well thank you for the invitation for

the kind introduction so thank you all

right so I guess without further ado let

me get started ok so I like to talk to

you about some problems in pattern

formation which I think on some level

are familiar to everybody so here are

some examples of patterns that form but

I why what I would call the simplest

interfacial instability which kind of

maps on to the mathematical problem of

laplacian growth which I'll talk to you

in just about in just a moment so we see

a couple different phenomenon here that

lead to similar patterns so on the Left

we have viscous fingering which I'll

talk about in a moment of phenomena

multi-phase flow where one fluid is

displacing another immiscible fluid in a

confined geometry the second one is

electro deposition so the growth of

copper metal by electrically depetro

deposition of copper ions from a

solution and on the right we have the

formation of snowflakes by

crystallization under the right

conditions the basic phenomena here that

leads to branch structures is the idea

roughly speaking that what something

sticks out it gets more flux and sticks

out more that's basically what happens

when you have an interfacial stability

instability and the source of it on a

very simple level is that around each of

these objects is some kind of laplacian

field or harmonic function in this case

it might be the pressure so we have a

pressure driven flow here in a point of

you know healy shah so as I'll explain

in this case it might be the

concentration of the electrolyte in this

case it might be the temperature as a

latent heat needs to be diffused away

from the interface and in each case if

you have a dynamics of an interface

which is sort of governed by the

laplacian field then you have this sort

of instability that leaves two fingers

or dendrites and the simplest way think

about it is that these these regions of

harmonic fields those are regions where

you have a high resistance and

resistance to transport and essentially

the interface forms a finger or a

protrusion because it's trying to find

path of least resistance so basically

rather than growing uniformly and sort

of fighting its way through the system

it finds a way to break through and sort

of more easily move ahead where it wants

to be okay

so just a little bit of a sort of

mathematical context as so this is a

problem that I worked on kind of wearing

a mathematicians hat 10 or 15 years ago

which is to look at laplacian growth

type problems not only with continuous

dynamics of interfaces but also

stochastic dynamics of interfaces and to

summarize a certain body of work I would

say that what I showed was that

transport quasi steady transport limited

growth is always unstable to forming

these dendrites or fingers and a certain

sense they're all in the same

universality of class and form the same

kinds of patterns and by transport

limited I mean a class of models we

imagine there's some set of variables ci

which could be concentration potential

temperature some kind of scalars and you

have fluxes which are driven by

gradients and the and you can even have

a nonlinear matrix of couplings so

basically this this K matrix could sort

of a couple of a potential feel to a

concentration variable all those kinds

of things but the important thing is

that these gradient driven fluxes are in

equilibrium so divergence of flux is

zero so those are equations of motion

that's your generalization of Laplace's

equation and when you have that kind of

dynamics where there's stochastic or the

street at least in two dimensions you

always form these kind of fractal

patterns in the case of stochastic

growth this leads to so-called dla

clusters or diffusion limited

aggregation you can also generate these

clusters by a Monte Carlo type

simulation where you have a random walk

of particles coming in from far away and

when they get near this cluster they

stick wherever they first touch the

cluster and again you get this branching

instability because the more samay

sticks out the higher probability it has

of catching the next particle and that

leads to this kind of fractal growth and

the main generalization which is

important for today's talk is that this

class of models includes things such as

dla growth in a fluid flow also dla or

other kind of stochastic growth on

curved surfaces all of them are in some

sense the same instability but you do

need

a couple of things do you need that it's

quasi-steady so there's no like dc/dt

this is just divergence of flux is zero

so it's quasi steady transport and also

the transport is only on one side it's

kind of ahead of you it's not behind and

so those two features of what leads you

to always get similar instabilities in

these kinds of problems now I'm going to

try to look at the intersection of those

kinds of problems with electro kinetics

so here are just some examples of

instabilities that occur in

electrokinetic systems so first if you

have if you have immiscible fluids for

example a liquid and air as in these two

cases or even two immiscible liquids

then it's well known that electric

fields can destabilize those interfaces

and cause undulations and patterns to

form or sprays also in liquid systems

with miscible liquid interfaces where

you have sailed a high concentration

stream of salt and a low salt stream in

our electric field in this direction you

can generate instabilities there as well

and then this is an example which I'll

come back to later in the talk of

electro dialysis this is a membrane

based separation process where you have

a flow in this direction a current

applied in this direction

through a stack of cation exchange

membranes and an exchange membranes and

that's to basically remove salt from the

water and there too when you have large

salt concentration gradients you have

some vortices forming and you have

electro kinetic instability so I'd like

to ask is sort of what is the

intersection of those problems that I

just described so is it possible that we

can use electro kinetic phenomena to

actually control this notorious

laplacian growth type instabilities and

of course I wouldn't ask that question

if there weren't at least somewhat of a

positive answer so yes and it leads us

to I think some pretty interesting new

types of problems that are kind of come

from combining those different fields so

one is to look at you know we're talking

about electrolytes they're confined by

solid surfaces but in particularly then

we're also looking at interfaces between

two immiscible liquid electrolytes or in

the case of electro deposition a moving

solid liquid interface but the key thing

again is sort of being under confinement

by charge surfaces so I'll give you

three examples of these this these types

of phenomenon so the first one is

electro kinetic control of viscous

fingering so

so first a viscous fingering which leads

to the pattern that I showed you on the

first slide is basically a the easiest

description of it is through so-called

Healy Shaw cell geometry so you

basically have two parallel plates which

are separated by a small gap which might

be sort of at the millimeter scale or

less and we have it initially filled

with one fluid the yellow fluid and we

inject a different fluid in there which

is immiscible and displace the first

fluid and do it in such a way that you

know we're not leaving behind to like a

wetting film but that the interface

actually can move so it's a partially

wetting situation so this is a classic

problem

the instability was actually first

analyzed well actually that was first

analyzed by some Russian authors

actually in 1945 by polar Bravo and

galine but the first description of this

sort of the nonlinear dynamics of the

instability that forms these fingers was

by Safin and Taylor and so this is an

example of their experiment of air

displacing glycerin but you very

generally see this when are you have a

less viscous fluid displacing a more

viscous fluid again the higher

resistance is ahead of you so when you

push the interface wants to go unstable

to allow sort of a path of least

resistance to breakthrough this is an

important problem in applications such

as oil recovery where you try to use

water to displace residual oil in the

ground of course fingering is bad

because then the water rushes through

and doesn't push out the oil that's just

one important example but it's a very

fundamental phenomenon a basically comes

back to us laplacian growth in this case

you get it because in a Haley Shaw cell

if you look at the depth average sort of

flow field here it's roughly a pause a

flow where the velocity is proportional

pressure gradient and divergence of

velocity zero so it's Laplace's equation

and if you have let's say air then it's

roughly like a constant pressure surface

and so you have basically all the

ingredients of laplacian growth so as I

said this was a classic problem analyzed

by Safa and Taylor and in fact others

and so if you ask it so the way linear

stability analysis works you start with

a base state which is it's a moving

interface which is flat and you ask if I

put a perturb Ridge perturbation of a

certain wavelength will it grow and if

it grows it's it's linearly unstable so

you can solve that problem in the case

a viscous fingering and what you find is

the stability is controlled by this

ratio M which is the ratio of

viscosities and it's exactly what I said

before if there if the fluid ahead of

you is more viscous it's unstable if it

one behind you is more viscous it's

stable and you also can predict what is

the wavelength instability in the sense

that this is the dispersion relation

that relates the growth rate of an

instability of wave number K to the

system parameters so the capillary

number comes in which brings you surface

tension because basically the laplacian

growth instability is incredibly

unstable as I said the smallest

perturbations actually grow the fastest

so it's incredibly unstable but surface

tension kind of brings it back down so

under the normal conditions when you

have unstable fingering as a function of

the wave number at high K out here you

have very you know small periods down

here you have a very low period and long

wavelength and you get here as a maximum

and that's the that's the mode that will

grow the fastest and so when the

instability develops initially that's

the pattern you're gonna see so so the

starting point here is to see what would

happen if you kind of repeat the

analysis that I did in in the first

mathematical work I showed you where it

was kind of one-sided vector laplacian

grow three is set of fields ahead of you

governing an air fashion motion but now

we have those fields on both sides

interface so it's a much more subtle

problem and in particular the vector

laplacian gothe comes in because i

already described you how flow field is

related to a pressure gradient through a

term here which could be considered the

hydraulic conductance similarly if I

have a grading of electric potential

that's electric field it gives you an

electrical current and the same movement

of ions through Ohm's law so you have a

you have a electrical conductivity an

electric kinetic phenomena linear

electric kinetic phenomena come from the

coupling between those terms so

basically there's a matrix that relates

kind of the two the two forces

thermodynamic forces pressure gradient

electric field two the two kinds of

fluxes in this problem which are the

flow and the electric current and

they're off diagonal terms by Ansari's

relations it has

be a symmetric matrix and it turns out

this is the electro osmotic coupling so

that gives you phenomenon such as

electro osmotic flow as streaming

current where that comes from is through

the fact that we have the system

confined between two plates which are

charged and so the electrolyte or the

liquid contains ions which on average

are an equal numbers positive and

negative so it's neutral but near the

surfaces which are charged we have

so-called luxury double airs where

there's an excess of opposite screening

charge and when you imagine for example

electric field which is parallel to this

interface in the bulk solution there's

no force on the interface or no strong

force because it's neutral but near the

interface in this double layer you see

you have a big push on the on the fluid

but there's also a you imagine a no slip

boundary condition and so you get sir a

balance of electrical force and viscous

drag and that gives you so called

electro osmotic flow that leads to this

factor here and just to sort of define

things epsilon is permittivity mu is

vers viscosity and Zeta is the so called

zeta potential it's the voltage drop

across this interface which somehow

measures the surface charge there's also

the inverse effect if I if I apply a

pressure I get a flow and that's what

imagine if I have sort of a pressure

driven flow through this system there'll

be a shear strain rate on the surface

which is going to move these ions and in

this double air there's a net charge so

again there's a flow of charge which is

a current and so the inverse effect is

there as well okay and we're here just

talking about linear responses so it's a

symmetric matrix okay but we have this

kind of matrix on both sides so if you

analyze that problem and you ask

yourself I have flat interface moving

and ahead and behind I have an arbitrary

like electro osmotic conductance matrix

governed by all those parameters the

stability criterion becomes this one

right here and this was work of my

postdoc Muhammad Mears a day and so you

have a very similar looking dispersion

Latian before it's got a linear and then

a kind of cubic term and it can be

either unstable or stable depending on

this criterion here and so again the

viscosity ratio is critical but what's

interesting now is you have this other

side here which involves parameters you

can control so what you have here

essentially is a dimensionless

it's the current to flow rate ratio with

some material parameters so you see like

an average of the two conductivity x'

average of the viscosity x' averages

zeta times the dielectric constant ratio

etc but basically by adjusting the

current or the flow rate you can play

with this instability and basically make

this be unstable have some growth modes

that are positive and also have it be

stable so here's some examples of that

so in the upper left you see what

happens if you just do viscous fingering

without any electric field applied this

is simulation of a healy saw cell and

you can see the onliest thing that's

changing is you go from here to here is

this parameter a is changing which is

basically that dimensionless current

that i mentioned so we given the sort of

charges we've assumed on the services

and the viscosity ratios and the

dielectric constant ratios and the

conductivity ratios all those parameters

when you move the current in this

direction you actually can stabilize the

interface and basically what's happening

is we're kind of like in a very simple

way flipping sort of where the highest

resistance is so if you have an unstable

situation again it's the more viscous

liquid is ahead of you so you're pushing

it and wants to break through but when

you Charlie electric field in the right

way it's making it so that the higher

sort of electro osmotic resistance is on

the back side basically so that if you

sort of have a perturbation like

sticking like this it kind of pushes it

back forward again you know so it's just

competing effects but it is kind of

complicated in each case to really

understand the instability because we do

have this coupling of different fields

anyway so this is the theory we

published this last year and in my lab

actually we've been trying to observe

this experimentally which frankly I

think would be very difficult to do if

we didn't have the theory because you

have to sort of find the right liquids

the right set of parameters to actually

see the phenomenon so here's an

experiment on just classic viscous

fingering the same as what I showed you

on the first slide in our own Keeley

shaw cell

oops sorry so here's yeah unstable

marginal and stable so those are the

different situations okay and so here

are the experimental results so here's

just an example of a stable case of oil

displacing water so in this case it's M

greater than one which means the oil is

more viscous but when we apply a current

in the right direction you can start to

see we get instability so we've just

taken

stable situation made it stable and if

we apply the same magnitude current in

the opposite direction it's equally or

we would say even more stable here's a

situation where it's actually unstable

even at zero current so this is a case

where you have a lower viscosity liquid

pushing a higher viscosity liquid and

you can see there's a little bit of an

instability there and sure enough when

we apply the correct current we're able

to stabilize that system and I should

say that there are some challenges

setting up these experiments so here's

what our system looks like we have a

Healey Shaw cell but we have to

integrate electrodes and as you can see

some of these colors here are coming

from the dye that we use actually reacts

electrochemically in at the at the

electrodes and so that's what you're

seeing is color changes here we hope

that that's not influencing too much the

dynamics but nevertheless we have in the

middle here what you're seeing is sort

of a conical shape reservoir where we

place one electrode and there can be

some water splitting reactions there for

example that are carrying that are you

know sustaining the current the direct

current and then we have a ring

electrode kind of on the outside and we

try to keep those electrodes far away

from the dynamics so we can apply a

current but whatever is happy electro we

hope is not influencing the dynamics and

so here are if you do a lot of

experiments that type you can start to

build up kind of a stability diagram so

this is for this particular choice of

materials we have a plot against the

current and the flow rate and the black

line is where we expect the stability

boundary to be according to the theory

and we see pretty good agreement to do

this plot here we take movies like this

and we digitize the images and identify

the boundary and we actually calculate

the roughness and the growth rate of the

roughness and so we actually plot the

dimensionless growth rate of the

roughness scale to the radius and

whenever that's kind of above a

threshold we could say the system is

unstable and so kind of the magnitude in

color and the size of these points tells

you how much of this kind of instability

is growing and you can see that sort of

below the theory line like things are

pretty stable there's still some

perturbations there but nothing growing

very quickly whereas above this line we

pretty clearly are seeing instability

and so at least we're getting a pretty

good fit there I should say there's no

fitting the only parameter that leads to

this range here is that we don't

actually know the zeta potential so

we've tried to measure them but there's

some uncertainty there but anyway does

it doesn't make it doesn't make a huge

difference if you kind of pick us or

reasonable ranges ated potentials for

these materials and otherwise the

material properties are all known the

the conductivity the dogger constants

and viscosity and so here actually is a

data collapse now for different

materials so now we basically play with

the viscosity liquids so one of the

liquids is a water glycerol mixture so

we can vary its viscosity and then we're

using a few different oils like octenol

for example and we're also using

different salts to vary the conductivity

'z materials also different dielectric

constants anyway so for every viscosity

ratio we can do a whole bunch of

different experiments of different

currents or different material

properties and here's all the different

dimensionless groups and again we have

this linear stability boundary which is

sort of plotted here in dimensionless

variables that should ideally collapse

all the data so everything above this

line should be unstable everything below

should be stable and so far you know it

looks like pretty good agreement up here

you see what the error bars look like so

it's not you know that that air by the

way is coming from the zeta potential is

we don't actually know for sure what the

zeta potential is are so that's the

range we're plotting so it seems to hold

up pretty well but not everything fits

so I just want to kind of be clear about

that so here you're in the stable regime

but if you go to really strong currents

in this direction you start to see

instability again and you see situations

like this where instead of kind of this

evil and even sort of linear stability

looking like nice viscous fingers

forming we're seeing like individual

bulges coming out we're not sure the

meaning of that but it might be from

that high currents we started to get our

Maxwell stresses so we've been

neglecting electro hydrodynamic switch I

showed you at the beginning the electro

spinning and all that this is just

electro kinetics away from the interface

so the forces and flows that are

generated there but the interface itself

feels electric fields that gets deform

in the field you can even form a cone

and that might be something we're seeing

here but I don't know yet and also in

this case we're supposed to be unstable

but sometimes it actually looks kind of

stable and

some of those cases it looks very blue

and some of those reaction products I

talked about maybe contaminating things

and so I don't claim we understand

everything here but certainly we have

like a reasonable picture at what we

have demonstrated is that definitely

electric fields electric annex can be

used to control viscous fingering okay

so now on to a different phenomena and

then at the end I'll try to bring these

things together actually so let me tell

you a bit about what I called ionization

shockwaves which are actually stabilized

again by electro kinetic gamma so this

comes back to the topic of electro

dialysis that I briefly mentioned at the

beginning so here's a classic sketch of

electro dialysis from the work of

protein and and Sonnen which is again

this method of desalination where you

flow a salty fluid into a stack of

membranes which have alternating

selectivity or polarity for ions so you

have a high anion selective membrane a

cation selective membrane etc you

alternate pass a current in this

direction and so it's indicated by these

arrows when you cross the membrane here

the current is let's say entirely

carried by cations for example sodium

and then at the next membrane it's

entirely carried by cation anions the

other way and that repeats itself and

since the system is really not at the

nano scale it's at the millimeter scale

let's say the liquid is more or less

neutral at least away from the

interfaces and so basically if you zoom

in on what's happening your interface

you have let's say cat is going through

anions get blocks even buildup of salt

concentration on one side on the other

side cation goes through anion goes away

you have a depletion of neutral salt so

overall the salt concentration stays

neutral

but you get this kind of rise on one

side and depletion on the other as you

flow through the system that takes the

form of boundary layers you know so you

see here you have a depletion layer

forming on this side on the other side

you have an enrichment layer forming

that's classic electro dialysis now one

of the fundamental questions in this

field for many many decades now actually

is why it is that you can see so-called

over limiting current

so if you do the simple theory of the

process I just described where you

imagine let's say outside this boundary

layer is a reservoir of uniform

concentration of salt and here's your

membrane if you require

electroneutrality

and you only have electro diffusion so

there's no other transport processes or

reactions going on then what you predict

is there's a diffusion limited current

at some point the current go that the

concentration goes to zero and that

would pretty it would look like this the

higher voltage do you apply you don't

get more current but in every experiment

you eventually do get more current so it

means that somehow the system is

breaking down so it's giving you more

current which can happen for example by

reactions you could be producing more

ions you could be breaking down the

membrane so they're not so less

selective all those things can happen

but also you've got new transport

processes and one of those is hydrogen

hammock instability so if you initially

say this is this assumption here is only

diffusion well if diffusion doesn't

carry the current then the liquid can't

start to flow and then you start to get

instability and that's the so call that

traumatic instability I showed you a

picture of that on my second one of my

early slides but I'm not gonna focus on

that for the moment actually so if you

had just a membrane in contact with a

bulk fluid as I showed you you do get

that electro osmotic instability it's

also called the Rubinstein Saltzman and

stability after actually my collaborator

is our Rubinstein and Saltzman on a lot

of this work and those are the guys that

initially develop the theory of that

instability but what I want to focus on

now which is the novel part is not think

about a bulk liquid in contact with a

membrane but rather a micro channel or

porous medium in contact with the

membranes so we have confinement with

charge surfaces I already showed you in

the first part of the talk that when you

have a Healy Shaw cell but the services

are charged and you're confined you can

totally change the dynamics of the

system with lectric kinetic phenomena

same thing is true here that if you

imagine now I have a membrane and let's

think about the situation where the

concentration is sort of appears to be

going to zero somewhere away from the

membrane so there's some very high

current we're over limiting and again

the classical picture there's some

instability happening near the membrane

but if you imagine now I have a micro

channel which dead ends at the membrane

you can also think of a porous medium

where this sort of kind of a pole

which like dead ends at the membrane and

that poor is charged and it's charged

with the correct surface charge and what

I mean by that is that if I want to

let's say pass positive ions to the

membrane I need to get them to transport

faster then I need to have negatively

charged walls so the double layer on the

on the on the less liquid side is

positively charged what that means is I

have a pathway to sort of short-circuit

the system and sort like a shunt

resistance almost that allows you to get

more current because basically this

region of depletion is a region of high

electric field so what happens is when

the concentration goes to zero or near

zero you can't rely on diffusion anymore

but this large field can do two things

it can in very small channels let's say

submicron the electric field is strong

enough to pull the ions in the double

layer and give you basically fast

transport by electro migration or a

surface so called surface conduction you

also have the effect which is stronger

in sort of middle range channels where

the sort of pulley lets you feel them as

ions generates flows and those flows

provide pathways for convection of salt

to the surface okay so those are two

mechanisms and these are described again

theoretically in this paper from 2011

and what it does it basically gives you

a sort of mechanism for over limiting

current so instead of having like a flat

current here you can go over limiting

another interesting aspect of that is

the dynamical part of it so the steady

state I just showed you but there's also

interesting dynamics that imagine like

now I apply an over limiting current but

I do it I I look at the dynamics of how

I get there

so a classic problem in sort of chemical

engineering which I teach in my

transport class I've got the diffusion

equation initial constant concentration

and at time equals zero I turn on a

constant flux which is like constant

slope so what happens is at constant

flux there's a diffusion layer the form

of the spreads like square root of time

and then the concentration goes down at

the surface also a square root of time

and at a certain time which is called

sans time which I will come back to the

concentration goes to zero and if you

have a neutral diffusion model at that

point the experiment has to stop because

you've now hit zero concentration you

can't

you going but somehow the system does

keep going and the way it happens is

that the simplest model of what I just

showed you with these a charge layers is

if I imagine homogenizing that surface

charge so Rho s is like a background

surface charge per volume then the

effect of electromigration is based the

electric field which is current divided

by conductivity times service charge and

it's basically a drift term so what you

have is is a diffusion equation with a

non-linear drift and those of you that

know a little bit of applied math know

that that kind of system leads to shock

waves basically whether it's gas

dynamics river waves like you name it

many systems lead to shocks as well the

nominee or drift and where it comes in

here is the fact that the conductivity

of the solution depends on concentration

and its concentration goes to zero

basically this electric field blows up

and it kind of turns on that drift term

and it basically does it causes kind of

this thing to sharpen and propagate as a

shock wave until you then reach the

steady state these shock waves were

actually observed

actually before I got started in fact

that's what got me into this field in

2009 Juan Santiago's group first

observed these shock waves

experimentally in microfluidic devices

so they made a device in an etched glass

micro channel which was if you take a

top view or side view of it was kind of

a thicker channel about a micron thick

and then it dropped to 100 nanometers

and then back to micron and water and

glass typically gives you a negative

surface charge and what they found is

when they passed a current through this

system they saw salt property of

propagation of salt concentration shock

waves away in both directions which are

kind of sketched here this by the way is

an experimental space-time plot so this

is space here's the mite here's the nano

channel constriction and these and this

is time and you can see kind of like a

light cone here which is basically the

shock waves propagating one is giving

you very strong depletion the one is

giving you a more modest enrichment of

the saw concentration anyway so that so

the interesting thing I want to focus on

is the propagation of these shock waves

so they focus on microphone expand what

I got interested in right at that time

was applying these ideas in porous media

and generating macroscopic shockwaves

I'll come to that in just a minute but

first here are some movies of the stuff

I don't want to show you that one first

I want to show you the other okay the

top one is a simulation we published in

2011 of a deionization shock wave here

you see the red are cations the bluer

anions the walls are negatively charged

and at the end of the channel you have a

membrane which is only passing cations

this is not like Rio mutter dynamics

actually we did this simulation just we

did this simulation just simply to make

this movie just to kind of visualize

what's happening we're not claiming this

is real all we did was just random

walking particles that feel the mean

electric field that and you know have

some kind of noise so I would not

claiming that's like the real system but

anyway what happened you see this sharp

front forming and you see a transition

from when this thing is out here it's

basically highly concentrated so you've

getting normal kind of electro diffusion

and back here there's a high electric

field and you're seeing that the the

current is carried by cations moving

along the surfaces by surface conduction

and these are some experiments done in

2015 by a group in Korea of seungjae Kim

which demonstrated this physics and also

the physics I described before of two

different mechanisms so first here isn't

a chant as a top view of a channel that

is 2 microns deep and we predicted that

would be in the service conduction

regimes I should look a lot like that

movie up there and sure enough these are

negatively charged tracer particles that

are fluorescing and you can see they're

forming a fairly sharp front which is

propagating this way there's no

significant flows affecting the shape

the important thing for today's talk is

that this thing is flat it's actually

stable now I'll explain you in a minute

why that is so stable in the sense that

doesn't want to form fingers or any kind

of vortices it just it's it wants to be

flat if you make a thicker channel then

you get electro osmotic flow and to be

honest actually we don't even know

actually how stable that is you look in

general sorry this is a zoom in of some

here so here you see look in this

channel there's much stronger electro

osmotic flow and so these particles are

moving all over the place but still a

shock wave goes by so this is the

concentrate regime and you'll see the

shock in just a second there it is

right about there and behind it is

heavily depleted but following behind

the shock are basically pairs of counter

rotating vortices like I showed you in

the steady state drawing so Bay

electro osmotic flow and the sidewalls

is generating some vortices that allow

you to kind of sweep the salt away

faster the diffusion that's what's

leading to that shock wave okay so

that's this font this phenomenon for

today's talk I want to focus on

stability so why do I say this

that I expect that the least diffusive

or let an electro migration shock wave

to be stable what in the case we don't

have the flow so it's because basically

it's another problem with placing growth

so basically we've talked about viscous

fingering where if we were to push a

less viscous fluid into more viscous

fluid like say air into water or air

into oil then the higher resistance is

ahead of the motion because it's

basically this pressure driven flow that

I talked about and so in that case where

you have a oh sorry sorry this is the

yes sir sorry in this direction it's

unstable because high resistance if

you're going the other way to be stable

the deionization shock wave is kind of

everything's flipped actually is because

the region of high salt you more or less

have constant electric potentials

there's very low electric field behind

the shock wave it's strongly depleted

salts there's high electric field so you

kind of have Laplace's equation on the

on the back side so as it's propagating

this way if there's a perturbation the

part that's left behind gets quickly

eaten away by higher fluxes and it

flattens out so it becomes stable

basically so and there's many other

examples in corrosion and other problems

where you see the same kind of physics

so basically it's like deposition is

stable corrosion is uncertain unstable

corrosion is stable and there's these

two directions basically so this is an

example of a stable process another way

I know it's stable even when some of

those electro osmotic flow is are

present is that we've actually built

devices that take advantage of this

phenomenon for desalination and

separation I this is a whole nother talk

I won't get into it but it's kind of the

experimental proof that we do have

stable shockwaves so I've been

developing for about eight not eight

years now this process called shock

electro dialysis in my lab and also in

theoretical work and it kind of a bit

like what I described earlier with

normal extra Dallas's where we have a

set of membranes and this is just one

one layer of a device

except instead of having alternating two

kinds of membranes we have just one so

for example a cation exchange membrane

on each side and then instead of having

an open channel we fill the channel with

a porous material which has the

appropriate surface charge to generate

that D ionization shockwave so in

particular I'm passing current this way

and imagine these sort of pores that

come through here they're all negatively

charged and they have a shock wave

propagating away as I showed you in that

movie okay right so it's kind of like in

this in this movie coming back to that

yeah we're not playing here but well

anyway there's the salty region and then

this sort of not salty region which you

also saw in the experiments imagine

separating this fluid from this fluid in

a cross flow that's basically what we're

doing yeah there it goes and so

basically what happens because we have a

cross flow the shock wave is propagating

this way but it's also flowing so you

reach a steady state where there's a

shock wave that kind of bends over and

then we split the two streams so it's

kind of like in this region anyway it's

a membraneless separation because no

physical barrier so by adjusting

currents we can move that you know shock

wave up or down and affect the amount of

desalination and so you see here for

example as we increase the current we

push the shock this way eventually

crosses the splitter and the Ayana Ruvo

gets to be very high and in fact this

deionized region which is by the way

that's 500 microns or half a millimeter

in this example is we're continuously

extracting highly d salient fluid so our

desalination factor our ratio of initial

to final saw concentration can actually

be like 10,000 fold actually we can

essentially remove all ions by this

process especially we don't start with

too high salt concentration we've also

found this is very good for doing ionic

separations as well but the important

thing is that we wouldn't be able to get

this sort of nice stable of data if this

shock we're like wildly flying around

like a turbulent flow it's obviously

just sitting there we get very stable

concentrations coming out of each of the

streams so it's kind of an experimental

proof that even in a porous medium with

all this complicated physics we still

are managing it to get stable shock

waves so let me now try to connect this

back to the original topic which is

stability of laplacian growth

and instead of talking about viscous

fingers I'm going to talk about electro

deposition so I showed you the copper

dendrites growing on the first slide as

well now what I want to ask now is if

the copper dendrites are growing in a

charged porous medium where I have

electro kinetic phenomena can I actually

control their stability and so that's

what I call and when we get to the shock

regime I call that shock electro

deposition and so I'll give you some

examples of that

in the next couple slides so first let's

start with a very simple porous material

here so this porous material is an

article lumina mach side it's a

commercially available material that has

non intersecting parallel almost

perfectly cylindrical pores with a

pretty well-defined size for example

this one I think is 2 or 300 nanometers

okay and can be like 50 microns long so

they're just very parallel you know

non-intersecting thin channels and those

are used in nanotechnology even for

fabrication of nano wires if you do

electro deposition into these materials

people do that you can dissolve the

template you're left with a bunch of

nano wires but what I think people have

not recognized either in the

nanotechnology field or even for that

matter in semiconductor like

microelectronic fabrication where they

also do a lot of deposition into porous

structures that electro kinetics matters

actually because normally that's not

considered and in particular we

demonstrated that through a bunch of

experiments where we controllably change

the surface charge so we take a porous

template like this and we either make

the surface as positive or negative by

basically grafting positively or

negatively charged polymers to the

surface and I should jihun Han is my

postdoc was actually a PhD in chemistry

it was very good at modifying surfaces

so that was essential for this work so

basically I'm going to show you a bunch

of experiments were the only thing we

change is the surface charge of the

template and as I said normal way of

thinking that should play no role in

electro deposition but it plays a huge

role in particular if I get into this

high current regime then imagine I have

so this is like the situation showed you

before of the G ionization shock wave

where if I have a negative charge

surface then the cations which are here

red would want to kind of they would be

able to go through the double air be

conducted

surface conduction to the surface so in

that case you would expect deposition to

still continue whereas in the case where

the surface charge is positive now the

surface conduction goes the wrong way

and so you can't get any extra

deposition so you're kind of shutting

off the current and here it's enix

parent we apply the same voltage same

saw concentration it's copper sulfate

we're trying to grow copper everything

the same but this is negative this is

positive in the case of positive the

copper barely wants to penetrate the

system effect it kind of grows

underneath and pushes away the template

so basically does not want to enter the

material where with the negative charge

it goes flying into the material and you

make essentially these sort of nano

wires and then a nice visual

demonstration of this physics here is

you can imagine if this surface current

is getting strong enough eventually all

the ions are coming along the surface so

the growth should also be along the

surface because the ions are being fed

along the surface and sure enough we see

a transition to making nanotubes where

the metal is growing along the surface

and the next slide shows that even more

dramatically here so if we do those

kinds experiments and we cut open the

system and then do imaging we can see

here's a sequence of experimental growth

at different applied potentials which

are in the overloading regime and again

the only difference between this set of

images and this one is the surface

charge so in the case of the positively

charged system that is this one here now

the surface current is going the wrong

way so the copper ions which are

positive really have no way of getting

easily to the surface here and so at

first they're kind of filling the

channel but if you notice they're not

really touching the walls and eventually

they're getting more and more rough but

they're avoiding the walls so we're

getting sort of dendritic growth but in

a way that is essentially avoiding the

walls and on the other hand in this

situation now the flux is coming along

the walls by service conduction you can

see some signs of it at this potential

but there's a transition here where you

start to see some rough growth entirely

along the surface or poor surface

dendrites and eventually that surface

flux is so strong you get the nanotubes

so basically the formation in attitudes

what people had seen before is actually

at least in these experiments a direct

consequence of electrokinetic

actually again which are not normally

considered in this kind of stuff but

again I want to come back to my main

focus here which is stability so an

interesting question now if I so

basically those experiments demonstrate

the service conduction is important and

that we should expect some of the same

phenomena here that we saw in the shock

electrodialysis

system but I showed you that in a in a

sir random porous material like the

shock which analysis was using a porous

silica glass frit I'm gonna show you

experiments with that in just a moment

the deionization shock wave wants to be

flat on the other hand as I explained

the beginning of the talk electro

deposition wants to grow dendrites if

it's transport limited because basically

you have a sort of a limitation of by

diffusion if something sticks out it

gets more flux that sticks out more so

it's unstable and as I said the D

ization shock wave is the opposite if

the shock waves propagating this way

then the higher resistance is behind the

shock wave and so actually that tends to

want to stabilize it because if

something sticks out like this it gets a

higher flux of ions away and it kind of

pushes that forward but each of these

are unsteady so coming back to the

theorem I mention at the beginning the

reason we might have some new behaviors

because this is on this is it's all

unsteady it's not this not so quasi

steady so the real question is what will

happen if we have both a Deanna's a

shock wave and metal growing so imagine

charged porous material I go to really

high currents where I form this shock

wave and metal growth is kind of

following it will it be flat or will we

have dendrites and so I say which effect

will win I say well we must experiments

partly actually because I never got

funding to do theory for this topic but

I got something funny for experiments

cuz people are very interested in

electro deposition these days for a

variety of reasons so short answer is

yes we can suppress dendritic growth

this way and in particular we can

certainly do it for copper as I should

because copper is very strongly feels

these effects so here you see again

copper growth from copper sulphate

solution this porous material is

cellulose nitride which has been treated

to be either positive negative or

positive and the case where it's

positive this is elemental mapping of

copper using yeah you

yes method this is STM imaging so you

see you can still form these kind of

rough structures in that case but when

it's a negative surface charge and we

believe that there is a dilation shock

wave which is proceeding this growth

because we're under over limiting

current you notice we get sort of very

flat interface here and you can kind of

see it there this is not the original

interface this is actually right there

so there's like a growth of copper

essentially if you see how much copper

was deposit you know the porosity you

really are filling the pores with copper

and it's pretty flat so it's just kind

of a wave of copper filling the porous

material and again the only difference

between this experiment is experiment

this potential plot is the same the salt

concentration everything is just the

surface charge of the porous template

okay so that's a nice result I put a

little asterisks here because I was

actually funded to do this by Bosch to

look at lithium metal people are very

excited about lithium metal for

batteries as of course we want to

control dendrites that's like one of the

big hot topics today and so the hope was

that maybe this would work by use we

could make design sort of charge

separators that would allow high ionic

currents but also would sort of in using

electric Kinetics would block the

instability that leads to metal

dendrites that can short-circuit a cell

unfortunately it doesn't necessarily

work for lithium but before I do that

let me briefly show you here the summary

of what I showed you so for copper I

showed you first this a OH material that

had these channels and there you saw

that when it was negatively charged I

could sort of form these like nanotubes

or these sort of surface growing things

but they were each one is kind of

growing into penally so you saw there

all the different lengths but in the

case of a random porous material where

instead of just a parallel pores they're

all kind of connected then you have this

more macroscopic behavior of a shock

wave of salt concentration followed by a

sort of flat a deposition front so the

question is does lithium actually grow

like copper so amazingly in this field

there's tons of papers including in very

high pack journals that claim to have

sort of solved the dendrite problem but

they actually never visualized dendrites

and they never specify what a dendrite

actually is so to me a dendrite means

something very specific it means those

fractal structures that we saw that are

growing by transport limitation if you

do not have transport limitation you can

have rough surfaces

but that is governed by some kind of

reactions that's not a dendrite in my

book anyway okay so I'm advocating for

changing that terminology by the way so

my former postdoc Poong by who's now a

professor at Washington versity devised

a very clever experiment to actually

visualize lithium growth under battery

relevant conditions so here he actually

used a pulled glass capillary where we

could just optically visualize the

system this is lithium metal it's in a

typical battery electrolyte we're

applying reasonable battery currents and

we just sort of watch what happens and

so if we do that this is a very typical

video of what happens so initially you

do get kind of a rough growth which is

certainly not flat but only it doesn't

look very much like the patterns I

showed you at the beginning it's very

dense and if you zoom in there's all

these little whiskers actually and sort

of complex structures and if you also

look at how it's growing it's not really

growing at the tips it's growing almost

like from the roots it's almost like

it's kind of bubbling over so it's

definitely not dendritic growth but then

at a certain moment boom that's a

dendrite it looks just like a dendrite

you know what you know it feels like a

dendrite right and and not only that at

this exact moment if you follow the

voltage the voltage blows up why because

you just hit transport limitation when

the concentration goes to zero you have

a roadblock and the current you know you

if you fix the current you get a high

voltage okay and so and we did lots

experiments demonstrating when is this

point it turns out it's exactly sands

time that I showed you before it's a

great homework problem for my transport

class basically is that I start with

constant concentration I apply a

constant flux at a certain time

concentration goes to zero and it comes

back now to this general discussion

stability when this interface has a

roadblock ahead of it it can do a couple

of things it can create more transport

which is all the electric kinetics I

talked about but if that's not working

it can also destabilize and push itself

forward so it's kind of like if the ions

can't come to you you go to the ions

okay that's what a dendrite is so they

see what's happening here is that it's

sort of that is that the interface is

moving very quickly and harvesting all

the ions everywhere they are and not

waiting for them to come okay that's

see what happened so it's transport

limitation and this kind of physics is

well known for copper but it wasn't

established whether or not this is

actually valid for lithium and what I'm

going to show you on the next slide is

that almost every experiment every

battery relevant situation for lithium

is usually in this regime not this one

so actually we pretty much are not

solving or not the dendrite problem

because you probably don't even have

Dondre it's actually another sense that

we're usually not transport limited so

it's like a totally different problem

and so what we define there is that I

told you about sands time if you're

applying a constant current then current

time stands time as sands capacities to

told them on a metal you're going to

deposit and for batteries you're

interested in sort of the trade-off of

like what kind of current do you want to

apply and what kind of capacity do you

want for your battery how much metal do

you want to deposit and dissolve you

know to store energy let's say for your

battery and so basically this formula

sands capacity is this redline and if

you you know kind of we and we visualize

in those capillary cells the transition

from sort of this mossy structure - more

like dendritic growth you know when you

cross that line and these are many

previous papers on the dendrite issue

and a lot of them are down here they're

either very low currents or very low

capacities or both which is still

interesting and you want it you

definitely want to control that

situation but it's not transport limited

and those are not dendrites there are

not these fractals that everybody draws

it's something quite different actually

you have lithium in close contact with a

separator there is very rough growth

going on there there are competing

reactions if you're in a liquid cell

anyway we also studied this idea of

interaction but between the dendrite in

the separator so there's also this

concept in the in the solid battery

field that your separator needs to quote

block dendrites like mechanically

there's a there's a sort of highly cited

paper of Monroe and Newman very nice

paper actually which looks at the growth

of a finger of metal into an elastic

medium and naturally if you make the

medium very hard it's not going to be

able to grow it's going to be

mechanically blocked so the idea is we

want hard separators and so a lot of the

research in this field has gone into

ceramic

separators that are good lithium ion

conductors because then you know the

idea is the dendrite nothing can poke

through you can't have the anode like

reach the cathode and have like an

explosion or a thermal runaway event for

example so it's safe but the problem

with hard separators is that if lithium

is changing its shape below you can't

maintain intimate contact between the

lithium metal and the separator and so

these wonderful lithium ion conductors

that are solids usually don't work that

well you actually need them to be more

soft and so we're also showing here that

you really don't necessarily need this

idea of a dense hard material to block

the growth so we set up a cell here

where we have like two compartments and

we're letting lithium grow in liquid

through different kinds of separators

and even the same aao material that I

showed you which has pretty big pores of

the scale say 200 or 300 nanometers is

pretty good at blocking growth depending

on the regime so if it's this regime of

sort of under limiting current that I

showed you that very dense growth that

mossy lithium tends to be pretty

thoroughly blocked actually however when

you reach transport limitation that's

when you very easily get metal growing

through the separator and you can get

short circuits you can control that with

surface charge as I showed you but you

may just want to stay away from that

regime actually because whenever your

transport limit you're always going to

trouble briefly this is what happens in

the relevant regime in liquid

electrolytes for lithium metal and this

is now departing from the rest of my

talk as I've been talking this whole

time about transport limitation but when

you have reaction limitation more things

can complicate things can happen in

particular lithium is different from

copper in that when you grow in typical

battery electrolytes the bet the the

solvent is decomposing and lithium is

reacting with solvent molecules to form

the so called s CI layer so there's kind

of a passivation film that grows it's

also true in lithium-ion batteries so

there's competing reactions and

basically when things are growing slowly

you kind of have a good se Aiye layer

kind of covering the surface but then

you can have situations that lead to

these sort of whiskers as I talked about

where there's a competing reaction of

lithium deposition and

and sei growth and sort of the thicker

this sei layer gets the more it starts

to block lithium metal and the place

where lithium most easily grows is

actually at the root that's where there

hasn't been enough of that as much of

that passivation layer growing so kind

of in this problem whatever something

starts growing it gets a lot of sei gets

kind of stopped and then lithium can

kind of come in underneath and so that's

how you get that growth from below

actually and so here's a movie actually

high resolution TEM imaging by my

colleague Julie at MIT showing actually

the growth of one of these fingers so

you can see it right there who pushes

out and it kind of we think is getting

sort of this sort of leaves mechanical

stresses that are building up below

actually where the lithium is being

inserted but if you look carefully you

can see that you know it's it's not

growing at the tip it's actually growing

at the root here's another movie of

lithium growing kind of in this more

like smaller over potential regime where

you're getting more like surface

undergrowth if you're getting used to

looking at these kind of patterns as a

physicist I would say that is kpz or

Eden growth where you have sort of

limited by surface reactions those

fractals I showed you on the first slide

those are dendrites that you're limited

by transport like genetically you see

that difference these guys are much more

debts because they're not limited by

transport they're limited by reactions

so basically to conclude I didn't really

talk much about engineering applications

everything I showed you today was basic

science and mathematics but motivated by

some applications so we talked about

controlling this is finger I mentioned

that's important potentially in oil

recovery and actually we're working on

trying to assess how electric fields

could be used in oil recovery and even

in fracturing so like hydraulic

fracturing is usually done by pushing

water into Rock you kind of break it

open and gas escapes but if you apply

electric fields at the same time you've

got some very interesting effects on

that process in terms of leaving more

water in the ground so you don't have as

much produced water what you have to

deal with and also in terms of

displacing more thoroughly the gas and

fracturing more thoroughly also electric

any farmer that could be influenced by

this multi-phase flow are involved in

soil remediation and also I should

mention a manufacturing in fact in

making certain kind of membranes you

sometimes apply to let your field in

order to make certain channels by

instabilities on this topic of growth

into

porous structures or templates it's

obviously important for things like

composite coatings and fabrication of

nanoparticles integrated circuits and

then finally for metal batteries

controlling and stability is very

important if you want to have any

electrode which is sort of changing its

shape by deposition and dissolution it's

extremely important to control that as

opposed to lithium ion batteries where

you're just inserting lithium in and out

of a fixed structure which is kind of

much more friendly you could say and

easier to control

unfortunately we would rather dilute the

metal because you don't have all the

waste it's are a dead volume and you'd

rather just use pure lithium metal if

you could but then you're stuck with

these kinds of issues so with that I'll

stop and take your questions thank you

[Applause]

[Music]

okay so that's a very good point so in

the viscous fingering experiments we're

using a glass that we have done some

characterization of and so we are we

believe at the pH as we are at that we

have a negatively charged glass when you

have the oil mixtures on there with

different viscosity additives like we

don't really know for sure that's why I

said there's the uncertainty about the

zeta potential I suppose one proof that

we have the zeta potential we think is

that the theory agrees rather well with

the experiment which assumes you know a

certain zeta potential solis the sign of

zeta potential probably is okay but I

agree with you that's a potentially

important issue and in fact the general

issue charge regulation what your

context is important so as you have

these especially large electric fields

and large depletions going on you change

the pH of the solution which then

definitely interacts in water very

strongly with most surfaces and changes

their charge so the point of zero H R is

very important and the shock

electrodialysis I didn't get into that

but a major area for our research is

actually controlling that very affect

actually because when you get the strong

depletion you get also big changes in pH

which often work against you so one of

the big inefficiencies for us at high

currents is that actually eventually

we're like splitting water and running

protons through the system and we

actually want to be removing salt so

there's kind of that's kind of what

happens when you run out of salt the

system is still going to do something

and what it starts doing is like

splitting water and moving charges

around we have not measured PCC yeah for

these but it's it's a good point we do

measure pH and the shock electrodialysis

experience we definitely are measuring

pH of every stream but we do not we do

not measure the PCC it's a very good

point and even for the experiments on

electro deposition at the end there we

have less reliable results when we rely

on the native surface charge of these

materials where we go

really nice results I showed is and we

do layer by layer deposition of charge

polymers you know following the work of

my colleague Paula Hammond where we lay

down sort of very stable highly charged

molecules basically that in the range of

PHS we're working we'll be you know very

reliably positive or negative you know

so we can don't address that issue it's

a very good question yeah that's a very

interesting question yes a very good

question so certainly in electro

deposition people use pulsed electric

deposition where sort of the average

current is the one you want but you

apply temporarily higher or lower

currents what that does is it fights

that depletion effect so the source of

the sort of instability in the dendrites

is really when the concentration is

getting low that's when you start to

have kind of the roadblock that you need

to break through and so when you

oscillate you allow you create like a

strong diffusion layer but then it kind

of gets a chance to relax and so you

control that depletion and so that

definitely is effective we have not

studied like the role of like

oscillating forcing in these kinds of

systems yet okay but to have a net for

which problem though for the oh and the

first scenario okay we haven't done that

yeah you know that would be that would

be interesting yeah you know there's

many variations one could do actually so

put what part of the elegance I think of

those experiments is just it's a simple

problem it's very well-defined and we

kind of don't want too much other

physics like I said if proof we've got

electro hydrodynamics of the interface

itself like forming tailor cones or jets

like that's not in our theory right now

and we don't want to put it in there

just yet so like you know you we try to

isolate certain physics you know but

that would certainly be interesting once

you kind of understand this to look at

the dynamical control of all all the

effects that I showed actually yeah

oh yeah here you so dendrite growth is

not good for the bad I'll repeat the

question

[Music]

so the cause of the dendrite is very

clearly established by this experiment

which is that it's transfer limitation

if the salt is getting depleted you can

get dendrites otherwise you won't

otherwise you have this totally

different growth like they described

here that we zoomed in on with forming

those little whiskers the mossy lithium

totally different growth so as long as

we avoid transfer limitation you're not

gonna get like what I call dendrites

that you might be talking about just

general you know rough growth and you're

also asking about well this is a point

okay fair enough but the other

experiments I showed like here we're

actually entire compartments you know

where there's it could be lots of metal

I mean there's it's not just a single

point actually so those experiments are

definitely looking at like a you know

big electrode like centimeters across

and how its stability looks but maybe

not answering your question

yes yeah that's true if all the kurma

focused at one point it would like a

breakthrough yeah that's that's

definitely true but generally like

there's no point even really looking at

that like in the sense that if you want

to design it better you want any

dendrites like as soon as it goes we're

talking about rechargeable battery it's

like a primary lithium batteries nobody

like if you want to dissolve the a no

that's a stable process so if you want

to use lithium metal and dissolve it and

then deposit at the cathode that's like

what was invented and put in your

hearing aids in the 1970s like that's a

primary lithium battery all the research

here is a rechargeable lithium battery

so we have to be able to deposit and

that we'd like to do it more than once

or there's really not much point to it

they want to do it like hundreds or

thousands of times so any kind of

instability interface is basically bad

so really understanding like the kind of

nonlinear dynamics of this growth that

you know after many cycles and whether

you have ten dendrites or 100 dendrites

like I don't think that's really that

important actually you want to make sure

like to avoid this situation entirely

yeah but the purpose of this study here

was again it was really to establish

like the mechanism for like when you

really get you know the tutor at least

two different kinds of growth here in

fact our more recent paper which I may

have cited no I didn't actually kind of

even identifies a third regime that's

kind of in the in the lower currents but

but anyway they're served a few

different mechanisms for controlling the

growth and once you understand those

then you can try to you know kind of

work around them but we do consider this

sans capacity as kind of like a safety

limit you know for batteries that

generally you don't want to deplete the

salt and have the risk of any kind of

instability of interface as I said in my

first slide the sort of laplacian growth

is stability that's one sided where the

transports only ahead of you is always

unstable when something sticks out it

gets more flux there's just like no way

around that you know so you never want

to be in the situation where you're

waiting for transport and it can't get

there fast enough with a moving

interface you're always gonna get

instability with the viscous finger I

was only able to flip it because I've

got transport out

both sides you know so basically by kind

of tuning the balance there with the

with an electric field I get I can

stabilize the system but for you know

electric that position that wouldn't be

so easy