Hello everyone, welcome to another lecture for Drug Delivery Engineering and
Principles. Let us do a quick recap of what we did in the last class.
(Refer Slide Time: 00:36)
So, we are continuing our discussions on hydrogels, as we discussed, what are
hydrogels? Hydrogels are these highly hydrophilic gels, that are made from hydrophilic
polymers and they have the capability to absorb lots and lots of water and they can swell.
So, why they are very important is because first of all they are very similar to how the
tissue normally is. So, if you look at the tissue lots of ECM components are hydrogels
themselves, they are extremely hydrophilic and they are encapsulating all kinds of cells
where cells then use various processes to synthesize more ECM as well as do their
So, within the hydrogels, we had discussed quite a lot of variations of them, we
discussed physical, we had discussed the chemical and then we also discussed ionic
hydrogels and then we further went ahead in the last class and we talked about in situ
crosslinking gels. And so what are these in situ crosslinking gels? It means, that they
crosslink at the site and interest of the site for our application is the human body or
So, in that sense; that means, that when we inject these hydrogel precursors, which are
typically in liquid formulations into the body, they will crosslink right then and there and
this could be on the basis of several thing, this could be on the basis of time, this could
be on the basis of temperature, so all of these are fairly feasible. And then we discussed a
paper in which maleimide was used as one of the crosslinking method for making
So, these are PEG based hydrogels that we talked about and maleimide was one of the
chemical group that was reacting with thiols. So, this was PEG which is reacting with
peptides, that had thiols on both ends and that and this PEG was of course, the four arm
PEG with maleimide at the four ends.
And that essentially led to crosslinking and what we found was basically there were
several chemistries that can lead to the production of the hydrogels, but among the four
five moieties that this group had compared, they found that maleimide was by far the
most efficient as well as well suited for any in situ applications as well because the gel
was then able to cross link and then adhere to the tissue as well.
So, now that we have described all this we are going to discuss some of the maths today
as to how we can compute what is the pore size, what is the different formulations that
leads to different pore sizes. So, how do we go about doing that?
(Refer Slide Time: 03:19)
So, for the purpose of this class we are only going to talk about neutral hydrogels and we
are going to define three of the terms and give the equations as we go along in today. So,
the first thing is the Flory Rehner theory and what it is? It is basically a theory that
defines how the crosslinking of the polymers gel happens. So, what they are proposing is
there is a Gaussian distribution the polymer chains that is found in nature and that is a
good assumption to make given what we see in the literature. And the degree to which a
polymer network will swell is going to be governed by the elastic retractive forces of the
polymer chains as well the thermodynamic compatibility of the polymer.
And, so what essentially this means is let us say I have a polymer that is crosslinked. So,
now, as I said these are hydrogels and they tend to absorb water, but there is some
limitations to it, of course, thermodynamically its favourable for the water to go in and
interact with these polymeric chains. So, that is where the thermodynamic compatibility
of the polymer comes from, whereas, as it continues to observe water, these chains here
will get stretched.
So, eventually this is going to get to a formation like this and let us say if they continue
to stretch they going to get to formation like this, but eventually these chains themselves
will start to have tensions because; obviously, they are sort of crosslinked at these sites
and there is a limitation to the amount that these can be stretched. So, that is where the
elastic retractive forces come in.
So, now these chains are going to try to bring this polymer gel back to its original state
and so its the force balance between these two forces the affinity for the water to get
absorbed into the system, as well as the elastic retracive forces of these chains to sort of
resist that and go back to their origin state. So, essentially that is what will define is at
what stage at equilibrium the hydrogen will stop right. So, now, that we have that
(Refer Slide Time: 05:42)
So, therefore, the total free energy upon swelling will be given by what? Will essentially
be the total free energy will be equal to the free energy because of the elastic forces as
well as the free energy because of the mixing. Now, with respect to the number of
solvent molecules in the system, we are saying at a constant temperature and pressure of
course, we change this, the final state of the hydrogen will change, if you increase the
temperature maybe it will swell more, if you change the pressure maybe it will swell
more or less depending on what you are doing with the pressure.
So, we can then essentially convert this deltaG to delta μ which is the chemical potential
change and we can basically say that whatever change we have seen in the chemical
potential will be equal to the change in potential due to elastic as well as the change in
potential due to mixing and the mixing is of course, the thermodynamic mixing of the
water with the polymer chains. So, again as I said the μ1 is a chemical potential of the
solvent in the polymer gel as well as μ1,0 is chemical potential in the pure solvents.
So, we are assuming that it was formed in the pure solvent and then it is mixed into this
new solvent now, within the polymer gel now. So, but we know at equilibrium, it would
not really change its state. So, basically it would have attained already this balance
between these two forces. So, we know that at equilibrium, the left hand term which is
the change in the potential will become zero. So, that means, then the elastic and the
mixing chemical potential change will have to be equal to, so that they balance each
(Refer Slide Time: 07:32)
So, now that we have established that, let us see how we can define the delta μ for
mixing. So, this is of course, related to the heat of mixing and entropy of mixing. So, this
can be defined directly as delta μ mixing is equal to the RT with this expression. I am not
going to go into the derivation of all of these equations. So, where we know that the chi
is the polymer solvent interaction so, how the polymer and solvent are interacting.
So, this is going to change if you change the polymer, this is going to change if you
change the solvent. And then V2s we have already defined before is nothing, but the
polymer volume fraction of the gel. What about the elastic chemical potential change?
So for crosslinked gel in absence of any solvent is calculated by that rubber elasticity
So, of course, this case with we saying in the absence of any solvent is sort of
nonexistent because the hydrogel will only form when there is some sort of solvent you
cannot expect just the polymer chains to just mix because there will be no diffusion, no
sort of interactions happening. So, this is just an hypothetical case for now, but the delta
μ elastic for these gels will essentially be calculated by the rubber elasticity theory and
this can be expressed as delta μ elastic is then expressed as this term where again now we
have defined couple of terms. So, this is the specific volume of the polymer.
So, what is the specific volume of the polymer itself V1 is the molar volume of the
solvent, so it then define this how much of the solvent is there. Mn is the molecular
weight of the linear polymer chain, so this is essentially how big your chain is of course,
that is going to help with that and what is Mc? Mc we had defined before is that distance
between the crosslink.
So, essentially as we discussed in the last slide, they have to balance out each other since
these two terms should balance at equilibrium. We can then equate and sort of solve for
what is the Mc with other terms and again Mc was the length between the cross links, the
average length between the cross links.
(Refer Slide Time: 09:56)
So, now that we have this mathematical term; as we already said in the previous slide
that this is in the absence of any solvent and this is not really any physiologically
relevant, this all the crosslinking in the hydrogel you will see is going to be in presence
of some solvent.
So, now if I add the solvent and try to correct this equation to reflect that, what I will do
is, this was done by modification to the Flory Rehner theory by Peppas and Merrill and
what is done is we have to include the volume fraction density of the polymer chains
during crosslinking. So, we have defined a new term called v2r which is essentially the
volume fraction of the polymer in that relaxed state and when I say relaxed state, what is
the relaxed state? So, let us say if I am doing the synthesis of the hydrogel in let us say
100 micro liter of a solution and the hydrogel is formed. Then as when it is immediately
formed what is the sort of the volume polymer fraction at that state.
So, this is a relaxed state, which essentially is just formed. Obviously, as more and more
time is given this, because it wants to interact more with the water, will essentially starts
to swell and at some equilibrium it will stop. So, we can call this one as a swollen state.
Or alternately what we can do is we can let it dry off, we can use some high vacuum or
we can just dry it in the air and so this is dry over time, this was time in water. So, if I let
it dry off these polymer chains will further collapse and this is actually nothing, but dry
So, we have just defined relaxed state, swollen state and dry state; obviously, in dry state
if we are saying that we completely dry it off, that there is no water present what will that
give you that will essentially give you what amount of weight did you actually started
with. So, so these are the couple of terms, but basically that previous equation for μ
elastic then gets corrected by this term of v2r, so which is then reflected here.
(Refer Slide Time: 12:53)
So, again relax state is basically immediately after crosslinking, but before swelling. So,
then we can now equate the delta μ mixing with the delta μ elastic like we did in the last
slide just because it has to be balancing each other out at equilibrium state and then what
we get is essentially, a modified equation which then defines Mc as a function of several
other parameters that we can compute or determine using various techniques.
(Refer Slide Time: 13:28)
So, continuing further with the rubber elasticity theory, so hydrogels, these are again
similar to natural rubbers. So, the ability to deform is high under low stresses and can be
changed, so they can be considered to be elastic essentially. So, right now we are talking
about how this rubber elasticity came up. So, we are saying that for deformations which
are less than 20 percent, we can say that these are elastic deformations and this was
originally developed by Treolar and Flory for rubber and then was later modified by the
Flory to reflect it for hydrogels.
And again as I said that, that theory was developed initially to reflect the change in
absence of solvents and then it was further modified by Peppas and Merrill and with the
rubber elasticity theory, they are saying that if tau is the stress applied to the polymer
sample and rho is the density of the polymer itself, then you can sort of equate them on
the basis of this equation.
So, now we have another equation, so tau is something that we are applying the stress
for, so we can use some instruments like rheometer and all to know what is the tau and
that will give us another equation along with the last equation to be able to get some idea
as to what are the different Mc and v2s and v2r. And then again alpha here represents the
elongation term, so this is basically the elongation due to the applied stress and as I said
the tensile testing apparatus can be used to measure these rubber elasticity for hydrogel
(Refer Slide Time: 15:17)
So, now to do what is the pore size calculation which is what eventually we want to get
at. So, again this is the one of the most important parameters in controlling the rate of
drug release from hydrogel right because if I have for the same drug and for the same
concentration if I change the pore size from let us say 10 nanometer to 15 nanometer,
now it will be much easier for drug to diffuse out from the hydrogel network, then it will
be when the pore size is low. And so that is one of the most important parameter that we
have to define. I cannot really try to deliver let us say a drug which is 15 nanometers in
diameter through a hydrogel that only has pores of 10 nanometer because those drug
molecules will not be able to even physically come out from the hydrogel.
So, this is something that we definitely need to define and then we can directly measure
that using some techniques like electron microscopy or light scattering, the problem with
the electron microscopy is this is typically done in high vacuum. So, even though you
will get some idea as to what is the pore size, but then since this is done in high vacuum,
this is essentially giving your pore size in dry state. However, when we put this hydrogel
in the body or use it at for any biological applications, its always going to be in aqueous
And now since it is in aqueous media this pore size will change because it has much
more water around it there is going to be swelling and all that and so the value you will
get is not going to be accurate, so you cannot really trust this high vacuum data for the
pore size. Even though you can then use this to compare between different hydrogels, but
that comparison will also have some caveats to it.
So, as we said the molecular weight between crosslinks is an indirect measure to the pore
size right I mean if I am saying that, if this length is Mc as this length increases the pore
size will also increase. So, it is some measure for that, so the mathematical expression of
the pore size is then given by
2 1/ 2
So, basically the root mean square of that which is the unperturbed end to end distance
between the polymer chains. So, essentially this is Mc itself, but define in some other
way, so that its always positive.
(Refer Slide Time: 17:55)
So, now assuming that and assuming that there is isotropic swelling of the gel, so; that
means, that let us say if this is the swelling, then its happening isotropically, which
means that all these lengths will increase rather than just having an increase like this, I
am saying that this is wrong, but this is correct, this is isotropic swelling and that is a
good assumption to make unless there is some inhomogeneity in the gel.
So, if I assume that then what we can say is that the alpha which is the elongation is
nothing, but it is going to be proportional to the cube root, the negative cube root of the
v2s which is the volume fraction in the solvent. And then the r naught is calculate
through the flory characteristic ratio N which is then defined empirically like this where l
is the bond length along the backbone.
So, typically 1.5 to 1.7 angstrom for different polymers, here is an example of 1.4
angstrom for vinyl polymers and N is the number of links per chain. So, if I know what
is the bond length I am saying there are N number of links per chain which is related to
the crosslinking density as well. So, M0 being the molecular weight of the monomer,
will come this will turn into
So, essentially we are saying if this is the chain and between the backbone, each bond is
let us say 1.54 angstrom this for the vinyl polymers and is the number of links per chain.
So, how many links of such are there, then we can sort of define N as the 2M c by M 0,
where M0 is basically the total molecular weight of the chain.
So, if I combine these two equations the pore size of the swollen networks and then we
expressed as, so, all I am doing is recombining the previous equations with these
equations and what I will get is a value for the pore size defined with some other terms
and that is how I can then find what is the pore size of my hydrogels.
So, quite a complex calculation up to here, but then I will give you an example now of
what it is typically done in the field if you are making a hydrogel, which will make it
(Refer Slide Time: 20:35)
So, here is a specific example. So this is specific example is looking at polymerization of
two different types of PEG. So, you have an oligo PEG fumarate and a PEG diacrylate, it
is again a radical reaction you use some initiator, so in this case ammonium persulfate.
And what will happen is these radical bonds will start to react with the PEG fumarate
essentially leading to a crosslinked network which is defined here.
So this is of course, example from Miko’s group at Rice University in US and what they
have done is they form hydrogels which they have called OPF hydrogels. And the PEG
diacrylates can be used by a radical initiator like ammonium persulfate ithout the need of
UV light and then the ester bonds ester bonds at a form which are again hydrolyzable.
So, all of these ester bonds, these can be degraded by water and the original purpose of
this hydrogel is to kind of use it as a tissue mimic sort of use as controlled release
potential as engineering matrixes at all.
(Refer Slide Time: 21:50)
So, let us see what they have done in terms of characterizing, what is the pore size of this
particular hydrogel that they have formed. So, what they did is for the PEG they have
different molecular weights PEG that they have use. So, they form three different types
of gels defined as PF10K, PF4K, PF1K and that is essentially telling you what is the
length of the PEG that they used.
And then what they did is just to form the gel they use these cell culture plates which
typically have wells and depending on how big that cell culture plate is, these wells can
be 10 millimeter, 16 millimeter or something else, in this case they have used 10 and 16
millimeter discs. And then once they have formed it, so that is the relaxed state and then
what they have done is then taken these discs because now the hydrogel acquires
whatever shape was there and this is a disc shape. So, they are now taken these discs and
placed it into a very large volume relatively compared to the gel size of water at 25
And then what they did is over time they have done experiments. So at the time the disc
were formed at time 0, they have weighed that and then they have continued to weigh it
at different times right and then they have also measured the diameter which is basically
using calipers already defined here is 10 millimeter or 16 millimeter.
So, now how do you calculate the swelling ratio, so you are saying that if it is swelled it
is because lots and lots of water had went in. So, we are saying that because the water
went in, after swelling there is a certain weight Ws and before swelling there is a certain
weight Wd and then the ratio is basically how much it has swelled. If it swelled twice,
then what has happened is now we have a swelling ratio of one, so it has become double
And so now they have determined and what they found is if they are using a 1K, 4K or
10K they find that the swelling ratio is different and notice how as you are increasing the
molecular weight of the chain, this swelling ratio is dramatically increasing, right, it is it
is become all the way up to 3 as you have increased the molecular weight of the chain
that you are using.
(Refer Slide Time: 24:13)
So, now they can apply the Peppas Merrill equation for Mc and again just reiterating it
here; here is the tau right. So, now, if they want to apply that there are several parameters
that they do not know, what is v2s, what is v2r, so these are easy to find out. So, what
they can do is, they can use the hanging pan and measure this, so v2r is nothing, but
weight of the polymer divided by the weight of this term and then similarly Vp which is
the one of the polymer is nothing, but how much you have added the water divided by
the density. So, once you apply all these equations and get some of the values from
different parameters reported in the literature, you can do this in different solvents as
well, you can do this in hexane.
(Refer Slide Time: 25:06)
And so what you end up with is essentially applying and fitting all these value you get an
Mc for different gels. So, you have PF1K, PF4K, PF10K and notice how the molecular
weight between crosslink has also increased. So, it does went all the way from 2000
grams per mole to 6250 grams per mole.
(Refer Slide Time: 25:33)
And then once you have that you can just directly put this into the pore size calculation
and essentially just apply this throughout, where we have 1.47 is the angstrom length
between these polymer bonds in the backbone.
(Refer Slide Time: 25:5)
So, if you apply that what you will find and Cn is basically taken from the literature and
what you will find is the pore sizes increases from 76 angstrom which is essentially 7.6
nanometer to 16 nanometer. So, this sort of reiterates if that if you increase the molecular
weight of the polymer itself you will get a higher pore size.
So, we will stop here, these are some of the terms that we have used to define what is the
pore size, what is the molecular weight between crosslink. Lots of complex equations
here we have assumed some of the things here as well as we have taken this equation
literature and not went into the derivation. So, this is just some of the ways that you can
determine what is the pore size of a hydrogel that you are making or using. We will
continue in the next class.
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