Disorder effects and antifragility in Coulomb fluids ! Rudolf Podgornik Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia and Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia Conceptual introduction to interactions in disordered colloids: ! • disorder and DLVO idealizations • disorder in the charge distribution • disorder effects with counterions and salt • disorder and antifragility
Disorder: Strong coupling ES: ! ! David S. Dean (Bordeaux) A. Naji (Cambridge) Ali Naji (Cambridge, Tehran) P. A. Pincus (UCSB) Ron R. Horgan (Cambridge) Y. S. Jho (UCSB) Yevgeni S. Mamasakhlisov (Erevan) Jan Forsman (Lund) Jalal Sarabadani (Helsinki) Matej Kanduč (Berlin) Malihe Ghodrat (Tehran)
I would like to acknowledge the seminal role of a discussion with Henri Orland at a meeting in Pohang, Korea (2003) for this work. Electrostatics of Soft and Disordered Matter (2014) Eds.: David S. Dean, Jure Dobnikar, Ali Naji, Rudolf Podgornik
- Coulomb Fluids: From Weak to Strong Coupling - Ions at Interfaces and in Nanoconfinement - Complex Colloids - Biological Systems and Macromolecular Interactions - Disorder Effects in Coulomb Interactions
Graphics: A. Šiber Coulomb law - ever present electrostatic interactions
- J. Priestley (1767): no force inside a charged sphere. - J. Robinson (1769): force falls off with second power - C. A. de Coulomb (1777): invented the torsion balance for measuring the force of magnetic and electrical attraction - H. Cavendish (1789): measured but did not publish (published by Kelvin in 1879) ! Opposites attract and likes repel! Coulomb (1777)
2 2 2 DNA: ~1.0 eo / nm Polipeptides: ~0.6 eo / nm Membranes: ~0.1 - 1 e0 / nm (C graphics: B. Brooks) Coulomb interactions and ever present fluctuations
In soft materials one always needs to consider thermal fluctuation effects. Van der Waals and Casimir forces (French et al. RMP (2010))
+ quantum fluctuations
(Image credit: Umar Mohideen, UC at Riverside)
+ thermal fluctuations
(Sticky gecko - Image credit: NSF)
Not the only ways that electrostatic and thermal effects combine. But also disorder apart from thermal noise they depends on the nature of the interacting bodies. Quenched surface charge disorder Mica surfaces coated by surfactants that self-assemble intowith random charge domains. Klein experiments on patchy interfaces.
AFM images of surfactant AFM images of a coated mica surface octadecyltrimethylammonium bromide on mica: covered with a random mosaic of The break-up of the initially uniform monolayer surfactant cetyltrimethylammonium into positively-charged bilayer domains on a bromide. Initially smooth monolayer negatively-charged substrate. rearranges into positively-charged (Silbert and Klein, in Dean et al. Electrostatics patches of bilayer surrounded by the of Soft and Disordered Matter negatively charged bare mica. (2014)). (Perkin et al. PRL 96, 038301 (2006)) ! ! ! Quenched surface potential disorder (patch effect) ! Polycrystalline niobium-doped strontium titanate (SrTiO3: Nb) and polycrystalline copper film. Variations in the work function according to grain orientation. The grain orientation is determined by electron backscattering diffraction (EBSD). Correlation between surface composition and orientation for several grains.
Luiz F. Zagonel et al. Surf. Interface Nicolas Gaillard et al.,Appl. Phys. Letts. Anal. 40,1709 (2008). 89, 154101 (2006)
Bare metallic surfaces are composed of crystallites, each of which constitutes a single patch, with the local surface voltage determined by the local work function. Measuring vdW - problems with parasitic LR interactions
Contact potentials in Casimir force setups: an experimental analysis S. de Man, K. Heeck, R. J. Wijngaarden, and D. Iannuzzi (2010) ! Systematic measurements of the contact potential difference between a gold coated sphere and a gold coated plate kept in air at sub-micron separation. The contact potential in Casimir force experiments can depend on both separation and time, and seems to indicate that electrostatic interaction between conducting surfaces in ambient conditions plays a role.
Patch effects are a source of concern for Casimir experiments and for other precision measurements of forces between neutral conductors at distances in the micrometer range and must be accounted for (Behunin et al. (2014)). Quenched vs. annealed charge disorder
Q A
Quenched disorder when parameters defining its behaviour are random variables, which do not evolve with time, i.e.: they are frozen. The random variables may not change their values and are set by the method of preparation of the sample. ! The average over the disorder and the thermal average may not be treated on the same footing. ! ! Annealed disorder when parameters entering its definition are random variables, but whose evolution is related to that of the degrees of freedom defining the system. The random variables may change their value. ! The average over the disorder and the thermal average may be treated on the same footing. Can quenched disorder be incorporated into the DLVO theory?
Disordered charges on the surfaces and in the bulk of the interacting bodies as well as disorder in the dielectric response of the interacting bodies.
Quenched vs. annealed disorder. We investigated the effects of both. ! Annealed: S. Panyukov and Y. Rabin, Phys. Rev. E 56, 7053 (1997). Safran (1998), D. Lukatsky and E. Shakhnovich (2003). Brewster, Pincus and Safran (2008). Interaction between surfaces with quenched charge disorder
Assumed decomposition of the disordered charge distribution into surface and bulk terms
surface volume
Disorder variance with short range correlations or just as well finite size correlations.
(monopolar disorder) Electrostatic interaction between randomly charged dielectrics
The disorder parametrization is:
The partition function for the classical Coulomb system with disordered sources:
where the average over these disordered sources is: vdW interactions and effects of randomness
The Casimir–van der Waals interactions are in this formalism given simply by:
Yielding for the planar geometry the force:
dielectric missmatch:
The free energy shows the relation between the thermal fluctuations (thermal Casimir–van der Waals interactions) and the interaction stemming from the disorder fluctuation effects:
Quenched:
Annealed:
Though formally similar they are wastly different. Charge disorder driven interactions: bulk disorder The force induced between the two semiinfinite dielectrics:
bulk disorder surface disorder zero frequency Matsubara ! (anomalous D dependence) (standard D dependence)
The physics involved is indeed subtle as the disorder terms result from the self- interaction of the charges with their images (only in a dielectrically inhomogeneous system) and not from dipolar interactions (which come from a multipolar expansion).
For relatively small surface disorder, the anomalous 1/D behavior is predicted to dominate the vdW 1/D^3 behavior beyond the crossover distance
estimated from a few hundreds of nm to several microns for reasonable value of impurity concs. Charge disorder driven interactions: bulk disorder (a simplified, more intuitive argument)
With the Green’s function of the electrostatic interaction and the quenched disorder variance is:
Then the electrostatic interaction energy of the quenched charge disorder in the system (assumed net neutral!) is:
The force due to two layers with quenched charge is finite and given by
Which is exactly what we derived before! Thus: disorder interaction is image self-interaction. Statistically speaking each charge on the average (as any other charge has an equal probability of being of the same or opposite sign) only sees its image, thus explaining the leading monopolar form in the net force. Interplay between image and disorder effects
The physics involved is indeed subtle as the disorder terms result from the self- interaction of the charges with their images (only in a dielectrically inhomogeneous system) and not from dipolar interactions (which come from a multipolar expansion) as is the case in the vdW interactions which contribute additively.
Statistically speaking each charge on the average (as any other charge has an equal probability of being of the same or opposite sign) only sees its image, thus explaining the leading monopolar form in the net force. ! By contrast, the force induced by annealed disorder in general combines with the underlying van der Waals forces in a nonadditive fashion, and the net force decays as an inverse cube law at large separations. Charge disorder driven interactions: surface disorder
The force induced between the two semi-infinite dielectrics:
bulk disorder surface disorder zero frequency Matsubara ! (anomalous D dependence) (standard D dependence)
For strong surface disorder one expects the 1/D^2 behavior to dominate beyond separation
The disorder effects are considered here on the zero- frequency level and the results are compared with the corresponding Casimir–van der Waals interaction. The precise correction presented by the higher-order Matsubara frequencies is very material specific, but its magnitude (relative to the zero-frequency term) remains negligible in comparison with the quenched disorder effects. Patch disorder effects
Thus, an effective scaling exponent (defined as 1/D^a) of a < 1 (consistent with recent experimental observation of an anomalous residual force scaling as a = 0.8, W. J. Kim et al., arXiv:0905.3421v1.) may be obtained in the quenched case, for bulk or surface disorder model. ! The patch potential with finite correlation length can also give a finite contribution (Speake & Trenkel, 2003) but appears to be smaller from the charge effect.
For disordered patch potentials: ! dipolar disorder as opposed to monopolar
The dipolar disorder effect is then subdominant to thermal fluctuations. Sample-to-sample fluctuations of disorder driven forces: surface disorder Three different realizations of the experiment, i.e. three different lateral positions of the tip above the substrate, are shown corresponding to three different samples of force data. Each sample would show a different measurement of the normal as well as lateral force with a sample-to-sample variance that can be explicitly calculated.
Lateral force: Normal force: Sample-to-sample fluctuations of disorder driven torques: surface disorder Torque acting on a randomly charged dielectric slab (or a sphere) mounted on a central axle next to another randomly charged slab. Although the resultant mean torque is zero, its sample-to- sample fluctuation exhibits a long-range behavior.
Torque fluctuations are connected with lateral force fluctuations:
Extensive in interaction area! A much stronger effect then force fluctuations. Experimental imprint of sample-to-sample fluctuations?
The sample-to-sample variation in the disorder generated force and torque is fundamentally different from the thermal force fluctuations in (pseudo) Casimir interactions as analyzed by D. Bartolo et al. Phys. Rev. Lett. 89 230601 (2002). ! In order to detect sample-to-sample variation one would have to perform many experiments and then look at the variation in the measured force and/or torque between them.
Relative sample-to-sample fluctuations for spheres in PFA.
The most promising effect appears to be the torques fluctuations that are extensive in the interaction area.
EPJ E Highlight paper (2012) 3
and that the surface and bulk variances are given by 2 gs(z) = e [g sδ(z + D/2) + g sδ(z D/2)], (6) 0 1 2 − 2 g be z < D/2, 1 0 − gb(z) = ⎧ 0 z < D/2, (7) ⎨ 2 | | g2be0 z > D/2. The lateral co⎩rrelation between two given points is typ- ically expected to decay with their separation over a finite correlation length (“patch size”), which could in general be highly material or sample specific. However, the main aspects of the patchy structure of the disorder can be in- FIG. 1: (Color online) We consider two semi-infinite net- vestigated by assuming simple generic models with, for neutral slabs (half-spaces) of dielectric constant ε1 and ε2 instance, a Gaussian or an exponentially decaying cor- interacting across a medium of dielectric constant εm. The relation function. Without loss of generality, we shall monopolar charge disorder (shown schematically by small choose an exponentially decaying correlation function ac- light and dark patches) is distributed as random patches of cording to the two-dimensional Yukawa form finite typical size (correlation length) in a layered structure in the bulk of the slabs and on the two bounding surfaces at x x 1 z = ±D/2. It may be either quenched or annealed. ciα( ) = 2 K0 | | , (8) 2πξiα %ξiα &
where ξiα represents the correlation length for the bulk or surface disorder (α = b, s) in the i-th slab (i = 1, 2). The We shall assume that the two dielectric slabs have a dis- case of a completely uncorrelated disorder [8] follows as a ordered monopolar charge distribution, ρ(r), which may special case for ξ 0 from our formalism. We should arise from randomly distributed charges residing on the iα emphasize that the →correlation assumed for the bulk dis- bounding surfaces [ρ (r)] and/or in the bulk [ρ (r)], i.e., s b order is only present in the plane of the slab surfaces and ρ(r) = ρ (r) + ρ (r). The charge disorder is assumed to Variations on sa themeb not in the direction perpendicular to them. This assump- be distributed according to a Gaussian weight with zero 3 tion is wholly justified only for layered materials. In all The situation would be different if the disordermean charges(i.e., th aree sl a“correlated”bs are net andneu formtral) “patches”, and the: two-point the other cases of the bulk disorder one would normally correlation function andCorrelatedthat the sur f a“patchy”ce and bu quenchedlk variances disorderare given by expect the same correlation length in the direction per- 2 gs(z) = e [g1sδ(z + D/2) + g2sδ(z D/2)], (6) ′ 0 ′ − ′ pendicular to the surfaces. We will deal with this model 2 ρ(r)ρ(r ) g1b=e (zϱ< ϱD/;2z, )δ(z z ), (2) 0 − in a separate publication. ⟨⟨ gb(z) = ⎧⟩0⟩ G z −< D/2, − (7) ⎨ 2 | | g2be0 z > D/2. where Assumedenotes anth (reasonable)e average o vansatz:er all realizations of ⟨⟨· ·T·h⟩⟩e lateral co⎩rrelation between two given points is typ- III. FORMALISM the charigcaellydeixspoercdteedrtoddisetcaryibwuitthiothne,irρse(pra)r.atiWoneovheraavfientitheus as- sumed tchoarrtelatthioenrelenag(rthxe ()n“poatscKph as0itzi(ea”l)x, cw/ohr icr)helcaotuilodnins giennetrahle per- be highly maGterial or sample sp|eci|fic. However, the main The partition function for the classical vdW interac- pendiculaasprecdtsiroefctthieopna,tchzy, stwruhcitluer,e oifnthtehdeisolradterercanl bdeiirne-ctions FIG. 1: (ColoRandomr online) W patchese consider oftw surfaceo semi-infi chargenitϱe n=et- (ofx ,theyve)s tpositive(igiantetdhbey orpasl sanegativeunmeinogfsitmh pcharges.eledgieenleercic trmiocdse)l,s wwiethh, afovre a fi- tion (the zero-frequency Matsubara modes of the elec- neutral slabs (half-spaces) of dielectric constant ε1 annidteε2statiinssttiacnacell,ya iGnvauasrsianntorcoanrreexlpaotnieonntiaflulyndcetciaoyninwg hcoor-se spe- tromagnetic field) may be written as a functional integral interacting across a medium of dielectric constant εm. The relation function. Without loss of generality, we shall r monopolar charge disorder (shown schematically bycisfimcallformchoomseaayn edxepponeenndtialolyndezcayaisngwcoerlrle.latTionhfiusncimtiopnlaice-s that over the scalar field φ( ), light and dark patches) is distributed as random patcthhees ocf harcgoredidngistoortdheertwios-ddimisetnrsiibonuatleYdukianwagfoernmeral as random finite typical size (correlation length) in a layered structure −βS[φ(r);ρ(r)] in the bulk of the slabs and on the two bounding surfa“cpesaattches” in a layered structure in xthe bulk of the slabs [ρ(r)] = [ φ(r)] e , (9) x 1 z = ±D/2. It may be either quenched or annealed. ciα( ) = 2 K0 | | , (8) Z ' D as well as on the bounding2πsξuiαrfac%esξi.α &
The twohtearle ξciαorrreperleasteinotsnthfeucnorcrteilaotnionclaenngtbh eforwthreitbtuelknoras the with β = 1/kBT and the effective action surface disorder (α = b, s) in the i-th slab (i = 1, 2). The We shall assume that the two dielectric slabs havesaudmis-of the surface (s) and bulk (b) contributions case of a completely uncorrelated disorder [8] follows as a ordered monopolar charge distribution, ρ(r), which may r r r 1 r r 2 r r special case for ξ 0 from our formalism. We should [φ( ); ρ( )] = d ε0ε( ) ( φ( )) + i ρ( )φ( ) . arise from randomly distributed charges residing on the iα 2 em′ phasize that the →correlati′on assumed for the bulk d′ is- S ' ∇ bounding surfaces [ρ (r)] and/or in the bulk [ρ (r)], i.e., ( ) s b (ϱ ϱor;dezr)is=onglys(pzre)scenst(ϱin theϱp;lazn)e +of tghbe(szla)bcbs(uϱrfacesϱan; dz). (3) (10) ρ(r)Interplay= ρ (r) + betweenρ (r). The thecha rdisorder-inducedge disorder is assum eandGd to van− der Waals interactions− lead to a variety −of s b not in the direction perpendicular to them. This assump- be dunusualistributed nonmonotonicaccording to a G ainteractionussian weigh profilest with ze rbetweeno the dielectric slabs. Limits: intervening The above partition function can be used to evaluate tion is wholly justified only for layered materials. In all mean (i.e., the slabs are net neutral), and the two-point averaged quantities such as the effective interaction be- medium has a larger dielectric constantF othanr th thee tsh ltwoeaboth gslabs,eerocmase standorfy,t hinwe ebbetweenuglkendiesorrad lethelryoan twoesswuo mslabs.ulde ntohramtaltlyhe lat- correlation function eral correxeplaecttiothne sfaumnectcioorrnelsatimonalyengbteh idn itffherdeirnetctifoonrpetrh- e two tween the dielectric bodies. However, since the charge pendicular to the surfaces. We will deal with this model r ′ ′ ′ slabs, i.e. distribution, ρ( ), is disordered, it is necessary to aver- ρ(r)ρ(r ) = (ϱ ϱ ; z)δ(z z ), (2) in a separate publication. ⟨⟨ ⟩⟩ G − − age the partition function over different realizations of the charge distribution. The averaging procedure differs where denotes the average over all realizations of c (x) z = D/2, the cha⟨⟨r·g·e· ⟩d⟩isorder distribution, ρ(r). We have thus as- c (x; z) =III. 1FsORMALISM (4) depending on the nature of the disorder. In what fol- s c (x) z = D−/2, sumed that there are no spatial correlations in the per- ! 2s lows, we consider two idealized cases of either completely pendicular direction, z, while, in the lateral directions The partition function for the classical vdW interac- tion (the zero-frequenccybM(xa)tsubazra
In colloidal systems we never have bare charges residing all alone on the surfaces, there are always mobile counterions or even mobile electroyte ions in the space between the surfaces.
DLVO theory = ES repulsions + vdW attractions
Typical composition of colloidal system: surface charges and an intervening electrolyte, with counterions and sat ions. Can be symmetric or asymmetric. Weak vs. strong coupling phenomenology
Bjerrum length Gouy - Chapman length
Coulomb’s law! and! kT Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions.
Weak coupling limit! Strong coupling limit! (Poisson - Boltzmann)! (Netz - Moreira)! Ξ➝ 0 Ξ➝ ∞ Coupling parameter
Collective description (Poisson - Boltzmann “N” description) Screened Debye-Hueckel vs. Single particle description (Strong Coupling “1” description)
M Kanduč, A Naji, J. Forsman and R Podgornik, JCP (2013). “Perspective” article. Interaction between equally charged planar charged surfaces
Weak coupling Strong coupling
Weak coupling ! - interactions in the symmetric case strictly repulsive and large (Confusion: Bowen & Sharif 1998). - fluctuation contribution in the symmetric case strictly attractive and small ! Strong coupling ! - interactions in the symmetric case mostly attractive and large - repulsive only at small separations Netz & Moreira, 2000. - Wigner cristal heuristic model (Shklovskii 1999) (simple limiting SC result, µ = GC length ) Patching together disorder & Coulomb interactions
Disorder in the surface charge distribution and the strength of the Coulomb interactions couple.
Perkin et al. (2005) Surface charge disorder.
Coupling parameter
Weak coupling limit Strong coupling limit (Poisson - Boltzmann) (Netz - Moreira) Ξ➝ 0 Disorder coupling parameter Ξ➝ ∞ ! ! ? ?
Partially annealed: one can define two temperatures. The mutual equilibration of fast (solution ions) and slow (surface charges) variables is hindered. Because of the wide time-scale gap, the slow dynamics of surface charges can exhibit a stationary regime at long (but not infinite) times, characterized by an equilibrium- type distribution at effective temperature T′. Weak coupling disorder effects Weak coupling (PB) theory and NO IMAGES. Assuming surface charge disorder:
Poisson - Boltzmann equation remains unchanged!
No effect! The Orland conjecture is valid. But only if there are no image interactions. ! A very surprising result indeed.
On the Debye-Huckel level with quenched charge disorder one remains with:
disorder effect vdW DH term
Where G(r, r’) is just the Debye-Huckel screened potential. Without dielectric discontinuities the disorder effects are non-existent! Weak coupling disorder effects with images
Pure disorder effect for 0 < κa < 1 and εm/εp = 0.2 to 10.0 (bottom to top).
Small separation:
Disorder does NOT renormalize charge. Attraction even if the surfaces carry no NET charge!!!
Large separation:
Disorder renormalizes zero frequency van der Waals interaction term. Strong coupling disorder effects
In this limit one can obtain very simple analytical results. Two charged surfaces with non-zero average and non-zero mean square average.
Disorder coupling parameter.
Disorder generated attraction is present even for surfaces that are neutral in average (σ = 0).
Disappearance of the entropic minimum for large enough disorder. Antifragile behavior
No, we do not expect this kind of behavior. ! Details of the competing effect of intrinsic thermal fluctuations and externally imposed disorder. The disorder can counter the thermal fluctuations actually making the system more ordered!?
vs.
energy entropy energy entropy
The factor before the ln D in the entropy term should correspond to a “temperature”. Apparently this “temperature” depends on the amount of surface disorder χ.
It is as if the surface disorder can change the sign of the “temperature”. This is just another way of saying that adding external disorder makes the system less disordered - antifragility. Origin of the antifragility Disorder generated enhanced adsorption of the counterions onto the surface. ! Surface adsorption of counter ions decreases their translational entropy in the solution, a deficit compensated by the configurational entropy gain due the presence of quenched randomness in the surface charge distribution, generated by different realization of the charge disorder.
Counterions & disorder. (no salt, no images)
Counterions, salt Counterions, salt, images & disorder. (no dielectric images) & disorder. The dielectric jump parameter is ∆ = 0.95 Charge inversion and overcharging Cumulative charge of multivalent counterions next to a semi-infinite, randomly charged dielectric slab
charge inversion
overcharging
(a) Cumulative charge of multivalent counterions next to a semi-infinite, randomly charged dielectric slab for different values of the disorder coupling parameter. (b) “Phase diagram” showing the minimal amount of multivalent counterion concentration, χ c̃ (in rescaled units), for charge inversion (main set) or overcharging (inset), as a function of the rescaled salt screening parameter with and without surface charge disorder. Fascinating world of Coulomb interactions
Coulomb says: Opposites attract and equals repel! In line with the common wisdom. Quenched disorder with no mobile charges says: Anomalously long range disorder-generated interactions that can dominate the standard vdW interactions.
Mobile charges:
Weak coupling says: Strong coupling says: Opposites attract and equals Equals attract but only if everybody is repel but not quite so much! very charged!
Quenched disorder Partially annealed disorder electrostatics says: electrostatics says: Neutrals can attract! Plus antifragility Same as quenched disorder, but no antifragility. FINIS