 back to the course on polymer chemistry and in today's class, we will discuss, we will basically continue our discussion we are having in polymers in solution and the topics we plan to cover today. This lecture is you know cohesive energy density and solubility parameter, how we can use the knowledge from solubility parameter and we can estimate it or we can predict the solubility behavior of a polymer in a solvent and then we will follow up our discussion with phase separation behavior of polymer solutions. Now, let us begin just recapping few minutes what we discussed in last class or last lecture. Now, this is what we found, we derived that Flory Huggins equation where you know this is this energy of mixing given by two terms. This term was coming from purely combinatorial entropy and this is always negative. So, this is a always favors mixing. This has always a gives you negative contribution towards gives energy of mixing. So, is always favors mixing and this is coming from a combination of enthalpy and entropy change due to polymer solvent contacts and it will decide, because it is always negative and always contribute in favor of mixing. So, the value of this basically determines where polymer will mix or become soluble in a solvent or not and if the value becomes less than 0 negative. Obviously, mixing will happen if it is positive and large no mixing and if it is slightly positive small I did not change from last. So, this is if it is slightly positive and the value is small then it depends between the competition of these two to we shall decide whether the polymer will go into solution or not. In dilute polymer solution we have found that the excess contribution excess chemical potential we got this relationship and at the temperature this term the excess this term the excess contribution of excess chemical potential becomes 0 and the solutions become ideally. If you look at this expression if this contributes negatively obviously mu 1 this difference would be lower than the ideal that means the polymer will the solution the solution or mixture of polymer and the solvent will be favored if this is positive then it will be higher than the value of ideal. So, it will be it will have effect against solution. So, theta temperature the this becomes 0. So, the solution become ideal solution and polymer exist in solution with un part of dimension. Now, we will come in in a next lecture what is un part of dimension, but what does it mean in brief is as if the polymers are present as such without no interference or no interaction on the other polymer molecule or the solvent molecule which will discuss what the meaning of un part of dimension in details in in next lecture. Obviously as I said that when theta is temperature is theta temperature value of polymer solvent interaction parameter chi becomes 0.5. So, the solution we have the polymer and solvent we have like ideal solution if this is the if the value of chi becomes lower than half then the coil expands the polymer expands in solution which promotes mixing. Chi was if you recollect that this was entropy parameter. Now, to have this term negative for a positive entropy factor or parameter T has to be higher than the theta temperature to have mixing. Whereas, if the entropy factor is negative then T has to be lower than the theta temperature to have mixing possible. And the other cases where the opposite case or reverse case where chi polymer solvent interaction parameter is higher than 0.5 it acts against mixing. Just to give few data for some polymer solvent the polymer solvent system having different values of i and theta. Let us look at some data of different polymers and with different solvents as we are mentioning. For this all these cases that interaction parameter value chi is less than 0.5 that means this is the all these examples we are talking about favoring mixing. So, as I mentioned if chi is positive then for mixing to happen temperature has to be higher than theta. And examples where there are strong interaction between the solvent and the polymer the entropy factor becomes negative. In that case the temperature has to be lower than the theta temperature to have the mixing possible. And these are the two examples where chi factor is negative and chi has to be lower than theta to get the mixing possible. Now, let us move to the one more topic on solubility which is determined by passive energy density and solubility parameter. Now, if you look back the Flory Huggins expression you have two terms. First term is from combinatorial entropy which you can predict theoretically if you know the volume fractions of the polymer and number of moles of the polymer molecules. The second term which actually determines whether the polymer will get solubilized in the solvent you need to find out the value of that. And there are some approach some some approaches from different different scientist has been made. And one of the important and important contribution from this towards this is by Hildebrand. Hildebrand who basically introduced the concept of cohesive energy density and relations to solubility parameter. This is a simple way to predict the solubility of polymer in a solvent. And it also can predict the miscibility behavior of polymer polymer system. And it also can because it if as it can determine or predict the solubility behavior of polymer it can also simultaneously detect the solvent resistance behavior of the polymer. It it mainly useful for as we will see that it is mainly useful for non polar solvent and non polar amorphous polymers. It is not very useful for polar molecules and and polar solvents. It it it is basically a semi-empirical approach as suggested by Hildebrand following the principle like dissolve like this means basically as we know that like the polymers which are a similar structure a similar similar sort of interaction behavior with a solvent will dissolve in that solvent. Quantitatively this cohesive energy density is is defined by this which is delta e v is the molar energy of vaporization. It can be written by this expression also where delta h v sub v is the molar energy of enthalpy of vaporization. This is molar energy of vaporization this is molar enthalpy of vaporization v is the molar volume of the substance. Now as you know cohesive energy is basically it it gives idea about the interaction or the the force of attraction between the similar type of molecule. So, if the cohesive energy density is higher than the attraction or the interaction between the similar molecules is higher. So, as we know from our previous understanding that cohesive energy is the energy of interaction between the similar molecules. So, higher cohesive energy density means higher attraction between the molecules. Now one Hildebrand defined the term solubility parameter which is basically square root of the cohesive energy density. So, it it is given by this expression and this is what we are looking for the enthalpy of mixing is given by if you know the values of the solubility parameter. Then the enthalpy of mixing can be obtained by this expression. We know the the contribution from the combinator entropy which is always favoring mixing. Now this is term will give you the enthalpy of mixing. The way this expression has been proposed it always gives the entropy of enthalpy of mixing. Enthalpy of mixing del H sub m is always positive or 0 if del down 1 is equals to del 2. So, it the drawback of this this approach is that it cannot predict any negative enthalpy of mixings which sometimes happen if there are strong specific interaction between the polymer and the solvent. The V m is the average molar volume of the polymer and and the solvent. So, if you if you if you think about the Gibbs energy of mixing which is delta H minus T delta S. So, obviously the lower is the value of delta H the better is the mixing and if it is negative then obviously it favors most. But in this case because it is positive or 0 then the case where delta H mixing would be 0 it will be most favorable for mixing and if it is positive then the lower is the value of delta H sub m it will be it will be better for mixing. So, as Hildebrand suggests that compatibility between component 1 which is the solvent and component 2 which is the solute in the polymer in this case arises as you have the less differences between the solubility parameter of the two components. So, if this they are same that is the base case where delta H m would be equals to 0. Now, we we have seen before that the polymer solvent interaction parameter can R is actually have two component the entropic component and enthalpy component and it can be shown that enthalpy component is can be expressed in by this expression as well. And this term entropic contribution towards the polymer solvent interaction parameter is most cases it has value of 0.2. So, if this is close to 0 or at least less than 0.3 then chi would be less than 0.5 which means it will promote mixing and which means the closer this the values of delta 1 and delta 2 then obviously the difference it will be nearer to 0 the chi H value would be nearer to 0. So, from this expression also you can you can you can predict that the polymer solvent interaction parameter would be lower than 0.5 which means the polymer will promote or it will dissolve in the solvent you are using. Now, as I said before that this works well for polymers or for a non-polar solvent and non-polar amorphous polymer it is not valid for crystalline polymers because you know this this expression or this approach we are talking about is talks about liquid solvent and liquid polymer means amorphous polymer. So, in if you if you want to dissolve a crystalline polymer then you have to also include the contribution for the enthalpy of crystallization. So, in this case that enthalpy of crystallization is not taken into account. So, that is why this is this approach is not valid for crystalline polymer this is valid only for amorphous polymer. In case of polar solvent you can have specific interaction between the polymer and the solvent. For example, it can have hydrogen bonding it can participate in electrostatic interaction and which we can think that if there is specific interaction like hydrogen bonding or electrostatic attraction between the polymer and solvent then there is a possibility that del H mixing would be negative and that as this approach solubility parameter approach does not predict any value of any negative value of delta H mixing. So, this is not valid for polymer non-polar as polar polymer and polar solvent usually it is usually valid for non-polar solvent and non-polar non-polar amorphous polymer and non-polar solvent. Now, how to get this delta value the value of the solubility parameter. Obviously, as we you can recollect this the delta can the delta is the square root of cohesive energy density and cohesive energy density can be found from molar energy of vaporization or molar enthalpy of vaporization. For liquids or the for solvents you can experimentally determine the molar enthalpy of vaporization and the molar volume. So, you can get the cohesive energy density and the value for the solubility parameter as well. But for polymers because they cannot be vaporizable vaporize or polymers are not vaporizable you cannot experimentally get the value of solubility parameter for the polymer because they are large in size. So, they are the value of delta the solubility parameter of polymers are typically determined indirectly and it is basically following the approach we took the delta H m becomes 0 as delta 1 and delta 2 becomes same. So, polymers are added if you are talking about linear polymers then the intensive viscosity of that polymer is measured in several solvents several common solvent. And the highest or the maximum intensive viscosity in the solvent which if the polymer gives we the solvents solubility parameter of the polymer is taken as the same as the solubility parameter of that particular solvent. So, basically this indirectly telling that the delta 1 and delta 2 are same. So, this gives you giving the maximum solubility or maximum intensive viscosity for that polymer. If the polymers are cross linked or gelled then we do the same experiment take the polymer and do a series of you know put the polymer network in series of solvents and find out the maximum swelling ability. So, if the where the interaction is maximum the maximum swelling happens. So, we take the solubility parameter of that particular solvent as the solubility parameter of the given polymer. The value of solubility parameter of polymers can be also indirectly obtained from group contribution method. As you know group contribution methods are utilized to find out theoretically the value for several other parameters, several other properties of polymers. And this solubility parameter also can be obtained by group contribution method and this is how it is done. There are particularly the solubility this is done theoretically delta is this f is the contribution form or the value for different groups. So, if you can divide or dissect the polymer into different parts that these values for f are available in literatures. So, you can add those and this is molar volume. So, you can also write this as in density and molecular weight. Now, f can be a f value of f is given by small and high and let us look at one example say we take PMMA polymethanethacrylate the structure of which is as you know this is the structure. So, how many groups it has you have to count groups there are two C H 3 groups. So, two C H 3 groups one C H 2 group. So, one C H 2 group one C O group and one C O group quaternicarbon. So, these are the groups there in this polymer and the values of these are supplied this the contribution of these groups are already tabulated or theoretically find out by scientist like small and high. So, you have to just take those values and add up to get the this value and then get the solubility parameter the f value which has been supplied in the literature for this is 3 0 3.4. So, this gives you 2 3 0 3.4 269. So, you sum up you get 69.5. So, delta would be given by 1 depleted the density 1.19 and why the molar mass mass of the repeat unit. Now, look at these values of the groups as I said that this is this gives this basically the solubility parameter is basically the square root of the cohesive energy density and cohesive energy density is nothing but a measure for the interaction or attractive interaction between the molecules. If there are polar molecules obviously, there are attraction between the the similar molecules become higher. So, the contribution become higher and if the volume becomes higher there is no polar groups the volume becomes higher there is more vulnerable forces between them which also increases the cohesive interaction between the groups. So, this C H 3 has higher than C H 2 than the quaternary carbon atom this is directly related to the volume the size of the group. So, as the size increases the Van der Waals forces increases. So, at Van der Waals attraction increases. So, the value of or contribution of these groups increases. So, like this you can get the value for this interaction parameter for any of the polymers because the value of these are supplied by available in the literature. Now, look at some of the values of the polymers and solvent common solvent like acetone, dintam and 20.3 C C N 6. Let us example of take few polymer like polystyrene this is some polymers polystyrene 18.5 and say this is only a polycarbonate is around 19.5 to 20. Now, obviously if you look at this number 19.5 to 20 this number 19.5 to 20 obviously look at this solvents. Now, as we have seen that if the delta value of the polymer is closer to the value of the solvent that of solvent then it will promote mixing or the polymer will be soluble in that solvent. So, polycarbonate will obviously become soluble in chloroform it is 19.5 and 19.19 and 19.5 obviously polycarbonate will not be soluble in methanol water or some of the very non-polar solvents. So, cyclohexane, cyc carbon tetrachloride, xylene and so on. Acetone 20.3 is basically this polycarbonate becomes soil you know it is not very good solvent for polycarbonate, but it is 80 it is not also very non-solvent either. So, the solubility of polycarbonate is very low, but it is not zero in acetone. So, it basically if you know the solvents and the value of polycarbonate you can predict you can estimate the whatever whatever you want you know which application for example, if you want to use some medical devices you know in medical devices like syringes and some other medical component where you in most cases before use they are sterilized by applying some spirit rectified spirit which contains alcohol. Now, if you know the solubility parameter of that alcohol then you know what are the you can estimate or you can basically zero down or narrow down the choices of your polymer which will not be soluble in that alcohol which will be used for wiping those or sterilizing those equipment. So, you can basically reduce your number of experiment which can which you can or your choice of polymers by a prior knowledge of solubility parameter and the sol value of solubility parameter solvent and polymer. Some cases this rubber gaskets are used for sealant in maybe in aircraft application where the sealing is very important. Now, obviously, they come or in case of car also the sealants are very important. So, they come in the contact of the petroleum vapors or they come in contact with this liquid fuel. Now, if those rubbers or the sealants which are made of rubbers actually get swollen then there would be leakage or there in that sealing method. So, you do not want your material rubber material to swell in that particular gasoline or liquid fuels. So, you should choose a polymer which will be non soluble polymer in that environment. So, if you have prior knowledge of your solubility parameter of the polymer and solvents then you can narrow down the polymers before you go for the exact trial. If you look at the polystyrene value of polystyrene obviously, polystyrene will not soluble in water and methanol, but it polystyrene almost soluble in most of the solvents. So, it actually gives you idea it does not give you exact sort of solubility behavior, but it this gives you guideline or help you to understand what is the solubility you know the solubility of a polymer in a solvent. Now, let us move to the next topic where we talk about the polymers phase separation behavior of polymer solution and again going back to the Flory Huggins expression, Flory Huggins equation. This is Gibbs free energy of mixing. Now, if you want to convert it to Gibbs free energy mixing per segment or per unit cell per lattice site then we will divide it by the number of segments total segments or number of lattice site N 1 is for the solvent and this is for the polymer. So, you divide that and you get this as another expression for the Flory Huggins expression where this Gibbs free energy of mixing express in terms of per segment of the polymer molecule. Now, this gives the behavior of or the how the Gibbs free energy of mixing per segment varies with say one of the composition. So, if you in this case phi 2, phi 1 can be converted to phi 2 easily. So, if you plot say del G mixing, if you plot del G mixing per segment you can get from this expression of what we just saw in this case. From this expression we will tell you that it will give two general type of variation of Gibbs free energy expression with in this case say phi 2, the volume fraction of the polymer. Now, let us consider this case the temperature is T 5. Now, in this case the temperature if you take any point say I take this point. Now, if this has to spontaneously phase separate to any of the other two composition say this composition and this composition two point. So, I consider a solution which corresponds to this particular phi 2. Now, if that has to spontaneously phase separate into two other composition. Obviously, one has to be polymer rich and this one and other has to be solvent rich left hand side phi 2 comes out. You can take any two point. Now, if that process of phase separation spontaneous phase separation happen the delta G mixing of that process would be given by the value of this which is the distance between the interaction point of the tie line. This is tie line. Now, tie line is the line straight line connecting two points in that curve and this line which is a vertical line parallel to the y axis. Now, this the value of this gives you the Gibbs free energy mixing I am telling you again that the value of this gives you the Gibbs free energy of mixing for the process of separating a composition corresponding to this to two composition corresponding to this and this. Now, this value if you can see for any of this points is positive which means that this process of spontaneous phase separation is thermodynamically not feasible which also means that each point or each solution in this line corresponds to different phi 2. They are all stable that means the polymer and the solvent is solvable across composition. So, if you take polymer and dissolve in the solvent it will be solvable for that particular temperature in this case T 5 for any at any composition. It will not phase separate to any two composition having polymer reach and solvent reach phases as that is thermodynamically not feasible. Now, consider another temperature T 1 you get this type of this is another general type of curve. Now, the region this side below this point say I write C and say point D below the left hand side C and above D is similar to this expression. So, all the composition will be stable. So, you get solvable polymer will get solubilize in the solvent in this range and in this range exactly like this. Now, look at another composition here any point here between this and this point any point in this curve this is the tie line. So, if this has to phase separate this solution having polymer mole fraction of this value if this has to phase separate to any two composition having polymer mole fraction correspondence to this and this the Gibbs phenyl gel mixing would be given by this value and that is negative. So, which means that between this point and this point between this point and this point any composition which immediately phase separate and come to this two points. So, if you have any solution having phi 2 value in this region immediately they will phase separate and form two equilibrium solution equilibrium one is polymer rich one is solvent rich. Now, look at a point here any point between this two line this point and this point and this point and this point. So, this is E F say any point between C and E and F and D. Now, if at if this has to phase separate to any of the slight changes into slight changes in the composition say this point in this point it will be similar to this curve similar same in this case, but if this has to phase separate if I consider this point which has this value of phi 2 if this has to separate phase separate into two phases having composition between this point and this point then obviously delta G mixing would be positive. So, that will be unfavorable, but if this composition is phase separate into a point here and here the delta G would be negative. So, it will spontaneously phase separate. So, if this any point in this region any composition any solution would be stable for small change in composition, but it is not stable for a large changes composition beyond this this point. So, this is a region of meta stable region where the composition are stable only for slight change in composition. So, this is meta stable region. So, this regions are meta stable C 2 E and F 2 D and any composition between E to F is completely unstable which will immediately phase separate into C and D and any composition between this point and C and D to N would be stable they will be clear solution. This is the inflection point this is the inflection point. Now, if I change some other temperature the solution will you can draw some other lines and then join this points this points to get a bell shaped curve which I just I will just come in a minute. Now, let us talk little more about these two points and these two points C D and E F. Now, delta G mix I can write delta mu 1 plus X 1 X 2 of the mole fraction delta mu 2 raise delta mu y is mu y minus mu y is 0. It can be write written as G is a extensive quantity. So, I can write X 2 delta mu 1 plus X 2 delta mu 2. So, if you plot delta m versus X 2 you get the slope as delta mu 2 minus delta mu 1 and the intercept will be at X equal to 0 delta mu 1 and at X 2 is equal to 1 you get delta mu 2 minus delta mu 2. Those are the intercepts intercept at X 2 is equal to 0 and X 2 is equal to 1. Now, if I want to do it for delta G m star which is first segment and phi 2 we can easily imagine that the slope would be slope at any point of that curve would be mu 1 and the inner intersect at phi 2 0 would be delta mu 1 which is mu 1 minus mu 0 and at this is a intercept at phi 2 is equal to 1 would be delta mu 2 X. This is a simple mathematics you can find out that. So, what means that if you plot this delta G m versus phi 2 then the slope will be given by this and the intercept would be given by these values. Now, if you look at these two point the tangent for those two point would meet the same points which means they have they must they must have a common tangent. So, these two point must have a common tangent. So, the and as the slope is given by this which will also mean that mu 1 for the solvent at point C will be equals to mu 1 at D and mu 2 at point C would be equals to mu 2 at this. These two point will have same slope and have a common tangent. If they have common tangent then the slope will be same and from the slope we have already seen you can write that mu 1 C equals to mu D. So, the chemical potential of the solvent in this point is same as in this point. So, as the chemical potential of the polymer in this point in this place which means that C and D are two equilibrium equilibrium composition and we name this as binodal binodal compositions or binodal points. Now, this is between C and D you can clearly see that C and D are the binodal points where is 0 between C C and E D and F the slope is changing in positive way. So, D phi 2 would be greater than 0 and between E and basically F this is the change of slope. This is slope and this is the rate of change of slope. This rate of change of is moving in positive direction here and negative direction here and in the inflection point which is E and F inflection point E and F be 0. These two points which are inflection points is called spinodal points, spinodal points or spinodal composition. Basically, it is the boundary between spinodal points or internal composition. This is the boundary between a metastable and an unstable composition. The inflection point E is the boundary between the metastable and the unstable compositions and this gives by this expression. Now, as I was telling little earlier that now you can draw similar lines like let me try and then join this spinodal points and also join the you can join this spinodal points and you can also join the inflection points to get something like this. I have already drawn. So, you can just see from here. So, you can get a curve like this. These are the locus of the binodal points at different temperature. At different temperature, you have different binodal points. One temperature you have this and a temperature you have this like this and one point they will merge. Similarly, this line is the loci for locus for the spinodal points. These are the different points for different temperature and at some points they converge. So, this point this is critical points is given by convergence of the points of common tangency and the inflection points. Common tangency points are the spinodal points and inflection common tangency points are binodal points and inflection points are spinodal points. So, by converging you get T c and so, you can write at that point at T c. So, you can get T c both are 0. So, you can get this. I hope you got it. These are this line this solid line is the just you have joined different binodal points obtains in different temperature. So, if for one temperature you have got binodal points here and here. Another points you have got binodal points here and here. The another one here and here. You just joined them and got a binodal curve. Similarly, this temperature you have spinodal points like here and here. This temperature you have here and here. Similarly, here and here you joined them to get a spinodal curve and this is the point they converge. So, what about this regions then? Any composition above this curve this solid curves will be completely soluble. They will be have single phase. So, any point in this diagram will be having single phase. Any point within this. So, basically this region will be completely insoluble they will phase separate out and any region any point here would be metastable point. So, in single curve symbol diagram you can get the complete phase diagram. It basically gives you the phase diagram of that polymer in that solvent. So, basically you are plotting at different temperature the composition of the solution. So, at any point in this region you get a single phase which is polymer soluble in the solvent. Any point this temperature and this composition you get two phase polymer pure poly and in this case you get metastable solution which will be stable towards small fluctuation in the composition, but not in the large fluctuation. So, this basically gives you the phase diagram. You are getting the phase diagram of a polymer in a solvent from that original Flory Huggins equation. This point you call U C S T upper critical solution temperature. So, above this temperature the polymer is soluble in that solvent at any composition. Below this temperature the polymer is only soluble in that solvent in this composition left hand side of this point and in this composition, but above this temperature the polymer is soluble in that solvent at any composition. That is called upper critical solution temperature and you can get the reverse also and in that case this point called L C S T is given called lower critical solution. So, basically below this temperature everything is soluble in a polymer is soluble at every composition. So, if I take that Flory Huggins expression this one and Flory Huggins expression this one and write the conditions from for spinodal is equal to 0. You get the spinodal points like 1 1 minus 5 2 and for the critical this is for the critical point you get again if you do this we have the expression for delta g m star per per segment. If you do this you will get 5 2 C for the critical 1 plus x 1 by half and from these two you can write the critical. So, this is just for the critical just pure mathematics from getting this expression from Flory Huggins expression which we got earlier for the per segment gives energy fixing per segment. So, you can get this expression what you will do is just in the next lecture we will take this and just complete the discussion of our phase separation of polymer solution and then we will look into it that how the molecular weight or the size of the polymer determine the solubility of polymer and which can be utilized to fractionate the polymer of different molecular weight. We will just start from this in next lecture.