with undeformed brush conditions. Finally, in section 5 (21) Brochard-Wyart, F. Europhys. Lett. 1995, 30,387. we summarize our results and conclude the paper with (22)Harden, J. L.; Cates, M. E. Physical R ev. E 1996, 53,3782. a discussion of the highlights and shortcomings of our (23) Harden, J . L.; Borisov, O. V.; Cates, M. E. Macrom olecules 1997, 30,1179. approach as well as a few suggestions for future experi- (24) Long, D.; Viovy, J . L.; Ajdari, A. Phys. R ev. Lett. 1996, 76,3858. mental and theoretical studies. (25) Long, D.; Ajdari A. Electrophoresis 1996, 17,1161. Mobility of Charged Soft Colloid

Experimentally realizable!

D/ 101 100

1 1

0 0 red

µ 1 1 Rcol =1µm 2 2

3 2 1 0 10 10 10 10

cs/M

Monday, July 7, 2014 Mobility of Charged Soft Colloid: Charge Effect

Simulation =0.1 1 Theory cs =0.5M red µ

0

450 350 250 150 50 0 40 Qnet/e

Monday, July 7, 2014 Conclusion

• Neutral soft colloids electrophorese at high salt concentration • Mobility reversal of a charged composites in monovalent salt • This can be used as a measure of the local salt concentration • Good agreement with the numerical results via Hill’s solver • Darcy-Brinkman is applicable to soft surfaces

Monday, July 7, 2014