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MICHAEL CIMA: But let's just start with looking at this spherical micelle
for a second. And we ended with this.
What's the typical size? Well a spherical micelle typically is on that
range of nanometers, 3 to 50 nanometers.
And, obviously, it depends on how big the surfactant is.
And what that means is that these spherical micelles have aggregation
numbers, where this means the number of molecules involved in them.
Anywhere from 10 to 100 or so molecules actually spontaneously get
together and aggregate. That's where these ionic surfactants--
remember we also talked about non-ionic surfactants, where there
isn't an ionized group at the end, but just a polar group like polyethylene
glycol or something like that. And those can get huge aggregates, thousands
of molecules in them. Now, of course, they may not be spherical
at that point. They may look more and more like this, as
we'll see in a second. And the other thing that's, in general, different
between these two types of surfactants is just how much you
need to create these micelles. The critical micelle concentration for an
ionic surfactant is anywhere from as small as about 10^-3 molar, or one millimolar,
to 10^-3, minus 2, 10 millimolars.
Whereas a non-ionic surfactant, you can get micelles that's extremely low
concentrations-- 10^-4 molar kind of thing.
OK, so what goes into determining these shapes? And I thought this was particularly interesting
because it's actually a very simple structural phenomenon.
So let's say we have a polar group. And this is the nonpolar group.
This little tadpole-looking like thing. And, well, if you think about it, this structure
here, each one of these polar groups occupies some sort of area on
the surface, exposed area a. And that's going to depend on how big this
water soluble or polar group is. And then I have to ask, well what volume is
in the center? And that's going to depend on the volume of
this group here. On my notes, I did it in color.
But let's do it here. Well, first off, this thing has some length.
I'll call it I_0. And it may occupy a volume that's not a simple
function of this, because this can fold up or something like that.
But we'll call that volume that it occupies V0.
So a_e is defined as the area per molecule at the surface of micelle--
one c. And V_0 is the tail volume.
And I_0's the tail length. So then it ends up being a pretty simple matter
to relate the radius of this micelle.
You get it. So this is, let's say, R, big R. Well the
volume of the core of the micelle is going to equal Z_A, the aggregation number,
times the volume of the tail. And that's got to equal to 4/3 pi R cubed.
Now the surface area has got to equal Z_A times a_e.
Right? The area for each one of the molecules on
the surface. And I think you know what I'm going to do.
It's going to be 4 pi R squared. And what I can do then is take this expression
divided by that one to solve for R. And you get that R has to equal
3 V_0 over a_e, just by dividing one equation by the other.
Or I can solve for the following ratio. That means that 3 equals R a_e over V_0.
Now, of course, this is all for a sphere. These equations are for a sphere.
Come back to that in a second. Now I have one another parameter here, this
length of the tail. And the length of the tail, I've stretched
it out there. But it isn't necessarily that long.
Right? It could fold over on one another.
And so that means R is less than or equal to I_0.
So if I plug this in, this inequality into here, you end up with--
in other words, if I'm going to make a sphere, I require that the head be
small enough that this ratio ends up being less than 1/3. Now it's not exact.
Right? Because it's a less than.
But it's interesting because if you do the same type of
analysis for a cylinder-- STUDENT: Professor?
MICHAEL CIMA: Yep? STUDENT: Why wouldn't R be less than or equal
to 2 times I0 if it's two different--
MICHAEL CIMA: Oh, did I do this wrong? STUDENT: [INAUDIBLE]
MICHAEL CIMA: What's that? STUDENT: [INAUDIBLE]
STUDENT: R is representing the diameter, not the radius.
MICHAEL CIMA: Oh, thank you. R is the diameter.
That's a silly thing. STUDENT: [INAUDIBLE]
MICHAEL CIMA: Oh, just make it like this. OK, 2R.
Thank you. All right.
Yes, can I help you? OK.
And then I'm going to do a bilayer here. STUDENT: Professor?
MICHAEL CIMA: Yep? STUDENT: Why is R greater than or equal to
I0 now? MICHAEL CIMA: Oh, I changed it twice?
Yeah, R is less than. Sorry.
I forgot to change it back. All right.
And so what am I doing here? Well what I did was, if you just go through
the same sort of simple argument for, instead of doing a sphere you
do a cylinder, you do a bilayer, you end up proving to yourself that
as a_e gets smaller, if everything else being the same, if a_e gets
smaller, then you can't satisfy this.
So instead of making spheres, you'll make a cylinder.
And then if a_e gets still smaller, you make layers.
And it's just simply packing everything into these shapes.
At some point, a_e compared with the volume of the tail will force you, as
you decrease its size, will force you to make a layered structure.
So in other words, the relative sizes of the polar and the non-polar group
determine what kind of micelle you produce. So, I mean, I guess you could do this the
other way, a_e getting bigger. The larger a_e is, the more likely you're
going to get spheres. And I guess you can do the same thing.
Keep everything else the same and increase V_0.
You'll go from spheres to cylinders to layers. And so that's going to lead into what causes
these structures. And these are actually solid structures.
They're not liquids.