Electrophoresis of Janus Particles: a Molecular Dynamics simulation study
aa r X i v : . [ c ond - m a t . s o f t ] D ec Electrophoresis of Janus Particles: a Molecular Dynamics simulation study
Taras Y. Molotilin, Vladimir Lobaskin, and Olga I. Vinogradova ∗ A.N. Frumkin Institute of Physical Chemistry and Electrochemistry,Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia School of Physics and Complex and Adaptive Systems Lab,University College Dublin, Belfield, Dublin 4, Ireland Department of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia DWI - Leibniz Institute for Interactive Materials,RWTH Aachen, Forckenbeckstraße 50, 52056 Aachen, Germany (Dated: June 12, 2018)In this work, we use Molecular Dynamics and Lattice-Boltzmann simulations to study the prop-erties of charged Janus particles in electric field. We show that for relatively small net charge andthick electrostatic diffuse layer mobilities of Janus particles and uniformly charged colloids of thesame net charge are identical. However, for higher charges and thinner diffuse layers Janus particlesalways show lower electrophoretic mobility. We also demonstrate that Janus particles align withthe electric field and the angular deviation from the field’s direction is related to their dipole mo-ment. We show that the latter is affected by the thickness of electrostatic diffuse layer and stronglycorrelates with the electrophoretic mobility.
I. Introduction
Electrophoresis is both a useful tool and a broad field ofresearch that has recently met its 200th anniversary.[1, 2]Since then, much work has been done, and today nu-merous applications exist, and often are even treated assomewhat routine.Until recently, most studies of electrophoresis have as-sumed that particles are uniformly charged. In sucha situation, the electrophoretic mobility µ , which re-lates the translational velocity v c of a particle of ra-dius R immersed in electrolyte solution of concentra-tion C Σ to the electric field v c = µ E , is given by µ ( ζ ).Here, ζ is the zeta potential, which for hydrophilic sur-faces is simply equal to surface electrostatic potentialdetermined by the charge density of the particle (butnote that for hydrophobic particles the situation is morecomplicated).[3] The exact µ ( ζ ) relation depends on thethickness of the electrostatic diffuse layer (EDL) via adimensionless quantity κR , where κ is the inverse De-bye length, κ = (4 πl B C Σ ) / with l B being the Bjerrumlength and C Σ for a 1:1 electrolyte is the total concentra-tion of ions in the system. The dimensionless mobility, e µ = 6 πηl B µ/e , where η stands for the dynamic viscos-ity of the solvent and e is the elementary charge, can beexpressed as e µ = f e ζ, (1)with dimensionless zeta potential e ζ = ζe/k B T (where k B T denotes the thermal energy), which can be deducedfrom measured e µ if f is known. Earlier models have pre-dicted f = 1 in the H¨uckel thick EDL limit[4] ( κR ≪ f = 3 / κR ≫ f depending on κR the ma- ∗ [email protected] FIG. 1. A snapshot of the simulation system: The ‘rasp-berry’ colloid particle is surrounded by cations (red) and an-ions (blue). The uncharged sector is shown by white. Thesolvent grid is not shown. jority of previous works have used the classical mean-field solution by O’Brien and White,[6] which is oftenreferred to as the
Standard Electrokinetic Model (SEM).This theory has played a major role in the interpretationof electrophoretic measurements over several decades.The assumption that particles are uniformly chargedbecomes unrealistic for such colloids as Janus particles(JP),[7, 8] which have opened a new field of investigationwith both fundamental and practical perspectives. SuchJPs can be used for optical nanoprobes,[9] E-paper dis-play technology,[10] or cargo transport.[11, 12] Their sus-pensions demonstrate rich phase behavior ranging fromcross-linked gels up to ferroelectric crystals.[13] In thecase of JPs, other factors like surface charge (or zeta po-tential) heterogeneity and anisotropy, come into play, sothey should become a very important consideration inelectrophoresis.The body of theoretical and experimental work investi-gating electrophoretic properties of this class of particlesis much less than that for uniform objects, and quan-titative understanding of electrophoresis of JPs is still
FIG. 2. Sketch of the model JP with the radius of inner shell R i , the effective cutoff radius for the WCA potential σ WCA ,and the radius of the outer shell R = R i + σ WCA (black dots)serving as the effective hydrodynamic radius. The chargedarea of the fraction φ is shown by filled red circles. Open cir-cles indicate uncharged region. Blue circles mark the closestdistance at which ions do not experience any WCA repulsionfrom the colloid beads. challenging, despite some recent advances. Previous the-oretical work[14] has shown that in the thin EDL limitthe electrophoretic mobility of JPs is well representedby the Smoluchowski model,[5] i.e. remains governed bythe average zeta potential, while the dipole moment ofJPs only affects the orientation of particles relative tothe external field. A recent numerical study [15] per-formed under the assumptions of SEM has shown thatat the same averaged surface potential, the mobility ofJPs in a spherical cavity of arbitrary size is generallysmaller than that of a uniformly charged particle and thedifference becomes more pronounced with the increasein non-uniformity. Molecular dynamics simulations havealso concluded that charge inhomogeneities could reducethe diffusion coefficient of nanoparticles in nanopores.[16]Nevertheless, the electrophoretic properties of JPs re-main largely unexplored outside the range of applicabilityof SEM, when one has to consider the finite size of ionsand particles’ own thermal wobble that disturbs the pre-ferred alignment to the external field. Furthermore, wewould like to point out that SEM assumes the constantsurface potential while often it is the charge density thatis kept constant. A recent study [17] has pointed it outfor the case of uniformly charged particles, but we are un-aware of any previous work that has applied a constantcharge condition for JPs.In this paper, we use a hybrid Molecular Dynamics -Lattice Boltzmann simulation and a SEM-based mean-field approach to study the electrophoresis of a singleJP. We are interested in effects arising from variationsof surface charge which occur both on the scale of par-ticle radius and over distances comparable to the De-bye length, so that we focus on intermediate values of κR = O (1), where quantitative understanding of elec-trophoresis remains especially challenging. Our results show that in at low total charges JPs can be charac-terized by area-averaged surface charge, but at highercharges and κR their electrophoretic mobility is reduced,being strongly affected by non-uniformity and anisotropyof surface charge.Our paper is organized as follows. In Section II wedescribe our methods and model of JPs. In section III,we present our data on electrophoretic mobility and ro-tational dynamics for JPs of different total charges andin various screening regimes. Our conclusions are sum-marized in Section IV. II. Simulation and numerical methods
For simulation of the dynamics of charged JPs, we usethe hybrid Lattice-Boltzmann (LB) - Molecular dynamics(MD) method combined with the primitive model of theelectrolyte.[18] All MD-LB simulations are performed us-ing ESPResSo.[19, 20] We model all the charged speciesand the particle surface elements explicitly as MD beads,while the medium is modelled as a viscous fluid of massdensity ρ and dynamic viscosity η at the level of the LBmethod. It is treated as a dielectric continuum charac-terized by the Bjerrum length l B .In our model, the MD beads – small ions in solutionand surface beads of the colloid – interact via the Weeks-Chandler-Anderson (WCA) potential U W CA ( r ) = ( ǫ ij (cid:16)(cid:0) σ ij r (cid:1) − (cid:0) σ ij r (cid:1) + (cid:17) , r < / σ ij , , r ≥ / σ ij (2)and the Coulomb potential U C ( r ) = l B k B T z i z j r . (3)Here, l B is the Bjerrum length, l B = e / (4 πǫ ǫk B T ),and z i , z j are ion valencies; ǫ and ǫ being the dielectricpermittivity of vacuum and dielectric constant of water,respectively. The bead size, σ ij , sets the unit length inour simulations and charachterstic energy scale is ǫ ij = 1. r is the distance between two MD beads. To facilitatea comparison to the mean-field theory we choose σ ij =2 − / ≃ . σ so that σ W CA = 1 . R i = R − σ W CA = . σ and holds the charged beads. Theouter shell’s radius is simply R = 4 . σ : it does not inter-act with other MD particles either via WCA or Coulombpotentials but only with the LB fluid, and thus servesto define the colloid’s hydrodynamic radius. Using twoshells shifted against one another is advantageous for tun-ing both ‘electrostatic’ and ‘hydrodynamic’ radii to thesame value which allows for more convenient comparison ˆ Z e µ FIG. 3. Electrophoretic mobility of Janus particles with φ = 1 .
00 (open triangles), 0 .
50 (filled squares), and 0 .
25 (filledcircles) as a function of net charge at κR = 1 .
0. Solid curveshows the predictions of SEM, dashed curves represent nu-merical mean-field calculations for JPs, dotted line shows theH¨uckel limit solution. The color of the curves matches thecolor of the symbols for the corresponding charge distribu-tion. of our results to the mean-field theory predictions. Thecoupling between the LB and MD subsystems is realizedvia dissipative interactions as introduced in Ref. [18].The viscous friction term, given by F f = − Γ ( u s − u f ),where u s , u f are velocities of the solute beads and thesolvent, respectively, acts on the solute particles – mi-croions and colloid surface beads. An opposite force isapplied to the solvent to ensure momentum conserva-tion, and Gaussian white noise F R with zero mean isadded that satisfies the fluctuation-dissipation theoremthrough h F α ( t ) F β ( t ′ ) i = 2 δ ( t − t ′ ) 2 δ αβ k B T Γ. Thiscoupling mechanism also works as a thermostat, keepingthe temperatures of MD particles and the LB fluid thesame. We choose the simulation units as k B T /ǫ ij = 1, ρ = 1 . σ − , η = 3 . √ mǫ ij σ − , l B = 1 . σ and LB latticespacing a = 1 σ . In comparison to the original ‘rasp-berry’ model,[21, 26] we here introduce two different fric-tion coefficients for the microions and the colloid surfacebeads: for the surface beads Γ is set to 20, at whichpoint the dependence of hydrodynamic radius on Γ issaturated enough to emulate no-slip boundary conditionat the hydrophilic surface[24]; while for microions Γ isset to 2 thus ensuring that the ionic atmosphere is fairlymobile compared to the colloid. The resulting reduceddiffusion constant for microions was 6 πηl B D/k B T ≃ L = 40 . σ , giving the R / L = . and the colloid volume fraction of 0.41%. The numberof monovalent ions N in the simulation box was set bythe number of background salt ions and counterions to . . . . . . . . κR e µ FIG. 4. Electrophoretic mobility of particles as a function of κR at high, ˆ Z = 9 . Z = 2 . φ = 1 .
00 (triangles), 0 .
50 (squares), and 0 .
25 (cir-cles). Solid curve shows predictions of SEM, dashed curvesrepresent numerical results for JPs, dotted line shows theH¨uckel limiting solution for uniformly charged colloid. Thecolor of the curves matches the color of the symbols for thecorresponding charge distribution. the colloid, N = 2 C Σ L + | Q/e | , where Q is the colloidcharge so that the system was overall electroneutral. Theelectrostatic interactions were evaluated using P3M im-plementation of the Ewald summation technique.[27] Wedescribe the ionic strength and the screening conditionsin the suspension by κR , where κ = (4 πl B N/L ) / , withthe total number of the (monovalent) ions in the simula-tion box N , and we vary κR from 0 . . . − . σ − . The external field E was modeled bya uniform force acting on each charged MD bead, andin all simulations we use E = 0 . k B T /σe . This fieldstrength belongs to the linear response regime for oursystems, which is confirmed by the linear dependence ofthe velocity on the field strength, i.e. constant mobility.At the same time, the field is sufficiently large to give anoticeable particle velocity and to facilitate the mobilitymeasurements. The ionic cloud at this field value is notsignificantly perturbed, while the external field E is lessthan the potential drop over the electrostatic diffuse layer κζ . The chosen field strength is also suitable to study theinterplay between the JPs’ thermal wobble and electro-static torque, which will be described in details in thefollowing sections.The area fraction of the charged surface, φ , (Figure 2)has been varied from 0 .
25 to 1, where the latter cor-responds to a uniformly charged particle, by keepingthe net charge of the colloid Q constant. The chargedpatch was always a spherical segment of a given height2 φR (Figure 2), so all the JPs had axial symmetry ofcharge distribution. We should stress that all the JPshad the same net charge and the same average chargedensity h q i = Q/ πR , but different local charge densi- FIG. 5. 2D radial axisymmetric distribution maps of counter-ions around particles of φ = 1 (a), 0 .
50 (b), 0 .
25 (c) calculated atˆ Z = 9 . κR = 1 . ties q = h q i /φ of their ‘patches’. Further in the text, weuse the scaled charge ˆ Z ≡ Q l B eR , which we vary from 2 . . Q via the charges of surface beads.We also employ a direct numerical solution of the elec-trophoretic problem that we describe in Appendix A. Wethe reader for a more in-depth look on this system in theoriginal publication[28]. We use such an approach forprecise control over the charge distribution around thecolloid, and accordingly we implement it to construct JPsin exactly the same manner as in MD-LB model, i.e. bydistributing fixed charge over fraction of the surface, φ . III. Results and discussionA. Electrophoretic mobility
We first investigated the effect of a charge hetero-geneity on the electrophoretic mobility. Figure 3 showsthe simulation results for electrophoretic mobility as afunction of ˆ Z obtained for κR = 1 . φ . Also included are numericalmean-field results and predictions of the H¨uckel theory e µ = ˆ Z/ (1 + κR ).[4] We see that when the particle isuniformly charged ( φ = 1), at relatively small chargesthe mobility is nearly equal to, while at high charges issmaller than that predicted in the H¨uckel limit. Thisconfirms that the electrophoretic mobility of a uniformlycharged particle is proportional to its charge only in theweak charge regime.[6, 17] Note that the simulation re-sults for a uniformly charged particle are in excellentagreement with predictions of the SEM, which demon-strates the predictive power of our simulation model. An-other result emerging from Figure 3 is that in the low ˆ Z regime simulation data at φ = 0 .
50 (a ‘balanced’ chargedistribution) and φ = 0 .
25 (highly concentrated chargeon a relatively small surface patch) practically coincidewith those obtained for the uniformly charged particle.This indicates that the electrophoretic mobility in thisregime is fully determined by the area-averaged charge(or zeta potential) in agreement with the SEM for uni-formly charged particles, as it is commonly assumed. Inthe high charge regime, the electrophoretic mobility of JPs is smaller than that of a uniformly charged particle,so that the SEM for the uniformly charged colloid signifi-cantly overestimates simulation results. This means thatthe mobility is no longer determined by the area-averagedcharge alone. Our observation – that the electrophoreticmobility of JPs decreases at high charges – is likely re-lated to the non-linear relation between e µ and ˆ Z . Sinceall the JPs we study bear the same net charge, their localcharge densities vary significantly. Hence in the regimeof high charges the mobilities of JPs divert from the lin-ear relation even more than it might be expected for theuniformly charged particles.The decrease in the electrophoretic mobility of JPsis also captured by our SEM calculations. We remark,however, that our numerical solutions show practicallyno difference in electrophoretic mobilities for particles of φ = 0 .
50 and 0 .
25, but the deviations of simulation datafrom the SEM are getting larger when φ = 0 .
25. We seethat the SEM and primitive model simulation results forJPs practically coincide at φ = 0 . Z ), but at φ = 0 .
25 the simulation data deviatefrom the numerical solution towards smaller mobilities.We can speculate that numerical solutions deviate fromthe MD-LB simulations (at high enough charges or strongenough screening) because of the limited resolution of thelattice that the mean-field solver we use provides. Indeed,when κR grows, the characteristic thickness of the elec-trostatic diffuse layer decreases, and with high chargesthe potential grows too rapidly in the close vicinity tothe charged surface. It is well-known[30] that Poisson-Boltzmann equation often may not grasp this rapid in-crease properly, thus decreasing the accuracy of the re-sults. The fact the our MD-LB results deviate even morefrom the numerical solution in case of φ = 0 .
25 is consis-tent with this suggestion, since the local density is fairlyhigh in this case to be precisely resolved by lattice-basedsolver.It is instructive now to focus on the role of κR . Fig-ure 4 shows numerical and simulation results obtained at ˆ Z d FIG. 6. Dipole moment of JPs with φ = 0 .
25 (circles) and 0 . κR = 1.Dash-dotted lines show dipole moment of unscreened parti-cles. Dashed curves are drawn only to guide the eye. low, ˆ Z = 2 .
5, and high, ˆ Z = 9 .
6, values of surface charge.A general conclusion from this plot is that in this rangeof parameters the electrophoretic mobility decays with κR , but the influence of charge non-uniformity is differ-ent for low and high net surface charge. In the case ofsmall charges, the mobility of JPs does not significantlydiffer from that of uniformly charged colloids, and thesimulation data are in agreement with mean-field theoryresults. One can therefore conclude that a simple H¨uckelmodel can safely be used to analyze the mobility data inthe studied range of κR . In the high charge regime, wesee that the simulation data for uniformly charged par-ticle are well fitted by the SEM, and are well below theH¨uckel solution. For JPs the numerical solution predictsslightly lower electrophoretic mobility, actually the samefor φ = 0 .
50 and 0 .
25. The simulation data deviate fromthese mean-field solutions towards the smaller mobilityvalues, especially at larger κR . We also note that thediscrepancy is larger for JPs of φ = 0 . B. Orientation and dipole moment
We present the counterion density maps near particlesof different φ in Figure 5. This plot demonstrates thatJP dipole moments are oriented along the external field.Note that our density maps account for the JPs own wob-ble, so that the thermal motion perturbs the preferredorientation of the particle, which means that counter-ioncloud also experiences orientational fluctuations. We alsoremark that the accumulation of counterions near JPs isstronger compared to a uniformly charged colloid, andincreases with a decrease in φ . This obviously reflectsthe fact that at the same ˆ Z the local charge density ofthe charged area is higher at smaller φ . These resultsindicate a non-negligible dipole moment of JPs, whichshould correlate with the decrement of electrophoreticmobility values, as both are caused by the charge screen-ing. Motivated by these observations below we studiedthe orientational torque of JPs in the electric field and e µ (a) ˆ Z e µ (b) FIG. 7. Electrophoretic mobility at κR = 1 calculated fromdipole moments of JPs with φ = 0 .
50 (a) and 0 .
25 (b) as afunction of net charge (empty symbols). The same data as inFigure 3 is shown by blue squares and red circles respectively.Dotted lines plot the H¨uckel limit solution (5), solid curvesare drawn only to guide the eye. its relation to the mobility.The JP’s rotational dynamics can be characterized bythe mean-squared angular displacement (cid:10) α ( τ ) (cid:11) of theparticle’s dipole moment within time τ . Our analyticalsolution is described in Appendix B, and can be presentedas ∞ X n =1 ( − n +1 (cid:10) α n (cid:11) (2 n − τ →∞ ≡ h α i Σ = k B TEd , (4)where d is the dipole moment, which can be deduced from h α i Σ , calculated by using the simulation data on α as afunction of time τ .In Figure 6, we plot the dipole moment of two JPsas a function of their net charge ˆ Z . We note that atlow ˆ Z the measured d coincides with the dipole momentof the unscreened particle, d , which can be defined as d = (1 − φ ) R ˆ Ze/l B . However, at high ˆ Z it is consid-erably smaller, which immediately suggests that counte-rions contribute to its effective value, likewise they con-tribute to the effective zeta potential and to the decre-ment of electrophoretic mobility.To test this assumption, we attempted to predict thevalues of electrophoretic mobility from the measureddipole moment. We introduce an effective charge asˆ Z eff = ˆ Zd/d , and then use it to compute the mobilityusing the H¨uckel limiting law for low ˆ Z and κR ˜ µ = ˆ Z κR . (5)In Figure 7, we plot the mobilities calculated at φ = 0 . .
25 and compare them with data from Figure 3.We see that for φ = 0 .
50 (Figure 7(a)) the two sets ofdata agree with each other quite well, so that the mea-surements of dipole moments can be used to evaluate thedecrement of electrophoretic mobility of JPs. However,for φ = 0 .
25 (Figure 7(b)) the effective charge approachunderestimates the mobility at high ˆ Z . Since even atthe highest local charge density the hydrodynamic ra-dius (calculated with Eq.(B11) of Appendix B) remainsthe same within a statistical error, such a discrepancycannot be related to counterion condensation. So, it islikely that the concept of single effective charge becomesunsuitable when the surface charge anisotropy is gettingvery large. IV. Conclusions
We have studied the electrophoretic mobility of JPsand have shown that it depends both on their netcharge and charge distribution. Namely, less homoge-neous charge distributions generally lead to lower mo-bilities, which is consistent with previous observationsmade for different systems.[31–33] The decrease in mo-bility as compared to that of uniformly charged particlesis negligibly small at low particle net charges and small κR . In this case, the electrophoretic mobility can berelated to the area-averaged charge (or zeta potential)thought the SEM, as it is commonly assumed. The de-viations from the SEM are becoming pronounced whenthe net charge and κR increase. Where the mobility issignificantly affected by charge heterogeneity, the mean-field predictions for JPs overestimate the mobility, andshould be used with care. Reversely, the zeta potentialor surface charge extracted from the mobility data inthe regime κR ≈ V. Acknowledgements
This research was supported by the Russian Academyof Sciences (priority programme ‘Assembly and Investiga-tion of Macromolecular Structures of New Generations’).The simulations were carried out using computationalresources at the Moscow State University (‘Lomonosov’and ‘Chebyshev’). We have benefited from discussionswith Jiajia Zhou, Salim Maduar and Alexander Dubov.We thank Roman Schmitz and Burkhard D¨unweg for ac-cess to their SEM solver.
A. Numerical solution of mean-field equations
The algorithm we apply is a solver for the followingPoisson-Boltzmann equation and a coupled set of Nernst-Plank and Stokes equations:0 = ∇ ψ + 1 ǫ e X i z i c i , (A1)0 = ∇ · (cid:18) D i ∇ c i + D i k B T ez i ( ∇ ψ ) c i − v c i (cid:19) , (A2)0 = −∇ p + η ∇ v − e ( ∇ ψ ) X i z i c i , (A3)0 = ∇ · v , (A4)where z i , c i and D i are valencies, concentrations anddiffusion constants of charged species i , ψ is the electro-static potential, p is the pressure and v is the velocityof liquid in the fixed colloid’s reference frame. In thismethod[34], instead of solving a set of partial differentialequations for electrostatics, the problem is reformulatedin terms of electric field rather than potential to obtaina free energy in the form of the following functional F = Z V f dV (A5) f = 12 E + X i c i ln c i − ψ ∇ · E − X i c i ! − X i µ i (cid:18) c i − N i V (cid:19) where are total numbers of charged species i , and V isthe system volume. The minimum of this functional cor-responds to the solution of a related Poisson-Boltzmannequation. Further discretization allows one to implementa version of this solver with charged species serving bothas ions in solution and the surface charged beads of acolloid, in a sense much like the MD implementation,albeit a lattice one. In order to minimize the free en-ergy functional the discrete charges are moved aroundthe lattice.The solution is used as an input to the set oflinearised Nernst-Plank and Stokes equations, and thisprocedure is repeated iteratively until the solution forthe fluid velocity in the frame of the colloid converges. B. Orientational dynamics of a Janus particle
For the case of a constant and uniform electric field E ,a dipole orientation satisfies the Boltzmann distribution W = Ae Ed cos α/k B T , (B1)where A = 4 π k B TEd sinh
Edk B T is a normalisation constant.The average value of the dipole moment component inthe direction of the field can be then calculated as d h cos α i = Z π Z π W ( α ) d cos α sin αdαdϕ = d (cid:18) coth Edk B T − k B TEd (cid:19) . (B2)In strong fields such that Edk B T ≫
1, the first term in thebracket turns unity and we have d h cos α i E →∞ = d · (cid:18) − k B TEd (cid:19) , (B3)whence, since α is small, we find h α i = k B TEd . (B4)We can write a 1D Langevin equation for the colloidorientation angle with respect to the field as I ¨ α = − Ed sin α − ζ R ˙ α + T rand ( t ) . (B5)Here I stands for the colloid’s moment of inertia, ζ R =8 πηR and dot and double dot denote the first andsecond time derivatives, respectively. Multiplying (B5)by α , expanding sin ( α ) in a series and also using that d ( α ˙ α ) /dt ≡ ˙ α + α ¨ α , we find I ddt ( α ˙ α ) − I ˙ α = − Ed ∞ X n =1 ( − n +1 α n (2 n )! − ζ R ˙ αα + T rand α, (B6)which after taking the ensemble average of both partsbecomes I d dt (cid:10) α (cid:11) + ζ R ddt (cid:10) α (cid:11) + Ed ∞ X n =1 ( − n +1 (cid:10) α n (cid:11) (2 n )! − k B T = 0 . (B7) Here we have used the equipartition theorem, T rand sym-metry and the fact that h α ˙ α i = ddt (cid:10) α (cid:11) . In the limit α →
0, we can further simplify Eq.(B7) by omittinghigher order terms and using A = (cid:10) α (cid:11) − k B T /Ed :¨ A + ζ R I ˙ A + 2 EdI A = 0 . (B8)This is a well-known differential equation for a dampedoscillator[35], in our case – over-damped as the dampingparameter ζ R / √ IEd is always larger than 1. Therefore,we can write the solution in the form of A = Ae − λt where λ is the smaller root of an auxiliary quadratic equation(we drop one term with large λ as it decays too fast andis negligible). Finally, we use the condition (cid:10) α (cid:11) t =0 = 0to get (cid:10) α (cid:11) ( τ ) = k B TEd (cid:0) − e − λτ (cid:1) . (B9)This equation describes the evolution of the JP’s orien-tation relative to the external field and at t > /λ theresult agrees with Eq.((B4)) derived from the statisticalviewpoint. Substituting the solution for (cid:10) α (cid:11) in equation(B7) we see that both derivatives vanish when t → ∞ and thus we can find the full solution in the equilibriumdistribution without the limitation α → ∞ X n =1 ( − n +1 (cid:10) α n (cid:11) (2 n )! τ →∞ ≡ h α i Σ = k B TEd . (B10)Since λ is a function of both ζ R and I we can rewrite thesolution of auxiliary equation in terms of R and omit the λ term R ≃ ( Ed πηλ ) / . (B11)Thus, the same h α i data allow one to calculate the hy-drodynamic radius of the particle. [1] R. Hunter , Zeta Potential in Colloidal Science , Aca-demic, London, 1981.[2]
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