Disorder effects and antifragility in Coulomb fluids

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)
Ali Naji (Cambridge, Tehran)
Ron R. Horgan (Cambridge)
Yevgeni S. Mamasakhlisov (Erevan)
Jalal Sarabadani (Helsinki)
Malihe Ghodrat (Tehran)
A. Naji (Cambridge)
P. A. Pincus (UCSB)
Y. S. Jho (UCSB)
Jan Forsman (Lund)
Matej Kanduč (Berlin)
!
!
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)
DNA: ~1.0 eo / nm2
Polipeptides: ~0.6 eo / nm2
(C graphics: B. Brooks)
Membranes: ~0.1 - 1 e0 / nm2
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
octadecyltrimethylammonium bromide on mica:
The break-up of the initially uniform monolayer
into positively-charged bilayer domains on a
negatively-charged substrate.
(Silbert and Klein, in Dean et al. Electrostatics
of Soft and Disordered Matter
(2014)).
!
AFM images of a coated mica surface
covered with a random mosaic of
surfactant cetyltrimethylammonium
bromide. Initially smooth monolayer
rearranges into positively-charged
patches of bilayer surrounded by the
negatively charged bare mica.
(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
Anal. 40,1709 (2008).
Nicolas Gaillard et al.,Appl. Phys. Letts.
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
A
Q
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
(anomalous D dependence)
zero frequency Matsubara
!
(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 selfinteraction 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
(anomalous D dependence)
zero frequency Matsubara
!
(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-tosample 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)
light and dark patches) is distributed as random patche
finite typical size (correlation length) in a layered struc
in the bulk of the slabs and on the two bounding surface
z = ±D/2. It may be either quenched or annealed.
We shall assume that the two dielectric slabs have a
ordered monopolar charge distribution, ρ(r), which
arise from randomly distributed charges residing on
bounding surfaces [ρs (r)] and/or in the bulk [ρb (r)],
ρ(r) on
= ρsa(r)
+ ρb (r). The charge disorder is assume
Variations
theme
be distributed according to a Gaussian weight with
3
The situation would be different if the disorder
“correlated”
form “patches”:
mean charges
(i.e., theareslabs
are netand
neutral),
and the two-p
correlation
function
and that
the surface and bulk variances are given by
Correlated “patchy” quenched disorder
gs (z) = e20 [g1s δ(z + D/2) + g2s δ(z − D/2)],
⎧
⟨⟨ρ(r)ρ(r′⎨)⟩⟩g1b=e20 G(ϱ
ϱ′ ; z)δ(z − z ′ ),
z <−−D/2,
|z| < D/2,
gb (z) = 0
⎩ g e2 z > D/2.
2b 0
(6)
(7)
(reasonable)
ansatz:
where ⟨⟨· ·The
· ⟩⟩ Assume
denotesanthe
average over
all realization
lateral correlation between two given points is typthe charge
Weover
have
thus
icallydisorder
expected todistribution,
decay with their ρ(r).
separation
a finite
correlation
which
G(x)
Ksize”),
) could in general
sumed that
therelength
are (“patch
no spatial
correlations
in the
0 (|x|/
be highly material or sample specific. However, the main
pendicular
direction, z, while, in the lateral direct
aspects of the patchy structure of the disorder can be inϱ net= (x,
y)
(in the
plane
ofsimple
the
dielectrics),
FIG. 1: (ColorRandom
online) We
consider of
two
semi-infinite
vestigated
by or
assuming
generic models we
with,have
for
patches
surface
charge
of the
positive
negative
charges.
neutral slabs (half-spaces) of dielectric constant ε1 and ε2
instance, a invariant
Gaussian orcorrelation
an exponentially
decaying
cornite statistically
function
whose
interacting across a medium of dielectric constant εm . The
relation function. Without loss of generality, we shall
may
depend ondecaying
z as well.
This
implies
monopolar charge disorder (shown schematically bycific
small form
choose
an exponentially
correlation
function
aclight and dark patches) is distributed as random patches
of
the charge
disorder
is distributed
in general
as rand
cording
to the two-dimensional
Yukawa
form
finite typical size (correlation length) in a layered structure
%
&
“patches”
in a layered structure
in|x|the bulk of the s
in the bulk of the slabs and on the two bounding surfaces
at
1
ciα (x) =
,
(8)
K
0
z = ±D/2. It may be either quenched or annealed. as well as on the bounding
2
surfaces.
2πξ
ξiα
iα
where ξcorrelation
correlationcan
length
the bulk oras
The total
function
beforwritten
iα represents the
surface disorder (α = b, s) in the i-th slab (i = 1, 2). The
We shall assume that the two dielectric slabs havesum
a dis- 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
special case for ξiα → 0 from our formalism. We should
arise from randomly distributed charges residing on the
emphasize that the correlation assumed for the bulk disbounding surfaces [ρs (r)] and/or in the bulk [ρb (r)],G(ϱ
i.e., − ϱ′ ; z) = gs (z)cs (ϱ − ϱ′ ; z) + gb (z)cb (ϱ − ϱ′ ; z).
order is only present in the plane of the slab surfaces and
disorder-induced
and
ρ(r)Interplay
= ρs (r) + between
ρb (r). Thethe
charge
disorder is assumed
to van der Waals interactions lead to a variety of
not inthe
thedielectric
direction perpendicular
to them.
This assumpunusual nonmonotonic
interaction
profiles
between
slabs. Limits:
intervening
be distributed
according to a Gaussian
weight
with zero
tion is wholly justified only for layered materials. In all
mean
(i.e., thehas
slabsa are
net neutral),
andconstant
the two-point
medium
larger
dielectric
than
thethe
two
slabs,
in
between
theone
two
slabs.
For
the
slab
geometry,
webulk
generally
assume
that the
other
casesand
of the
disorder
would
normally
correlation function
expect the same
correlation
length
the direction
eral correlation
functions
may
be indiﬀerent
forperthe
pendicular to the surfaces. We will deal with this model
slabs,
⟨⟨ρ(r)ρ(r′ )⟩⟩ = G(ϱ − ϱ′ ; z)δ(z − z ′ ),
(2) i.e.
in a separate publication.
where ⟨⟨· · · ⟩⟩ denotes the average over all realizations of
the charge disorder distribution, ρ(r). We have thus assumed that there are no spatial correlations in the perpendicular direction, z, while, in the lateral directions
ϱ = (x, y) (in the plane of the dielectrics), we have a finite statistically invariant correlation function whose specific form may depend on z as well. This implies that
the charge disorder is distributed in general as random
“patches” in a layered structure in the bulk of the slabs
!
c FORMALISM
(x) z = −D/2,
cs (x; z) = III. 1s
c2s (x) z = D/2,
⎧
The partition function for the classical vdW interac(x) z <modes
−D/2,
⎨ c1b Matsubara
tion (the zero-frequency
of the electromagnetic
field)
may
be
written
as
a
functional
|z| < D/2, integral
cb (x; z) = 0
over the scalar field⎩φ(r),
z > D/2,
'c2b (x)
Z[ρ(r)] =
[Dφ(r)] e−βS[φ(r);ρ(r)] ,
(9)
Charge disorder effects in the presence of counterions and salt
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!
(Poisson - Boltzmann)!
Ξ➝ 0
Strong coupling limit!
(Netz - Moreira)!
Coupling parameter
Ξ➝
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
!
Netz & Moreira, 2000.
- interactions in the symmetric case
mostly attractive and large
- repulsive only at small separations
- 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
(Poisson - Boltzmann)
Ξ➝ 0
!
?
Disorder coupling parameter
Strong coupling limit
(Netz - Moreira)
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
(no dielectric images) & disorder.
Counterions, salt, 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 ̃c (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
repel but not quite so much!
Equals attract but only if everybody is
very charged!
Quenched disorder
electrostatics says:
Partially annealed disorder
electrostatics says:
Neutrals can attract!
Plus antifragility
Same as quenched disorder, but no antifragility.
FINIS

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