>>[Slide 1] Thank you for

joining me again. This is week two and

we’re discussing how to beat the diffusion limit. In the first week and even for

the first lecture in this week, I said that diffusion is

a fundamental process, not only for biosensing but

for organism of all sort, that the molecules must

diffuse to the sensor’s surface and on the organism’s surface. And in that case, it is often

the geometry of the structure that determines how

efficient the capture is. But no matter what you do

with the geometry, at the end, there is a certain limit

beyond which you cannot go. And so, of course,

that’s the concern. And we are finding ways to go

around that “fundamental limit.” I have already discussed one

approach based on bio-barcode, essentially throwing a lot

of policemen, putting a lot of policemen in search

of suspects. And I explained to you how that dramatically reduces

the capture time. Here’s a second approach

that we’ll be talking about. [Slide 2] And the second approach

is based on the notion of droplet evaporation. I will explain the theory and

I will tell you a little bit about how this is actually

made, how the device that does this droplet-based

measurements or sensing how they are made because I think you’ll

find it very interesting. [Slide 3] Remember, where we are in terms

of this big “Mendeleev’s table,” we have understood the

diffusion limit associated with these types of–

various types of sensors and organize them according

to how they respond. And in fact, if you take

any modern sensors and don’t do anything special,

you can immediately analyze it in terms of, in terms

of their geometry. In fact, this is the first thing

we should do anytime we have a new sensor, understand

its fundamental limits, diffusion limits. Now, I haven’t explained to you

how the bio-barcodes sensor work and today, I want to explain to you how the droplet-based

sensor works is that as I said, very elegant and beautiful. [Slide 4] So, the three approaches

that we have discussed, the one that we focused

today, right now, is this approach

based on the droplet. The key idea is this. That we’ll be beating

the diffusion limit or making the tau,

diffusion limited tau smaller by making the L, the

length or the size of the box essentially

gradually progressively smaller. This is what happens when

you concentrate something. So the molecule might be

originally sitting anywhere within this box, which

is the blue molecule. And then, if you didn’t

do anything special, the molecule will walk around

and take a long time to get to the sensor’s surface. What can happen instead, if you

allow the water to evaporate, of course, the molecule

cannot evaporate, the molecule is sort of trapped. It’s a big molecule. So, water evaporates,

the molecule doesn’t. So, it is increasingly confined

to smaller and smaller boxes until it is sort of captured

by the sensor itself. And this time dependent L, increasingly smaller

L makes tau small. So, let’s discuss that. This would describe

that approach. [Slide 5] Now, before I get there, this is

a very nice example of how sort of ideas from biology or biomimetics essentially

inspires this type of droplet based sensing. In some way, you could say, even

the diffusion limited approach that I discussed previously

in the last week and also in the beginning of this week

also has the same biomimetic approach, the analogy to

biology, diffusion limit and all allow that to

understand the sensors. And this time, the

analogy to biology, this biomimetics will also

essentially also tell us how to beat the diffusion limit. So, nature teaches us both how

to analyze and how to overcome. So here, what I wanted

to tell you. So, the analogy that

one should– droplet and everyone can

think of is this lotus leaf. You can think– we all

know that a drop of water on this lotus leaf is very

spherical, essentially it sits on the surface and any particle

that you put in here, let’s say, you put a biomolecule or

a bacteria or something, it will always be

confined by the trap, by the cage of the droplet. Now, of course, one problem of

this droplet is it moves around. So, that’s the problem. But there’s another thing that

we also see, also sort of goes with this general idea about

trapping is the coffee ring. You know, in the morning, when

you pick the coffee, coffee cup, you will often see

that after a while, an unwashed coffee cup may

actually develop a ring like this. What happened? You see, what happened was

there is a fluid like this and essentially, because

the surface was not superhydrophobic, as

the fluid evaporated, this essentially

deposited the molecules of the coffee around the ring. So somehow, if we could

marry this concept of superhydrophobic surface

or hydrophobic surface which keeps the molecule

together and once it shrinks, it sort of drops the

biomolecule in the center. And this concept of

concentration of the analyte, if we could combine them,

then we would be able to beat the diffusion limit. So, let me explain. [Slide 6] So, this is how it works. This is cartoon, let’s say, if

it were a teaspoonful full of seawater, there’d be

millions of biomolecules, millions of bacteria in here. But even if it’s a

small quantity of blood, you may have a– within this

system, you may have at a– femtomolar concentration, quite

a few copies of the biomolecule. Now, if you waited for them

to diffuse down and get to the bottom of the

sensor’s surface as you know, you’d have to wait a long time. We have discussed that already. Instead, you could

do the following. For example, if you put

this droplet on a surface and these dots are

special, I’ll come back to that in a little bit. After 10 minutes, let’s say

the diameter is something like 500 micron or the

radius is about 500 micron, now the water evaporates. It becomes little smaller,

smaller and by 35 minutes, which is half an hour,

essentially it has, the size has reduced to

five micron, let’s say. How many force have the

concentration increased do you think? Well, if this was 500 micron

and if this is 5 micron, then 4 by 3 pi r cube. So, 500 micron divided by

q divided by 5 micron q, almost as I say, million fold. So, any concentration that

you had that was let’s say, an at attomolar concentration

here, at this point, the concentration

has become picomolar. Now, classical sensors can

subtly detect a picomolar. That’s not a problem and although the original

concentration was an attomolar. So, this ability to

concentrate has given us a way to beat the diffusion limit. And in fact, once the

molecule is deposited, and this picture is taken

from this particular paper, you can image it and detect

where the molecule itself was. [Slide 7] So, how does it work? How long should I wait? You know, we want to be– I know the droplet

will evaporate quickly. But, you know, droplet of

all sizes will not work. If you make a humongous droplet, that will also take a long time

before it evaporates, right? If you have a swimming pool, it doesn’t simply evaporate

overnight and gives you the– all that residues on the bottom

of the pool the next day. So, we have to sort of have

a theory of how long it– how long it takes to evaporate– given size of droplet to see whether we’ll have

any advantage or not. So, the idea is this. Consider this droplet

of radius R, and assume that the

concentration on the surface is Cs,

concentration of the vapor. And at infinity, the

concentration is zero. As the droplet is

evaporating, the diffusion of the water molecules is

going away and it’s going away through the diffusion equation because the particles

are living. They are sort of

randomly walking around and getting lost to infinity. So, inverse problem of the

captured by a nanosphere. Here, the particles are going

away because the source is here and sink is at the infinity. Now, you could immediately get

the solution of this problem. The amount of– the amount

of water lost is dm, dt, and that must be given by the

difference in the analyte– the difference of waters

vapor concentration on the surface and at infinity. And the CD,ss, because we are

solving the diffusion equation, here comes the CD,ss again

but in a different context. And the capacitance,

equivalent– diffusion equivalent

capacitance is 2 pi Dr because it’s a sphere. So therefore, if you go back to

the formula of the capacitance for a sphere, you

will see this formula. Now the m, the amount of

mass lost is essentially lost a shell of a shell of the droplet

at a given time, delta t. So, you will write it as 4 pi r

squared Dr and that’s the amount of mass lost multiplied by

the density of water, right? That’s how much you lost and

the only unknown variable in this particular case is r

and t. And so, we can connect r with t and be done with it. And once you have done it, it

tells you how long the droplet– it will take for the droplet

to go from the initial radius of Ri, let’s say 500

micron to the final radius, which would be five

micron or even smaller. And if you will sort of neglect

this quantity with respect to this, assume the

concentration of vapor at infinite distance, infinite, meaning let’s say

100 micron, is zero. So, then it immediately

tells you that your– in order to make

it go very fast, then your initial radius

has to be relatively small. It helps if you have

a lower density because it will go faster. It helps if the water molecules, the vapor can diffuse

fast in the environment. And it helps to have a

higher– lower vaporization. But the bottom line

is this r squared. Is the reason why a pool

doesn’t evaporate overnight but a little droplet

on the order of micron squared or 100 micron

q essentially will evaporate very, very fast. And this is where

the nanotechnology of biosensing helps you to

beat the diffusion limit. Now, the cS, the called

surface concentration depends on the pressure, the temperature

and the universal gas constant and so on and so forth. But for the time being,

just let’s focus on t and r that the r has to be small in

order for the diffusion time, the concentration time to be

faster than the diffusion limit. All right. [Slide 8] So, how do you do this? How do you make such a device? You can make it relatively

easily. You can have an array

of electrodes that will create a

sensor’s surface. You see, we don’t have

lotus leaf and how would– even if you had a lotus leaf, how would you actually put

contact within the lotus leaf. So, we have to make

our own lotus leaf to beat the diffusion limit. And this is a way of making

the lotus leaf, I will– as I will explain

in a little bit. So, what is done is that you

make a series of pillars, certain electrodes and you can

electrically connect them in order to measure the

analyte concentration. And from a top view,

it looks like a coffin. So, you can see as if

somebody is lying around there but essentially, this is a

droplet, elongated droplet because the pillars are along in a particular direction,

so that elongates it. In a lotus leaf, it would be– the structure would be

symmetric making it spherical. But essentially, the

physics will essentially, generally be the same. So then therefore, if you start

with such structure like this, within a few minutes

essentially, you can evaporate the sensor’s

surface, evaporate the droplet. And while you’re evaporating

the droplet, you can measure for the analyte concentration. And that will give you the– allow you to beat

the diffusion limit. And it’s very similar

to the picture that you had seen before. Now, let me tell you

a little bit about how to create this superhydrophobic

or hydrophobic surface because that is sort of essential element

to this strategy. [Slide 9] So, if you think about a water

droplet sitting on a surface, most of the time or often, depending on the

standard surface, it may look slightly elongated. It’s not hydrophobic enough

to sort of roll up or ball up. Why not? Think about

these three systems. So you have a vapor, you

have liquid and a solid, and for the time being,

forget about this evaporation. So, we have a time independent

droplet sitting somewhere. So, how– what determines

its shape? Well, what determines its shape

is essentially at this point, at the corner, you can

essentially look what the forces are. If the forces are very

strong, then it will sort of get elongated, forces between

the solid and the liquid. If they like each other very

much, it will get elongated. If they repel each other,

then it will be sort of pushed in and it does– the droplet

will become spherical. And so therefore, you

should be looking at– we should be looking

at three surfaces. One is the top half is the

air, bottom half is the liquid, that sort of gives you the force

on this side of the interphase. In here, bottom side is

solid, top side is liquid. So in this corner, you will

have essentially balance of these two forces and on

this side, when going away, you have solid on the bottom,

vapor on the top and the balance of these two forces will

determine the net force in this particular case. So, balance of these three

forces will determine what type of shape do you actually

have for such a system. [Slide 10] Now, this is something Young

has developed a long time ago. Again, the sphere,

the hemisphere is– a sort of cut sphere is

representing the droplet. The radius r is the radius of

curvature and the angle theta, you can convince yourself that this angle theta also

defines the angle theta of the droplet with respect

to the sensor’s surface. And as you can see, bigger the

theta, more spherical the shape. The volume of a truncated sphere

is given by, in terms of theta and r, by this formula. The surface area is

given by R sin theta which gives you the radius and

pi r squared gives you the area. And the top surface area is

given by the following formula. And so, you can see the

total energy of the system. LV stands for liquid to vapor. So on the top surface, we have

liquid on the bottom and vapor on the top, so you

multiply with S, that gives you the

total surface energy. This is surface tension. And then on the bottom,

we have solid and liquid, that’s the interphase and their

spread over A. And finally, solid to vapor is

everything that is not A. So, all these areas outside,

A infinity minus A, is essentially the

energy associated with the solid to,

solid to the vapor. And therefore, if you calculate,

you know, this S depends on R and theta and you can

replace everything therefore, after you do the substitutions,

you can replace R and express G, the energy G, in

terms of theta alone. Once you know theta and once

you know the volume, right, once you know these two things, you know the total

energy of the system. Now, of course, you may say that

the theta could be variable. For a given V, I can

have all sorts oftheta. Yes, but this energy

must be minimized for the shape to be stable. And once you minimize this

energy with respect to theta, you try out different

theta in order to see where the energy is minimized, you correspondingly

get Young’s equation which relates the theta

associated with the fluid equate to the surface tension between

solid to liquid, solid to, you know, liquid

to vapor and solid to vapor using this

simple relationship. Now, in most surfaces, it

will not be superhydrophobic or hydrophobic. So therefore, our

original strategy of beating the diffusion

limit will not work. So, we need to work

a little bit harder as I’ll show you

in the next slide. [Slide 11] So, what we need to do

is that in many cases, the angle may be small

because this solid to liquid tension may be very,

very large and therefore, it can elongate,

elongate the droplet. So, this strategy is–

in the following. You can cross multiply. And then you realized

on the bottom surface, essentially there’s a

competition between solid to vapor and solid to liquid. Somehow, we have to make the

solid to vapor quantity larger in order to make

the theta larger. How do you do that? Well, what you can do is

that essentially cut holes, drill holes in the surface

and therefore, all the points that previously had solid to liquid will now

have vapor to liquid. So phi is the sort of the periodicity

associated with this. So, the solid to vapor

and solid to liquid, the original surface has been

reduced to phi, only the top wire it’s touching it, and

one minus phi has been replaced with liquid to vapor

because this is liquid and this hole has vapor in it. And so there correspondingly, the theta will rise

dramatically. And that is how you make a

superhydrophobic surface. That is the purpose of the

pillars that we talked about. [Slide 12] So very quickly then,

what you do, you put down, you create the pillars with

the right configuration so that it creates a droplet. Once you have the droplet, you have the anode

and then the cathode. So, they not only create

the superhydrophobic surface but also the sensor in between. So, you can measure

the impedance between one set anode– one

set of electrode to the next and the impedance

is plotted here. And as the droplet, as the

droplet, and you have initially, we have one droplet that

gives you a certain impedance and as the droplet

is evaporating, your impedance continues

to decrease. And from this value of the

impedance and these values as the function of

time, you can detect the analyte concentration. And in fact, if your analyte

concentration is lower then, of course, your impedance

would be a little bit lower. And– But the point is,

our impedance would be, if your analyte concentration is

higher then your impedance would be lower. But the bottom line is, using

this approach, you may be able to detect tens of attomolar

approach because this has sort of allowed you to

concentrate on a single point, thereby beating the

diffusion limit. And this actually

gives fantastic, fantastic sensitivity. [Slide 13] So, let me summarize then,

bringing us back to the old plot that the way the droplets with evaporation allows

enhancement of sensitivity is that it starts at a very

low concentration. Evaporation essentially

pushes it– pushes the concentration

at much higher value so then you can use the planar

sensor or a nanowire sensor. It doesn’t matter. In that case, you’ll have very

high sensitivity beating the diffusion limit. [Slide 14] So again, this is a summary

which sort of tells us that these approaches

essentially allows you to approach very low

concentration levels that classical diffusion

limit prevents these classical sensors, standard

nanobiosensors from reaching. [Slide 15] So let me conclude. I told you about

superhydrophobic surface. Superhydrophobic surface

simply means something is so hydrophobic that the

ball is almost rolled up so that when the biomolecule is

brought closer to the surface, they’re all drop

at a single point. Unlike the coffee

ring which sort of spreads it around. Because then, you have to

put sensors all around. Here is much better because

it will bring everything down to the point where you

really want to sense it. Now, I told you about how

droplet allows pre-concentration to beat the diffusion limit. And these type of superhydrophobic surfaces

are relatively easy to design and therefore, this type of sensors can be pretty

interesting in terms of being able to beat

the diffusion limit. So, that’s it. In this lecture, I told

you about one strategy, the second strategy of

beating the diffusion limit. In the next lecture, I

will tell you a little bit about that how there

are other approaches that beats the diffusion limit. But you may have notice that

I have not really told you how if you have a fluid

flow, flow the fluid, bring the analyte faster to

the sensor’s surface whether that can give you a significant

enhancement in sensitivity. The surprise is that it doesn’t. And I’ll explain to

you in the next slide that while it beats the

diffusion limit to some extent, there’s not dramatic

enhancement. And from that then

onward, we’ll be able to understand how the

different techniques of beating the diffusion

limit compare. So until next time. Take care.