WEBVTT

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PROFESSOR: So let's look
at a little more detail

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at the equilibrium size
of respiratory droplets

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that are emitted
during breathing,

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or coughing, or speaking.

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And the key idea is that these
droplets are not pure liquid.

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As explained in the wells
curve, pure droplets

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that are small enough
will shrink completely

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and evaporate.

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However, these droplets
contain a significant amount

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of solutes.

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And those solutes, in the case
of mucus coming from your lungs

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or from your vocal cords,
your nasal pharynx,

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are full of proteins and other
macromolecules, carbohydrates.

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And also there are always
in bodily fluids plenty

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of dissolved salts such as
sodium and chloride or calcium

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or potassium ions.

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Also, in saliva, many of
these species are present,

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although it's not quite
as thick of a liquid.

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And of course, virions as well
will find themselves in here,

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and it also constitutes solutes.

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So the idea is that we don't
just have a pure liquid.

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So there is some
initial volume fraction

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of solute in the liquid.

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And in addition to that,
most of these liquids --

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sorry, the solutes
I should say are

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charged and thus hygroscopic.

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What that means is that,
of course, the salt,

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those ions are literally charged
species, but also the proteins

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and other
macromolecules that many

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charge residues and sites
along the molecule, which

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attract water.

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And essentially,
there is a layer

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of bound water solvating all of
these species I just mentioned,

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including the virus.

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So I'll just sketch
that there's lots

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of bound water, which is
surrounding each species,

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including the virus.

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There's essentially
a layer of water

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mostly around these molecules
here and other species.

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So it's solutes
plus the bound water

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that's coming from solvation
of these molecules in liquid.

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So that water is
pretty firmly attached.

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And even if you dry the
material, a lot of that water

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will still be left over.

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It takes a significant amount
of energy to remove it.

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And so if we think that --

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if we describe there is initial
volume of the droplet V_O,

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and a radius R_0, so
let's say it's initially

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a circular droplet, there
is an initial amount

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of solid, V_s, which is
phi_s^0 times V_0.

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So there's a certain amount of
solutes in there which cannot

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be removed.

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So the water can evaporate,
but the solutes will not.

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So let's think a
little bit about what

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the consequences of that are.

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So let me do a brief
derivation here

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for looking at the
thermodynamics of this system.

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And the key idea is just
to get to the final result.

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I don't want to dwell on the
details of thermodynamics.

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But an important concept
here is the relative humidity

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of the air.

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So there's moisture in the air.

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There's water vapor.

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And it's at a certain level.

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So we often write that
rh for relative humidity.

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And that can be defined as
the concentration of vapor,

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water vapor in the air relative
to the vapor concentration

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that would be in equilibrium
with pure liquid.

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So when the concentration
water vapor gets high enough,

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eventually you start to
nucleate water droplets.

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And you start to have
condensation water.

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That's essentially how
rain forms from the clouds.

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So that's that ratio.

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So relative humidity
is telling you

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how close you are to basically
having, water liquid water come

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out of the air.

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OK, now, the relative
humidity also tells us

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something about how far you
are from that phase transition

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point.

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And there's a very
simple approximation.

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I'll put approximate here.

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We can also write that
this is -- scales with, and is --

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can be in fact close
to the liquid volume

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fraction in equilibrium
inside the drop.

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So that'll be the water volume
fraction of water liquid

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inside the droplet.

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At least you can see here in
this relationship when this

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volume fraction is
one -- in other words,

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we have pure water --

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then the relative
humidity is 100%.

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OK, and on the other hand, when
you have, let's say, only 50%

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water, over here, that's like
having relative humidity 50%.

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This can be derived by
more careful consideration

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of the ideal entropy of
mixing where essentially

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this term here is take
into account the excluded

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volume and the fact that all
the sites in this droplet

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are not available for the water.

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So they're being excluded by all
the solutes and the bound water

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that are present.

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And similarly, we have
a buildup of free energy

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in the bulk as well.

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So basically, this comes
from some thermodynamic

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considerations of equilibrium
between water vapor

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and water liquid.

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We can write this as 1
minus the volume fraction

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in equilibrium of the solid.

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OK, now the thing is that
we can now write this.

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So if we multiply through, we
can write this as 1 minus the --

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so what is the volume -- so
when we get to equilibrium,

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this droplet is going
to change its shape.

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It's going to reach
a new shape, which

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we're going to calculate our
new volume, V_equilibrium.

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And so what this would
would be V_solid,

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which is phi_s^0*V_0
divided by V_equilibrium.

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So it's going to be new
volume, V_equilibrium, which

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will be achieved then.

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And then we'll end up with the
equilibrium volume fraction.

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So if I take these
equations here,

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and I solve, I get a
fundamental result, which

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is that the equilibrium
volume of the droplet

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relative to the initial
volume is equal to,

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well, we have put this
on the other side.

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That'll be (1-RH).

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And we divide that out.

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And we find that it's the
initial volume fractions

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solutes divided by (1-RH).

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That is our key result. And
let's plot what this looks

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like.

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So if we prop the
relative humidity

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on the horizontal axis,
from 0 to 100%, so at 100%,

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the water vapor is
saturating the air.

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And you would start to
nucleate and condensed water

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liquid from that.

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At 0, the air is completely dry.

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And there is essentially
no water vapor present.

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So that's the range.

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And typical comfortable rooms
have a relative humidity

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around 50%.

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This is a typical number.

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And let's plot on this axis
the equilibrium volume.

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It could also be the
equilibrium radius

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because I should say that
if there are spheres,

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that this is also equal to
R_equilibrium divided by (R_0)^3.

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So I can also take
a cube root of this.

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And I would have
the ratio of radii.

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So we would know if we started
a certain radius, what's

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the final radius.

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OK, so we can talk about volume.

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We can talk about radius.

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So here is the initial
size of the drop.

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And somewhere down
here is V_s,

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which is the solute volume,
which is phi_s^0*V_0.

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Now what is this value?

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It depends on the
kind of liquid.

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So saliva is mostly
water with some salt

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and a few other molecules.

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But in saliva, the
volume fraction phi_s^0

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is 0.5% in saliva.

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OK, so that's just
gives you a sense.

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So this is quite
far down, right?

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But then, if you
look in mucus, it

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depends which mucus
you're talking about.

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But the mucus that comes from
the lungs or from the pharynx,

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it can vary.

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But what has also been
measured in droplets

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that are emitted by breathing
is that this can range anywhere

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from 5% to 10%.

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So a fairly significant amount
of the volume of the droplet

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is containing all these
molecules and the bound water

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around that.

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Now we know that because
mucus is very sticky.

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It's a non-Newtonian fluid.

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It doesn't maintain a
nice round shape even.

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It can have a regular shape.

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It flows slowly.

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It has a high viscosity.

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And that's because
it has a large amount

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of these hygroscopic solutes.

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So mucus might be a
little bit higher up.

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But in any case,
what you then find is

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we can sketch different
regions of this plot now.

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So this curve this
formula drive here,

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when the relative humidity
is zero, we start here.

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So that's saying when
there's no water in the air,

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you completely dry the droplet.

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And you're left just with the
molecules, the solid molecules,

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and possibly the
bound water around it,

00:10:19.300 --> 00:10:21.790
depending on how dry
the air actually is.

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And then it rises up
and blows up at 100%.

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So when you get to
100%, then droplets

00:10:29.260 --> 00:10:31.660
are getting really large.

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And if you actually
hit 100%, then you

00:10:33.250 --> 00:10:35.050
can't really speak of
an equilibrium size

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because you'll just start to
get lots and lots of water.

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So that's that limit.

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And so now we can look at
three different regimes

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of the kinds of droplets
that we'd expect to see.

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So down here at 0%
or close to zero,

00:10:52.420 --> 00:10:59.770
we have a dried droplet
nuclei as they're called

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in the public health field.

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These respiratory aerosols,
if they completely dry out,

00:11:05.350 --> 00:11:07.600
and you're left with just
these solutes, then that's

00:11:07.600 --> 00:11:08.640
called a droplet nucleus.

00:11:08.640 --> 00:11:10.100
So it doesn't
necessarily mean it's

00:11:10.100 --> 00:11:11.710
a nucleus for phase
transformation

00:11:11.710 --> 00:11:15.100
as we use that term in, say,
engineering or in physics.

00:11:15.100 --> 00:11:18.190
But it's really just
refers to the core

00:11:18.190 --> 00:11:21.190
of just the hydrated solutes.

00:11:21.190 --> 00:11:25.530
So if I could sketch what that
looks like, that would be--

00:11:25.530 --> 00:11:27.160
for example, all
those molecules I just

00:11:27.160 --> 00:11:29.830
sketched there
might be condensed

00:11:29.830 --> 00:11:31.930
into some little blob,
which, by the way,

00:11:31.930 --> 00:11:34.450
could include a virion.

00:11:34.450 --> 00:11:36.880
In fact, it could even
be just one virion

00:11:36.880 --> 00:11:39.260
if that were all
that were in there.

00:11:39.260 --> 00:11:43.570
And you would have a little
bit of bound water around it.

00:11:43.570 --> 00:11:48.610
But you essentially have a dried
up blob of just the solutes.

00:11:48.610 --> 00:11:50.560
OK, and so that --
and then, of course,

00:11:50.560 --> 00:11:54.730
the smallest volume you can
get is just the initial solute

00:11:54.730 --> 00:11:58.690
volume that you started
with, plus the bound water.

00:11:58.690 --> 00:12:03.130
On the other end, if we are near
100% relative humidity, then

00:12:03.130 --> 00:12:06.100
the fact that these are
hygroscopic solutes, which

00:12:06.100 --> 00:12:09.400
like to have water near them,
will form as a nucleation site

00:12:09.400 --> 00:12:11.440
to actually cause
more and more water

00:12:11.440 --> 00:12:14.560
to grow and be absorbed
into this droplet.

00:12:14.560 --> 00:12:17.710
And not only do the
droplets not shrink,

00:12:17.710 --> 00:12:21.970
as predicted by the Wells
curve for a pure liquid,

00:12:21.970 --> 00:12:23.480
but they can actually grow.

00:12:23.480 --> 00:12:25.880
So if the size here is
small enough to begin with,

00:12:25.880 --> 00:12:29.190
let's say it were a several-micron
droplet to begin with,

00:12:29.190 --> 00:12:30.940
but it contains a lot
of solutes, and we're

00:12:30.940 --> 00:12:34.480
at very high humidity,
actually the particle can grow.

00:12:34.480 --> 00:12:40.350
So over here, we could end up
with an even larger droplet

00:12:40.350 --> 00:12:43.210
than we started with where
now because the humidity is

00:12:43.210 --> 00:12:45.630
so high, and we have the same
number of molecules in there

00:12:45.630 --> 00:12:51.280
that I sketched before,
that's more dilute now.

00:12:51.280 --> 00:12:54.660
And there's maybe a virus
or a virion here and there.

00:12:54.660 --> 00:12:58.680
And of course, there's
also some salt.

00:12:58.680 --> 00:13:00.640
But basically, the
droplet is growing.

00:13:00.640 --> 00:13:11.290
So here we have
hygroscopic growth

00:13:11.290 --> 00:13:18.500
and also we have what's
called deliquescence, which

00:13:18.500 --> 00:13:21.950
refers to water that's absorbing
around these salt molecules

00:13:21.950 --> 00:13:24.410
and even causing some
other molecules or charges

00:13:24.410 --> 00:13:26.800
on these macromolecules
to dissolve into solution

00:13:26.800 --> 00:13:28.220
because it's more and more
water present

00:13:28.220 --> 00:13:30.450
and it can solvate more species.

00:13:30.450 --> 00:13:33.260
And so whereas
hygroscopic growth

00:13:33.260 --> 00:13:35.720
refers to water being
absorbed into a more

00:13:35.720 --> 00:13:37.340
solid-like framework,
you can also

00:13:37.340 --> 00:13:40.910
be generating more aqueous
solution, which is deliquescence.

00:13:40.910 --> 00:13:42.980
So basically, the droplet
can actually grow.

00:13:42.980 --> 00:13:46.790
And that would be like when
you're here, let's just say.

00:13:46.790 --> 00:13:49.680
And this might be
when you're here.

00:13:49.680 --> 00:13:52.530
And then, of course, when
you're at 50% relative humidity,

00:13:52.530 --> 00:13:54.360
you can see the
droplet has shrunken

00:13:54.360 --> 00:13:58.230
but not all the way down to the
initial solute volume fraction,

00:13:58.230 --> 00:13:59.760
but something larger.

00:13:59.760 --> 00:14:02.400
And in fact, if the
relative humidity is 50%,

00:14:02.400 --> 00:14:05.650
you end up at exactly twice
the solid volume fraction.

00:14:05.650 --> 00:14:09.070
So if the solid volume
fraction of mucus is 10%,

00:14:09.070 --> 00:14:10.950
you may end up
with a droplet that

00:14:10.950 --> 00:14:12.550
is maybe 20% of the volume.

00:14:12.550 --> 00:14:15.900
So maybe it looks
something like this.

00:14:15.900 --> 00:14:21.200
OK, and so we have a little
bit of shrinking going on.

00:14:21.200 --> 00:14:24.260
And maybe there's even a
virus in there as well.

00:14:24.260 --> 00:14:26.870
But there is still
plenty of water.

00:14:26.870 --> 00:14:31.040
And so you can see also now
the value of having solutes

00:14:31.040 --> 00:14:35.810
in mucus in terms of making the
virions more viable and more

00:14:35.810 --> 00:14:38.960
easily transmittable because
they hold onto the water.

00:14:38.960 --> 00:14:41.820
So that the virion is
in a stable environment.

00:14:41.820 --> 00:14:44.990
So that when it ends up being
inhaled into someone else's

00:14:44.990 --> 00:14:47.600
lungs that it can
then more easily

00:14:47.600 --> 00:14:52.220
diffuse out of that region
and infect the host cells.

00:14:52.220 --> 00:14:55.010
In contrast, if you have
a nearly pure liquid

00:14:55.010 --> 00:14:58.290
that the virion is in, let's
say pure water or even saliva,

00:14:58.290 --> 00:15:00.620
which is actually
mostly water, then

00:15:00.620 --> 00:15:04.010
the droplet will shrink
by a factor of 100.

00:15:04.010 --> 00:15:07.520
And it might be just literally
a virion with a [couple of

00:15:07.520 --> 00:15:10.270
ions] just enveloped

00:15:10.270 --> 00:15:11.390
with a tiny bit of water.

00:15:11.390 --> 00:15:16.640
And maybe that, in some
cases, would be not as viable

00:15:16.640 --> 00:15:18.860
of a situation for the virions.

00:15:18.860 --> 00:15:21.170
So basically the
mucus fragments are

00:15:21.170 --> 00:15:25.460
likely to be the more common
source of the aerosols that

00:15:25.460 --> 00:15:29.440
will stay in the air
and remain infectious.