Question 1
Q 1 – A and B:
In order to use Miscibility Checker to predict whether or not a given system (homo-polymer + copolymer or copolymer +
copolymer) is miscible, we must determine what type of copolymer we are dealing with; there are three major cases for
random copolymers:
1) Type 1 - Chemically, both segments are similar and contain similar interacting sites. Examples include:
− Poly(methyl methacrylate-stat- butyl methacrylate)
− poly(styrene-butadiene)
2) Type 2 - Molecular segments have chemically different interacting sites with one segment being able to form
favorable intermolecular interactions while the other is effectively a diluent. Examples include:
− Poly(vinyl acetate –stat-ethylene)
− Poly(vinyl phenol–stat- styrene)
3) Type 3 - Complicated segments with multiple interacting sites. Different segments of the copolymer are capable
of forming favorable intermolecular interactions with varying strengths. Examples include:
− Poly(acrylic acid-stat- butyl methacrylate)
− Poly(vinyl phenol–stat- Methyl methacrylate)
Combinations of polymer pairs in blends that are handled by the "Miscibility calculation tools" are type 1 and type 2.The
"Miscibility calculation tools" give a big deviation when dealing with type 3.
Q 1 – C:
The miscibility window and map are based on the type of random copolymer, strength and type of non-specific
interactions (Van der Vals) and specific interactions (Hydrogen bonds). They differ mainly in the way the data is
displayed, and in the way the system is structured:
1) Miscibility window – for homo-polymer + copolymer. The relation between solubility parameters and
%concentration of a component in a copolymer
2) Miscibility map – for two random copolymers. Changes in miscibility regions as a function of concentration of
one component in the copolymer
miscibility window calculator for homo-polymer with copolymer
miscibility map calculator for two copolymers
Q 1 – D:
Average repeat unit – solubility parameter - The average repeat unit is the random copolymer repeating unit according
to the component fractions:
a) Type 1 - where both of the segments contain similar interacting sites is determined according to the component
fractions: molar, weight or volume.
b) Type 2 - By increasing the diluent's concentration, the number of interacting sites per unit volume decreases
and the solubility parameter for the average unit decreases. Furthermore, the repeat unit's molar volume
increases as the diluent's concentration increases.
Question 2
Following pair mixtures (blend) of random-copolymers:
First case: Poly (ethylene-co-vinyl alcohol) + Poly (propylene-co-methyl methacrylate)
Poly(ethylene-co-vinyl alcohol) - EVOH
Polyethylene (PE) - nonpolar
Polyvinyl alcohol (PVOH) - polar
Type 2 copolymer
– nonpolar + polar
1
𝛿𝑃𝐸
Poly(propylene-co-methyl methacrylate)
Polypropylene (PP) - nonpolar
Poly methyl methacrylate (PMMA) - polar
Type 2 copolymer
1
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
𝛿𝑃𝑉𝑂𝐻
𝑐𝑎𝑙 2
= 9.8 [ 3 ]
𝑐𝑚
– nonpolar + polar
1
𝛿𝑃𝑃
1
𝑐𝑎𝑙 2
= 7.4 [ 3 ]
𝑐𝑚
𝛿𝑃𝑀𝑀𝐴
𝑐𝑎𝑙 2
= 9.1 [ 3 ]
𝑐𝑚
Second case: Poly (ethylene-co-vinyl alcohol) + Poly (caprolactone-co-ethylene oxide)
Poly(ethylene-co-vinyl alcohol) - EVOH
Polyethylene (PE) - nonpolar
Polyvinyl alcohol (PVOH) - polar
Type 2 copolymer
– nonpolar + polar
1
𝛿𝑃𝐸
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
Type 2 copolymer
1
𝛿𝑃𝑉𝑂𝐻 = 9.8 [
Poly (caprolactone-co-ethylene oxide)
Polycaprolactone (PCL) - polar
polyethylene oxide (PEO) - nonpolar
𝑐𝑎𝑙 2
]
𝑐𝑚3
– nonpolar + polar
1
𝑐𝑎𝑙 2
𝛿𝑃𝐶𝐿 = 9.4 [ 3 ]
𝑐𝑚
1
𝛿𝑃𝐸𝑂 = 9.4 [
Solubility parameter calculation:
PVOH-co-PE + PMMA-co- PP
First case
PVOH-co-PE + PCL-co-PEO
Second case
𝑐𝑎𝑙 2
]
𝑐𝑚3
Q 2 – A and B:
According to "polymer blend miscibility checker" - The type of interaction in which Poly (ethylene-co-vinyl alcohol) EVOH and Poly (propylene-co-methyl methacrylate)/ Poly (caprolactone-co-ethylene oxide) random copolymer are
miscible In relation to the molar percentage of diluent (non-polar):
𝟏
Polymer blend (%molar)
solubility
𝛿0%𝑃𝐸−𝑐𝑜−100%𝑃𝑉𝑂𝐻
𝛿0%𝑃𝑃−𝑐𝑜−100%𝑃𝑀𝑀𝐴
𝛿0%𝑃𝐸−𝑐𝑜−100%𝑃𝑉𝑂𝐻
𝛿0%𝑃𝐶𝐿−𝑐𝑜−100%𝑃𝐸𝑂
𝛿10%𝑃𝐸−𝑐𝑜−90%𝑃𝑉𝑂𝐻
𝛿10%𝑃𝑃−𝑐𝑜−90%𝑃𝑀𝑀𝐴
𝛿10%𝑃𝐸−𝑐𝑜−90%𝑃𝑉𝑂𝐻
𝛿10%𝑃𝐶𝐿−𝑐𝑜−90%𝑃𝐸𝑂
𝛿30%𝑃𝐸−𝑐𝑜−70%𝑃𝑉𝑂𝐻
𝛿30%𝑃𝑃−𝑐𝑜−70%𝑃𝑀𝑀𝐴
𝛿30%𝑃𝐸−𝑐𝑜−70%𝑃𝑉𝑂𝐻
𝛿30%𝑃𝐶𝐿−𝑐𝑜−70%𝑃𝐸𝑂
𝛿60%𝑃𝐸−𝑐𝑜−40%𝑃𝑉𝑂𝐻
𝛿60%𝑃𝑃−𝑐𝑜−40%𝑃𝑀𝑀𝐴
𝛿60%𝑃𝐸−𝑐𝑜−40%𝑃𝑉𝑂𝐻
𝛿60%𝑃𝐶𝐿−𝑐𝑜−40%𝑃𝐸𝑂
𝛿90%𝑃𝐸−𝑐𝑜−10%𝑃𝑉𝑂𝐻
𝛿90%𝑃𝑃−𝑐𝑜−10%𝑃𝑀𝑀𝐴
𝛿90%𝑃𝐸−𝑐𝑜−10%𝑃𝑉𝑂𝐻
𝛿90%𝑃𝐶𝐿−𝑐𝑜−10%𝑃𝐸𝑂
𝒄𝒂𝒍
parameter [𝒄𝒎𝟑]𝟐
9.8
9.1
9.8
9.4
9.7
9
9.7
9.4
9.3
8.8
9.3
9.4
8.8
8.3
8.8
9.4
8.2
7.7
8.2
9.4
Type of interaction in which polymer blend will be miscible
weak
Polar – the two (co)polymer may be miscible but caution errors
are large for polar forces
weak
Polar – the two (co)polymer may be miscible but caution errors
are large for polar forces
weak
Polar – the two (co)polymer may be miscible but caution errors
are large for polar forces
weak
Weak to medium
Medium
Immiscible even in the presence of strong forces
According to "miscibility map calculator for two copolymers" - Miscible zones are determined by molar percentages of
diluent (non-polar) and the kind of interaction between Poly (ethylene–vinyl alcohol)–EVOH and Poly (propylene–comethyl methacrylate)/Poly (caprolactone–co-ethylene oxide) random copolymers.
Polymer blend →
Type of interaction ↓
weak polar forces (Dipole – Dipole)
Weak
Medium
Strong
PVOH-co-PE + PMMA-co- PP
PVOH-co-PE + PCL-co-PEO
Percentage of diluent (non-polar) where we will see miscible zones
Minimum %molar Maximum %molar
Minimum %molar Maximum %molar
PE -10% : PP-0%
PE -65% : PP-0%
PE - 0% : PEO - 0% PE - 45% : PEO - 60%
PE -0% : PP-0%
PE -85% : PP-30%
PE - 0% : PEO -0% PE - 60% : PEO - 100%
PE -0% : PP-0%
PE -100% : PP-65% PE - 0% : PEO - 0% PE - 70% : PEO - 100%
PE -0% : PP-0%
PE -100% : PP-85% PE - 0% : PEO - 0% PE - 80% : PEO - 100%
Comment on the results:
It can be seen that the larger the difference between solubility parameter, the less likely a blend will be miscible.
Therefore, for a blend to be miscible, strong interaction must exist.
Example of calculation:
I used two different approaches to solve this question
1. polymer blend miscibility checker
2. miscibility map calculator for two copolymers
According to "polymer blend miscibility checker"
molar percentage of diluent (non-polar)
0
10
30
60
90
PVOH-co-PE + PMMA-co- PP
PVOH-co-PE + PCL-co-PEO
Keep going with the calculation - According to "miscibility map calculator for two copolymers"
Type of interaction
weak polar forces
(Dipole – Dipole)
Weak
Medium
Strong
PVOH-co-PE + PMMA-co- PP
PVOH-co-PE + PCL-co-PEO
Question 3
In this question, I will use the "miscibility window calculator for homo-polymers with copolymers" and the "miscibility
map calculator for two copolymers", for the following Polymer blend:
Polymer blend
Polyvinyl chloride (PVC) + ethylene methyl acrylate (EMA)
Ethylene-vinyl acetate (EVA) + Acrylonitrile butadiene rubber (NBR)
Ethylene-vinyl acetate (EVA) + Poly(styrene-co-vinyl phenol) (SVPh)
Blend type
Homo-polymer + Copolymer
Copolymer + Copolymer
Copolymer + Copolymer
First case: Polyvinyl chloride (PVC) + ethylene methyl acrylate (EMA)
Poly(ethylene-co- methyl acrylate) (EMA)
Polyethylene (PE) - nonpolar
Poly(methyl acrylate) (PMA) - polar
Polyvinyl chloride (PVC) - polar
Type 2 copolymer
1
𝛿𝑃𝑉𝐶
– nonpolar + polar
1
𝑐𝑎𝑙 2
= 9.9 [ 3 ]
𝑐𝑚
𝛿𝑃𝐸
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
1
𝛿𝑃𝑀𝐴
𝑐𝑎𝑙 2
= 9.6 [ 3 ]
𝑐𝑚
Second case: Ethylene-vinyl acetate (EVA) + Acrylonitrile butadiene rubber (NBR)
Ethylene-vinyl acetate (EVA)
Polyethylene (PE) - nonpolar
Polyvinyl acetate (PVA) - polar
Type 2 copolymer
– nonpolar + polar
1
𝛿𝑃𝐸
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
Acrylonitrile butadiene rubber (NBR)
Polybutadiene (BR) - nonpolar
Polyacrylonitrile (PAN) - polar
Type 2 copolymer
1
𝛿𝑃𝑉𝐴
𝑐𝑎𝑙 2
= 9.6 [ 3 ]
𝑐𝑚
– nonpolar + polar
1
𝛿𝐵𝑅
𝑐𝑎𝑙 2
= 8.1 [ 3 ]
𝑐𝑚
1
𝛿𝑃𝐴𝑁
𝑐𝑎𝑙 2
= 13.8 [ 3 ]
𝑐𝑚
Third case: Ethylene-vinyl acetate (EVA) + Poly (styrene-co-vinyl phenol) (SVPh)
Ethylene-vinyl acetate (EVA)
Polyethylene (PE) - nonpolar
Polyvinyl acetate (PVA) - polar
Type 2 copolymer
1
𝛿𝑃𝐸 = 8 [
𝑐𝑎𝑙 2
]
𝑐𝑚3
Poly(styrene-co-vinyl phenol) (SVPh)
polystyrene (PS)- nonpolar
poly(vinyl phenol) (PVPh) - polar
– nonpolar + polar
Type 2 copolymer
1
𝑐𝑎𝑙 2
𝛿𝑃𝑉𝐴 = 9.6 [ 3 ]
𝑐𝑚
– nonpolar + polar
1
𝛿𝑃𝑆 = 9.5 [
𝑐𝑎𝑙 2
]
𝑐𝑚3
1
𝛿𝑃𝑉𝑃ℎ = 9.8 [
𝑐𝑎𝑙 2
]
𝑐𝑚3
For polymer blends, here are the solubility parameters:
First case
Second case
Third case
Polyvinyl chloride (PVC) + ethylene methyl
acrylate (EMA)
Ethylene-vinyl acetate (EVA) +
Acrylonitrile butadiene rubber (NBR)
Ethylene-vinyl acetate (EVA) +
Poly(styrene-co-vinyl phenol) (SVPh)
∆𝛿[𝑃𝑀𝐴 + 𝑃𝐸] = 9.6 − 8 = 1.6
∆𝛿[𝑃𝑉𝐴 + 𝑃𝐸 ] = 9.6 − 8 = 1.6
∆𝛿[ 𝑃𝐴𝑁 + 𝐵𝑅 ] = 13.8 − 8.1 = 5.7
∆𝛿[𝑃𝑉𝐴 + 𝑃𝐸 ] = 9.6 − 8 = 1.6
∆𝛿[ 𝑃𝑉𝑃ℎ + 𝑃𝑆 ] = 9.8 − 9.5 = 0.3
Comment on the results:
Because the software does not take into account the order of the bonds, the solubility parameters for Polyvinyl acetate
(PVA) and Poly (methyl acrylate) (PMA) will be the same. Because C-O has a stronger effect on type of interaction than
C-C, I expect the PVA (C-O) to have a lower solubility parameter than PMA (C-C).
PMA : C-C-O
PVA : C-O-C
In general, when two polymers (BR and PAN) have a large difference in their solubility parameters, and we wish to lower
the solubility parameters, we increase the proportion of the polymer with a lower solubility parameter (BR), thus
lowering the overall solubility parameter. By doing this, we will not only reduce the solubility parameter for the two
polymers (BR and PAN) but also ensure that the value is more or less comparable to the solubility parameter for the
other two polymers (PVA and PE), so that a system (EVA and NBR) will be miscible in a wider range for the same type of
interaction.
In case of PVPh and PS - Whenever the polymer's self-association bonds is stronger, it prevents its ability to form inter
association links, resulting in a decrease in its solubility. To avoid self-association bonds and increase the solubility of the
polymer, a diluent (nonpolar segment) is usually added. Non-polar diluents reduce the ability of the copolymer to
interact with itself and improve its ability to receive miscibility. A diluting agent, however, also lowers the strength of the
inter association links (specific interactions (hydrogen bonds)) and the bonds obtained in solution are less stable.
Choosing the type of forces depends on the stricter criteria, i.e. the smallest ∆δcrit.
For example: in a polar system and a nonpolar system mixed together, the solubility criterion is the lower of the two,
and ∆δcrit is the critical value of a system with only depressive forces ∆δcrit = 0.1.
PVC
weak
PE
dispersion
forces
PMA
weak polar
forces
PVA
weak polar
forces
BR
dispersion
forces
PAN
weak
PS
dispersion
forces
PVPh
strong
Example of calculation:
Type of interaction
Dispersive forces
only
weak polar forces
(Dipole – Dipole)
Weak
Strong
First case
Second case
Third case
Question 4
Q 4 – A:
In the original Flory-Huggins theory, it was assumed that the two polymers would mix randomly. If there are strong
polar forces or specific interactions between the components of the blend, such as hydrogen bonds, this assumption
doesn't hold.
The general expression for the Flory-Huggins theory of free energy of mixing (which is applicable to polymers-solvents
and polymers-polymers) is as follows:
′
∆𝐺𝑚𝑖𝑥
∆𝐺𝑚𝑖𝑥 𝑉𝑟
𝜙𝐴
𝜙𝐵
=
∗ = ( ) ∗ ln 𝜙𝐴 + ( ) ∗ ln 𝜙𝐵 + 𝜒𝐹𝐻 𝜙𝐴 𝜙𝐵
𝑅𝑇
𝑅𝑇
𝑉
𝑁𝐴
𝑁𝐵
𝜙𝐴
𝜙𝐵
) ∗ ln 𝜙𝐴 + ( ) ∗ ln 𝜙𝐵 ] → 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑏𝑜𝑡ℎ 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠
𝑁𝐴
𝑁𝐵
Δ𝐻𝑚𝑖𝑥 (𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦) = 𝑅𝑇𝜒𝐹𝐻 𝜙𝐴 𝜙𝐵 → 𝑃ℎ𝑦𝑠𝑖𝑐𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 𝑎𝑐𝑡𝑖𝑛𝑔 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠
Δ𝑆𝑚𝑖𝑥 (𝑒𝑛𝑡𝑟𝑜𝑝𝑦) = −𝑅 [(
Hydrogen bonds (intra- or inter-molecular) are not random, so the enthalpy (ΔH) term would consist of a much more
complex structure. In addition, hydrogen bond formation would result in favorable changes in enthalpy (ΔH), but also
impose constraints on the degree of freedom of orientation and translation, which affect the entropy (ΔS) change.
Intermolecular hydrogen bonding between polymers depends on several factors:
a) If one or both polymers contain intramolecular hydrogen bonding (self-association) in pure state, then
intermolecular hydrogen bonding will be limited.
b) flexible chain can bend back on itself to avoid intermolecular interactions, a process known as intramolecular
screening
c) Accessibility of the functional groups that form intermolecular interactions is another factor determining the
extent of the intermolecular hydrogen bonding
Painter-Coleman group have developed an association model approach for the thermodynamics of mixing of two
polymers that have strong hydrogen bonding capabilities:
∆𝐺𝑚𝑖𝑥
𝜙𝐴
𝜙𝐵
Δ𝐺𝐻
= ( ) ∗ ln 𝜙𝐴 + ( ) ∗ ln 𝜙𝐵 + 𝜒𝐹𝐻 𝜙𝐴 𝜙𝐵 +
𝑅𝑇
𝑁𝐴
𝑁𝐵
𝑅𝑇
Where ∆GH is a free energy term that imposes the constraints due to hydrogen bonding and represents chemical forces
that have favorable, negative valued contribution to the free energy of mixing
Due to the fact that weak and strong interactions have different compositions and temperature dependences, it is
necessary to separate them.
Factors that impact the magnitude of ∆GH:
a) if inter-association between two different components is more favorable than self-association within the pure
components, then this trend is favorable for miscibility
b) Number of specific interaction sites per unit volume of the blend. For example, if the number of specific
interaction sites per unit volume is decreased in a system, this would result in lower ∆GH in the system
compared to the original state
Association model are mixtures where the first component self-associates (i.e. has functional groups, such as –OH, that
can form hydrogen bonds with one another in the pure state), while the second component does not self-associate, but
has a functional group that can form hydrogen bonds with the first component. Accordingly, the free energy of
hydrogen bond formation in the mixture can be described as:
′
∆𝐺𝑚𝑖𝑥
𝜙𝐴1
𝜙𝐵1
ℎ
ℎ
= 𝑛𝐴 ∗ ln (
) + 𝑛𝐵 ∗ ln (
) + 𝑛𝐵𝐵
+ 𝑛𝐴𝐵
𝑅𝑇
𝜙𝐴
𝜙𝐵
Where 𝑛𝐴 , 𝑛𝐵 are the number of A and B type segments;
𝜙𝐴1
𝜙𝐴
,
𝜙𝐵1
𝜙𝐵
are the fractions of ‘free’ (non-hydrogen-bonded) A
ℎ
ℎ
and B segments; and 𝑛𝐵𝐵
, 𝑛𝐴𝐵
are the number of B--B and A--B hydrogen bonds, respectively.
Q 4 – B:
Types of complexes
Self-association can lead to the formation of two main types of complexes: linear or cyclic complexes:
•
•
Linear - Polymers containing amide groups (HNCO), urethanes groups (HNCOO) or hydroxyl groups (OH) form
linear chains upon hydrogen bond formation.
Cyclic - possible with hydroxyl groups, but is not favored from an energical point of view. Cyclic hydrogen
bonded structures are favored in molecules containing carboxylic acid and urazole functional groups
hydroxyl groups (not
favored)
Formation of
cyclic dimers
carboxylic acid groups
Hydroxyl groups
Linear chains
The equilibrium constants are a link between the stoichiometry, the concentration of the hydrogen bonded species
present in the mixture and the free energy of mixing.
I'll answer this question in two ways:
1. hydrogen bonds - self-associating, It is divided into two types of connections:
• intermolecular H-bonds (relationships between two or more molecules)
• intramolecular H-bonds (relationships within the same molecule)
2. Inter-association - hydrogen bonding between two different functional groups. can occurs between hydroxyl
group of phenol and carbonyl group
Schematic representation of hydrogen bonded structures:
Self-association
Inter-association
Schematic representation of self-association of phenol
Schematic representation of inter-association in phenol/
ethyl propionate mixtures
When we are dealing with mixtures where one component, B, self-associates while the second, A, does not, but has a
functional group capable of forming hydrogen bonds with B. The self-associating components contain groups having
both "donor" and "acceptor" parts.
The self-association equilibrium can be written as a linear condensation polymerization:
𝐾2
𝐵1 + 𝐵1 ↔ 𝐵2 𝑤𝑖𝑡ℎ 𝐾2 =
𝐾3
𝐵2 + 𝐵1 ↔ 𝐵3 𝑤𝑖𝑡ℎ 𝐾3 =
↓
↓
𝜙𝐵2
2
2 ∙ 𝜙𝐵1
𝜙𝐵3
2
∙
3 𝜙𝐵2 ∙ 𝜙𝐵1
↓
𝐾ℎ 𝑜𝑟 𝐾𝐵
𝐵ℎ + 𝐵1 ↔
𝐵ℎ+1 (ℎ ≥ 2) 𝑤𝑖𝑡ℎ 𝐾ℎ 𝑜𝑟 𝐵 =
𝜙𝐵ℎ+1
ℎ
∙
ℎ + 1 𝜙𝐵ℎ ∙ 𝜙𝐵1
The equilibrium constant of inter-association equilibrium is written as:
𝐾𝐴
𝐵ℎ + 𝐴1 → 𝐵ℎ ∙ 𝐴 𝑤𝑖𝑡ℎ 𝐾𝐴 =
𝜙𝐵ℎ 𝐴
ℎ∙𝑟
∙
ℎ + 𝑟 𝜙𝐵ℎ ∙ 𝜙𝐴1
The equilibrium constant of self-association to the formation of cyclic species equilibrium is written as:
𝐾𝐷
𝐵1 + 𝐵1 ↔ 𝐵2 𝑤𝑖𝑡ℎ 𝐾𝐷 =
𝜙𝐵2
2
2 ∙ 𝜙𝐵1
𝑜𝑟 𝐾𝐷 =
1 − 𝑓𝐹𝐶=𝑂
2 ∙ [𝑓𝐹𝐶=𝑂 ]2
The volume fraction of all A and B units can be written as follows:
𝜙𝐵 = 𝜙𝐵1 ∙ [(1 −
𝐾𝐴 ∙ 𝜙𝐴1
𝐾2
𝐾2
1
)+
∙(
)] ∙ [1 +
]
2
𝐾𝐵
𝐾𝐵
𝑟
(1 − 𝐾𝐵 ∙ 𝜙𝐵1 )
𝜙𝐴 = 𝜙𝐴1 + 𝐾𝐴 ∙ 𝜙𝐴1 ∙ 𝜙𝐵1 ∙ [(1 −
𝐾2
𝐾2
1
)+
∙(
2 )]
𝐾𝐵
𝐾𝐵
(1 − 𝐾𝐵 ∙ 𝜙𝐵1 )
2
𝜙𝐵2 = 2 ∙ 𝐾2 ∙ 𝜙𝐵1
𝜙𝐵3 =
3
∙𝐾 ∙𝜙 ∙𝜙
2 𝐵 𝐵2 𝐵1
𝜙𝐵ℎ = (
ℎ
) ∙ 𝐾𝐵 ∙ 𝜙𝐵ℎ−1 ∙ 𝜙𝐵1
ℎ−1
K2 and KB values at different temperatures are used to calculate the enthalpies of hydrogen bond formation, H2 and HB,
from the slope of lnK(2 and B) vs. 1/T plots.
𝐻 = (−) ∙ (𝑠𝑙𝑜𝑝𝑒) ∙ (𝑅) → 𝑠𝑖𝑛𝑐𝑒
𝜕(ln 𝐾)
1
𝜕( )
𝑇
=
−𝐻
𝑅
𝑤ℎ𝑒𝑟𝑒 𝑅 𝑖𝑠 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Complexes A-H...B are formed upon hydrogen bonding. Due to the small strength of the hydrogen bond, the H...B
stretching mode appears at very low frequency and cannot be measured by fundamental infrared spectroscopy.
Information about the nature of the hydrogen bond can be obtained from the analysis of the A-H stretching mode.
Measurement of the band shifts in the A-H stretching mode in the system A-H....B is a good measure of the average
strength of the hydrogen bond.
Frequency shifts of the A-H stretching mode upon hydrogen bond formation:
Strength of the
hydrogen bond
Weak
Medium
Intermediate
Strong
IR frequency shift [cm-1]
Enthalpy of the bond [kcal/mol]
Examples
10-50
300
600
800-2000
1
5
6-8
>8
PVC-polyesters
-OH; Amide, urethanes
-COOH
Acid salts
The bond connecting the B atom can also have some sensitivity to hydrogen bonding. This is the case when hydrogen
bonds form between carbonyl groups (C=O) and N-H or O-H groups.
Changes in temperature and concentration are two factors that affect the distribution of monomers, dimers and
multimers.
Look at the following two cases in more detail:
1. Infrared spectra of hydroxyl (O-H-O) stretching regions of 2-propanol in cyclohexane (only self-association of 2propanol in this system and there isn’t any inter- association between 2-propanol and cyclohexane):
• Different concentrations of 2-propanol
• Different temperatures
2. Infrared spectra of carbonyl (C=O-H) stretching region of ethyl phenol (EPh)/ethyl isobutyrate (EIB) mixtures as
a function of composition:
• carbonyl (C=O-H) stretching region of the infrared spectra show two bands at 1736 and 1707cm-1 that
are assigned to free and hydrogen bonded carbonyl groups, respectively.
carbonyl (C=O-H) stretching region of
EPh/EIB mixtures of different
compositions
Hydroxyl (O-H-O) stretching regions of 2-propanol in cyclohexane
Different concentrations of 2-propanol
Different temperatures
Lastly, I would like to analyze the results for the case 2-propanol in cyclohexane:
Effect of 2-propanol concentration:
1. Increased concentration from 0.02M to 0.09M - The monomer peak at 3630cm-1 gradually decreases in
intensity and two new peaks appear at 3530 and 3350cm-1, indicating that dimers and multimers form at higher
concentrations.
2. Increased concentration from 0.09M to 0.3M - Monomer peak intensity decreases even more while dimer and
multimer peak intensity increase at 3530 and 3350cm-1.
Effect of temperatures:
As the temperature is decreased, the monomers of 2-propanol forms dimers and multimers, therefore the intensity of
the peak at 3630cm-1 decreases, whereas the intensity of the peaks at 3530 and3350cm-1 increases at lower
temperatures.
This information from infrared studies is the basis for determination of equilibrium constants and enthalpies (ΔH) of
hydrogen bond formation for self-association of low molecular weight molecules.
According to the Beer-Lambert law:
𝐼 =𝜀∙𝑙∙𝑐
•
•
•
•
𝐼 - Absorbance (intensity of the isolated hydroxyl band)
𝜀 - Absorptivity coefficient
𝑐 - Concentration
𝑙 - Path length
Experimental fraction of free monomers (𝑓𝑚𝑂𝐻 ) at any given concentration is then given by:
𝑓𝑚𝑂𝐻 =
𝐼
𝐼0
𝑓𝑚𝑂𝐻 =
−1
𝜙𝐵1
𝐾2
𝐾2
1
= [(1 − ) +
∙(
)]
Φ𝐵
𝐾𝐵
𝐾𝐵 (1 − 𝐾𝐵 ∙ Φ𝐵1 )2
The fraction of free carbonyl groups (𝑓𝐹𝐶=𝑂 ) is determined using the relative absorption coefficients and the relative
intensities of the two bands from the IR spectra:
𝑓𝐹𝐶=𝑂 =
𝜙𝐴1
1
=
𝐾
𝐾
1
𝜙𝐴
{1 + 𝐾𝐴 ∙ 𝜙𝐵1 ∙ [(1 − 𝐾2 ) + (𝐾2 ) ∙ (1 − 𝐾 ∙ 𝜙 )]}
𝐵
𝐵
𝐵
𝐵1
Appendix - Symbols Used in Q4:
𝐴 − 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑒𝑎𝑡 𝑢𝑛𝑖𝑡 𝑜𝑓 𝑛𝑜𝑛 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑛𝑔 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒
𝐵 − 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑟𝑒𝑝𝑒𝑎𝑡 𝑢𝑛𝑖𝑡 𝑜𝑓 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑛𝑔 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒
ℎ − 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒𝑑 𝑢𝑛𝑖𝑡𝑠 𝑙𝑖𝑛𝑘𝑒𝑑 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
𝑟 − 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑚𝑜𝑙𝑎𝑟 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 ‘𝐴’ 𝑡𝑜 ‘𝐵’ (𝑉𝐴/𝑉𝐵)
𝐾𝐵 − 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑜𝑟 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐ℎ𝑎𝑖𝑛 − 𝑙𝑖𝑘𝑒 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑢𝑛𝑖𝑡𝑠
𝐾2 − 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑜𝑟 𝑑𝑖𝑚𝑒𝑟 𝑡𝑦𝑝𝑒 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑢𝑛𝑖𝑡𝑠
𝐾𝐴 − 𝐴𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑜𝑟 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑏𝑜𝑛𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑎𝑛𝑑 𝐴 𝑢𝑛𝑖𝑡𝑠
ℎ𝐴 − 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑜𝑓 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑏𝑜𝑛𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑎𝑛𝑑 𝐴 𝑢𝑛𝑖𝑡𝑠
ℎ𝐵 − 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑜𝑓 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑏𝑜𝑛𝑑 𝑓𝑜𝑟 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐ℎ𝑎𝑖𝑛 − 𝑙𝑖𝑘𝑒 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑢𝑛𝑖𝑡𝑠
ℎ2 − 𝑒𝑛𝑡ℎ𝑎𝑙𝑝𝑦 𝑜𝑓 𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑏𝑜𝑛𝑑 𝑓𝑜𝑟 𝑑𝑖𝑚𝑒𝑟 𝑡𝑦𝑝𝑒 𝑠𝑒𝑙𝑓 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐵 𝑢𝑛𝑖𝑡𝑠
𝜙𝐵1 − 𝑡ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑛𝑜𝑚𝑒𝑟𝑠 (𝑛𝑜𝑛 − ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑏𝑜𝑛𝑑𝑒𝑑 𝑔𝑟𝑜𝑢𝑝𝑠)
𝜙𝐵2 , 𝜙𝐵3 , 𝜙𝐵ℎ 𝑎𝑛𝑑 𝜙𝐵ℎ+1 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛𝑠 𝑤𝑖𝑡ℎ 2, 3, ℎ 𝑎𝑛𝑑 ℎ + 1 𝑚𝑒𝑟𝑠, 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦.
𝜙𝐴1 − 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴 𝑢𝑛𝑖𝑡𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑚𝑎𝑖𝑛 𝑢𝑛𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒𝑑
𝜙𝐵ℎ 𝐴 − 𝑖𝑛𝑡𝑒𝑟 − 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑒𝑑 𝐵 𝑎𝑛𝑑 𝐴 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
Question 5
Q 5 – A and B:
Intra-molecular Screening Parameter:
Screening in polymers is caused by chain connectivity that prevents hydrogen bonds from forming:
Covalent linkages between polymer segments result in an increase of same-chain contacts over that calculated based on
random segment mixing, since the chain bends back on itself both locally and through long-range effects, consequently
preventing hydrogen bonding and thereby decreasing the number of possible intermolecular interactions.
Schematics of intra-molecular screening in long-chain polymers:
The condition that actually occurs is chain connectivity that blocks the interactions and prevents them from forming a
bond with additional chains.
The formation of intra-molecular screening has an effect on association model:
∆𝐺𝑚𝑖𝑥
𝑅𝑇
𝜙
𝜙
= (𝑁𝐴 ) ∗ ln 𝜙𝐴 + (𝑁𝐵 ) ∗ ln 𝜙𝐵 + 𝜒𝐹𝐻 𝜙𝐴 𝜙𝐵 +
𝐴
𝐵
Δ𝐺𝐻
𝑅𝑇
∆𝐺𝑚𝑖𝑥
𝜙𝐴
𝜙𝐵
Δ𝐺𝐻
= ( ) ∗ ln 𝜙𝐴 + ( ) ∗ ln 𝜙𝐵 + 𝜒𝐹𝐻 𝜙𝐴 𝜙𝐵 ∙ (1 − 𝛾) +
𝑅𝑇
𝑁𝐴
𝑁𝐵
𝑅𝑇
Δ𝐺𝐻
𝑅𝑇
This term would have a lower positive
value, so the free energy of hydrogen
bonding needs to overcome that
Becomes more positive than negative - The reason for this is that it considers the specific interactions favorable to
mixing, most commonly hydrogen bonds. In a lattice model, these interactions alter the number of possible
configurations.
Intramolecular screening 𝛾 not only modifies the enthalpic term, but indirectly modifies the free energy of hydrogen
bonding
Δ𝐺𝐻
𝑅𝑇
terms through the modification of self- and inter-association equilibrium constants.
As a function of intramolecular screening parameter, equilibrium constants for self- and inter-associations are modified:
𝛾 + (1 − 𝛾) ∙ 𝜙𝐵
𝐾2𝐵 = 𝐾2 ∙ [
]
ϕ𝐵
𝛾 + (1 − 𝛾) ∙ 𝜙𝐵
𝐾𝐵𝐵 = 𝐾𝐵 ∙ [
]
ϕ𝐵
𝐾𝐴𝐵 = 𝐾𝐴 ∙ (1 − 𝛾)
Where K2B and KBB are the modified self-association equilibrium constants for di-mer and multi-mer formation,
respectively; and KAB is the modified inter-association equilibrium constant, all involving the intra-molecular screening
effect.
Absence of intra-molecular screening, KA is higher than KB, which indicates that inter association between two polymers
is more favorable than self-association of polymer B. When screening is taken into account, KBB increases whereas KAB
decreases. This indicates that when screening is taken into account, the favorability of inter-molecular interactions
decreases and the self-association of polymer B increases, which would be unfavorable effect for miscibility of polymers
A and B.
Taking into account that intramolecular screening has an effect on miscibility, it could contribute to a more realistic
miscibility prediction. Based on experimental data, the values of γ between 0.25 and 0.35, average value of 𝛾 = 0.30 is
being accepted for most polymer systems.
Q 5 – C:
A polymer chain is significantly different from a molecule with low molar mass because of intramolecular screening and
access to functional groups.
Because of the self-bending back of the polymer chain, there is an increase in the number of contacts made by a given
polymer chain. This means the number of inter-association hydrogen bonds per volume in the polymer blend will be less
than that in the model compound.
Moreover, the spacing between functional groups along a polymer chain and bulky side groups reduces the hydrogen
bonds per unit volume as well, due to the so-called functional-group accessibility effect.
Therefore, the inter-association equilibrium constant (KA) between the polymer blend and the model compound should
be different after considering the intramolecular screening, functional group accessibility effects, and polymer-chain
architecture.
It's possible to get good correlation between methods when you consider intramolecular screening and functional group
accessibility.
Question 6
Q 6 – A:
In chain association models, the "phase calculator" computing tool is important for increasing the accuracy of the
calculations by taking into account the process that occurs in the system.
Molecule labeled as ‘B’ can self-associate in the pure state through hydrogen bonding. The monomer
(Repeating unit) of B molecules is represented by B1. Two monomers of B form a dimer which is represented by B2,
where K2 is the equilibrium constant describing formation of dimers. Bh is the hth order multimer of B molecules (h
monomers forming a h-mer), where Kh is the equilibrium constant describing formation of h-mers.
Molecules of ‘B’ are mixed with molecules of ‘A’, where ‘A’ is also non polymeric and do not self-associate in pure state,
but has a functional group that is an “acceptor” for the proton “donor” of the OH group in molecule ‘B’.
where A1 is the monomer of A that has the functional group that can make hydrogen bond with B molecules; BhA
represents the inter-associated B and A molecules; and KA is the equilibrium constant of inter-association
Model type
Equilibrium schemes
𝐾ℎ 𝑜𝑟 𝐾𝐵
𝐵ℎ + 𝐵1 ↔
𝐵ℎ+1 (ℎ ≥ 2)
Description
self-associate of unhindered phenols – hydrogen
bonded of higher multimers
𝐾𝐴
Inter-association of phenols with aliphatic esters
𝐾2
self-associate of unhindered phenols – hydrogen
bonded dimers
𝐵ℎ + 𝐴1 → 𝐵ℎ ∙ 𝐴
𝐵1 + 𝐵1 ↔ 𝐵2
𝐾ℎ 𝑜𝑟 𝐾𝐵
𝐵ℎ + 𝐵1 ↔
𝐵ℎ+1 (ℎ ≥ 2)
self-associate of unhindered phenols – hydrogen
bonded of higher multimers
𝐾𝐴
Inter-association of phenols with aliphatic esters
𝐾2
self-associate of unhindered phenols – hydrogen
bonded dimers
𝐾𝐴
Inter-association of phenols with aliphatic esters
𝐵ℎ + 𝐴1 → 𝐵ℎ ∙ 𝐴
𝐵1 + 𝐵1 ↔ 𝐵2
𝐵ℎ + 𝐴1 → 𝐵ℎ ∙ 𝐴
Note: for other systems involving say carboxylic acids, amides, urethane etc. other equilibrium schemes may be
appropriate.
Formation of hydrogen bonds in a variety of chemical systems:
Oxygen- Oxygen
𝑂 − 𝐻 ⋯ 𝑂 = 𝐶⋱⋰
𝑂 − 𝐻 ⋯ 𝑂⋱⋰
Oxygen-Nitrogen
𝑂 − 𝐻 ⋯ 𝑁 ⋯⋰⋱
𝑂 − 𝐻 ⋯ 𝑁⋱⋰⋰
𝑁 − 𝐻 ⋯ 𝑂 = 𝐶⋱⋰
𝑁 − 𝐻 ⋯ 𝑂⋱⋰
Donor
Acceptor
Phenol
cholesterol
Water
Phenol
Phenol
Acetone
Glyceryl Triacetate
1,4-Dioxane is a heterocyclic organic compound
1,4-Dioxane is a heterocyclic organic compound
Dibutyl Ether
Phenol
Phenol
ɣ-Butyrolactam
ɑ-Pyridone
Aniline
Trimethylamine (TMA)
Pyridine is a basic heterocyclic organic compound
ɣ-Butyrolactam
ɑ-Pyridone
THF (Tetrahydrofuran)
Q 6 – B:
The main difference between the terms "miscibility window" and "miscibility map", as they appear in the "miscibility
checker tool" or “phase calculator tool”:
✓ "Miscibility checker tool" - Is a tool based on how different its components are in terms of solubility parameters,
the size of intermolecular interactions, and the number of active sites per unit volume.
✓ “Phase calculator tool” - Fourier-transform infrared spectroscopy (FTIR), used to determine the hydrogen
bonding equilibrium constants as a function of temperature. In this model, the free energy of mixing is given as
a function of the equilibrium constants of hydrogen bond formation, the Flory-Huggins interaction parameter,
the composition of the mixture and specific molecular characteristics of the components of the mixture.
BACKGROUND (Q7 and Q8) - Thermodynamics of Mixing in Polymer Systems:
The first condition for miscibility of one component in another is obtaining a negative change in the free energy of
mixing and the second condition required for miscibility is to have a positive second derivative:
a) Free enthalpy of mixing negative:
∆𝐺𝑚𝑖𝑥 = ∆𝐻𝑚𝑖𝑥 − 𝑅𝑇∆𝑆𝑚𝑖𝑥 < 0
b) Second derivative of ∆𝐺𝑚𝑖𝑥 positive (the phase is stable against small perturbations):
𝑠𝑝𝑖𝑛𝑜𝑑𝑎𝑙 (𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒 𝑟𝑒𝑔𝑖𝑜𝑛) − (
𝜕2 ∆𝐺𝑚𝑖𝑥
2 )
𝜕𝜙𝐴
>0
𝑇,𝑃
On this curve, any point (for example, point Q) has a lower free energy than any two phase system with the same overall
composition. As shown by P1 and P2, the free energy of a hypothetical phase separated mixture, Q*, is proportional to
their composition weighted free energy and is greater than the free energy of a miscible mixture, Q. On these plots, the
second derivative of free energy of mixing is positive over the entire composition range.
In contrast, if the free energy of mixing against composition plots show concave downward portions, then the blend
components are not miscible in that particular composition range, even though the free energy of mixing is negative.
Phase separated systems with compositions B1 and B2, which are the double tangent to the free energy curve, have a
lower free energy. Free energy curve portions from B1 to S1, as well as those from B2 to S2 are still concave upward.
Metastable mixtures are characterized by compositions between B1-S1 and B2-S2 points but are unstable against phase
separation at compositions B1 and B2. The free energy curve between S1 and S2 is concave downward, unstable and
spontaneously separated.
For a blend to be miscible, it is important to consider both the free energy of mixing and the second derivative of free
energy of mixing.
Question 7
PVPh - self-associating polymer that can form hydrogen bonds in pure state:
a) PVPh + PVPh (O…H-O) – Hydroxyl- Hydroxyl hydrogen bonding
b) PVPh + PBMA (O-H…O=C) – Hydroxyl-Carbonyl hydrogen bonding. PBMA non-self-associating.
Strong self-association of PVPh, intramolecular screening, and functional group accessibility effects on the blend system.
That means PVPh will prefer to bond with each other, rather than with PBMA → self-associate multimers
Poly (styrene-co-vinyl phenol) (SVPh) – PVPh + PS (diluent) - The diluting agent (PS) reduces the ability of the PVPh to
interact with itself, and it gives the blend of SVPh + PEMA better miscibility → Inter-association
poly(butyl methacrylate) (PBMA)
poly(vinyl phenol) (PVPh)
1
1
𝑐𝑎𝑙 2
𝛿𝑃𝐵𝑀𝐴 = 8.3 [ 3 ]
𝑐𝑚
𝑐𝑎𝑙 2
𝛿𝑃𝑉𝑃ℎ = 10.6 [ 3 ]
𝑐𝑚
Poly(ethyl methacrylate) (PEMA) - homo-polymer
PEMA
Poly(styrene-co-vinyl phenol) (SVPh) - random copolymer
PS
PVPh
1
𝛿𝑃𝐸𝑀𝐴
𝑐𝑎𝑙 2
= 8.9 [ 3 ]
𝑐𝑚
1
𝛿𝑃𝑆
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
1
𝛿𝑃𝑉𝑃ℎ
𝑐𝑎𝑙 2
= 10.6 [ 3 ]
𝑐𝑚
The miscible system will be created by combining negative free energy and positive second derivative of free
energy. The Painter-Coleman hydrogen bonding association model has limited applicability for carbohydrate systems
containing carbon (C), hydrogen (H) and oxygen (O) atoms.
Due to the fact that the model was designed in such a way that the repeating unit of the A polymer has a functional
group, such as OH, that can form self-association with itself, and that the repeating unit of the B polymer has one
functional group that is capable of forming a H-bond with the first polymer. In any polymer blend component containing
multiple types of functional groups or that self-associates in the pure state, calculation of hydrogen bonding
contribution to free energy becomes more complicated, and the association model is only able to provide a crude
approximation to predict miscibility.
Example of calculation:
1. For PVPh + PBMA, I only used the "Phase calculator for hydrogen-bonded systems".
2. For SVPh + PEMA, the calculation was performed in two stages:
1) SVPh was calculated with "copolymer average repeat calculator", when weight% PS was different for
two cases
• 50% PVPh + 50% PS
• 75% PVPh + 25% PS
50% PVPh + 50% PS
𝑀𝑜𝑙𝑎𝑟 𝑉𝑜𝑙𝑢𝑚𝑒 = 208 [
𝑐𝑚3
𝑚𝑜𝑙𝑒
75% PVPh + 25% PS
] ; 𝑀𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑊𝑒𝑖𝑔ℎ𝑡 = 240 [
𝑆𝑜𝑙𝑢𝑏𝑖𝑙𝑖𝑡𝑦 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 = 10 [
0.5
𝑐𝑎𝑙
]
𝑐𝑚3
𝑔
𝑚𝑜𝑙𝑒
]
𝑀𝑜𝑙𝑎𝑟 𝑉𝑜𝑙𝑢𝑚𝑒 = 136 [
𝑐𝑚3
𝑚𝑜𝑙𝑒
] ; 𝑀𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑊𝑒𝑖𝑔ℎ𝑡 = 160 [
0.5
𝑔
𝑚𝑜𝑙𝑒
]
𝑐𝑎𝑙
𝑆𝑜𝑙𝑢𝑏𝑖𝑙𝑖𝑡𝑦 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 = 10.3 [ 3 ]
𝑐𝑚
2) Next, I entered the values obtained from "copolymer average repeat calculator" into "Phase calculator
for hydrogen-bonded systems" for SVPh(25%) + PEMA and SVPh(45%) + PEMA
PVPh + PBMA
SVPh(50%PS) + PEMA
SVPh(25%PS) + PEMA
Continued - Example of calculation:
Phase calculator
PVPh + PBMA
SVPh(50%PS) + PEMA
SVPh(25%PS) + PEMA
Free Energy at 100
[C]
Free Energy (2nd
derivative) at 100
[C]
fraction of HB
associated groups
at 100 [C]
Phase Diagram
(spinodal) - -100 [C]
– 300 [C]
PBMA
PEMA
Question 8
Q 8 – A:
Two cases are covered by the 'Miscibility window':
1) Homo-polymer (B): Strong self-association + Copolymer (AC) – A: Non self-association, C: diluent
2) Homo-polymer (A): Non self-association + Copolymer (BC) – B: Strong self-association, C: diluent
Before we begin, we will split the mixture into two groups of polymers, based on the cases mentioned above:
PVPh/EVA
Case 1
PVPh/STBMA
PHMA/STVPh
Case 2
PBMA/STVPh
PVPh - poly(vinyl phenol)
EVA - Ethylene-vinyl acetate
PVPh - poly(vinyl phenol)
STBMA - Poly(styrene-b- methacrylic acid)
PHMA – poly(Hexyl methacrylate)
STVPh - Poly(styrene-co-vinyl phenol)
PBMA - poly(butyl methacrylate)
STVPh - Poly(styrene-co-vinyl phenol)
Homo-polymer
Copolymer
Homo-polymer
Copolymer
Homo-polymer
Copolymer
Homo-polymer
Copolymer
Strong self-association
Non self-association + diluent
Strong self-association
Non self-association + diluent
Non self-association
Strong self-association + diluent
Non self-association
Strong self-association + diluent
Case 1
PVPh - poly(vinyl phenol) - Homo-polymer
Strong self-association
EVA - Ethylene-vinyl acetate - Copolymer
Non self-association + diluent (PE)
1
𝛿𝑃𝑉𝑃ℎ
𝑐𝑎𝑙 2
= 10.6 [ 3 ]
𝑐𝑚
1
𝛿Ethylene (𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
PVPh - poly(vinyl phenol) - Homo-polymer
Strong self-association
𝛿𝑃𝑉𝑃ℎ
1
𝛿𝑉𝐴
𝑐𝑎𝑙 2
= 9.6 [ 3 ]
𝑐𝑚
STBMA - Poly(styrene-b- methacrylic acid) - Copolymer
Non self-association + diluent (PS)
1
𝑐𝑎𝑙 2
= 10.6 [ 3 ]
𝑐𝑚
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
1
𝛿PS
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
1
𝛿MA
𝑐𝑎𝑙 2
= 10.4 [ 3 ]
𝑐𝑚
Case 2
PHMA – poly(Hexyl methacrylate) - Homo-polymer
Non self-association
STVPh - Poly(styrene-co-vinyl phenol) - Copolymer
Strong self-association + diluent (PS)
1
𝛿𝑃𝐻𝑀𝐴
𝑐𝑎𝑙 2
= 8.5 [ 3 ]
𝑐𝑚
1
𝛿PS
PBMA - poly(butyl methacrylate) - Homo-polymer
Non self-association
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
1
𝛿𝑃𝑉𝑃ℎ
𝑐𝑎𝑙 2
= 10.6 [ 3 ]
𝑐𝑚
STVPh - Poly(styrene-co-vinyl phenol) - Copolymer
Strong self-association + diluent (PS)
1
𝑐𝑎𝑙 2
𝛿𝑃𝐵𝑀𝐴 = 8.7 [ 3 ]
𝑐𝑚
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
1
𝛿PS
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
1
𝑐𝑎𝑙 2
𝛿𝑃𝑉𝑃ℎ = 10.6 [ 3 ]
𝑐𝑚
Comment on the results:
In order to get miscibility blend two conditions must be met - compatibility is improved by matching solubility
parameters and miscibility is achieved only when specific interactions are present
Case 1 – PVPh/STBMA blend has a wider miscibility window area than PVPh/EVA blend, which can be explained by the
fact that STBMA's solubility parameter is smaller (0.9), than EVA's (1.6).
Case 2 - The miscibility window of PBMA/STVPh blend is larger than that of PHMA/STVPh blend, which is explained by
the fact that STVPh's solubility parameter is closer to PBMA (8.7) than to PHMA (8.5).
Example of calculation:
Homo-polymer (B): Strong self-association + Copolymer (AC) – A: Non self-association, C:
diluent
PVPh/EVA
PVPh/STBMA
Miscibility
window
Miscibility window
Homo-polymer (A): Non self-association + Copolymer (BC) – B: Strong self-association, C:
diluent
PHMA/STVPh
PBMA/STVPh
Miscibility
window
Miscibility
window
Q 8 – B:
Miscibility map calculator for hydrogen bonding systems of copolymer at a given temp:
−
−
BC – B: Strong self-association and C: diluent
AD – A: Non self-association and D: diluent
VPh - vinyl phenol
BR - Butadiene Rubber
MMA - Methyl methacrylate
ST – Polystyrene (PS)
VPh - vinyl phenol
ST – Polystyrene (PS)
VA- vinyl acetate
Ethylene
BC
VPh-co-BR/ MMA-co- ST
AD
BC
VPh-co- ST/E-co-VA
AD
VPh-co-BR
BC
MMA-co- ST
AD
1
𝛿PVPh = 10.6 [
1
𝑐𝑎𝑙 2
]
𝑐𝑚3
𝛿BR
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 8.1 [ 3 ]
𝑐𝑚
1
1
𝑐𝑎𝑙 2
𝛿PMMA = 9.1 [ 3 ]
𝑐𝑚
𝛿PS
VPh-co- ST
BC
𝛿PVPh
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
E-co-VA
AD
1
𝑐𝑎𝑙 2
= 10.6 [ 3 ]
𝑐𝑚
Strong self-association
diluent
Non self-association
diluent
Strong self-association
diluent
Non self-association
diluent
1
𝛿PS
(𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 9.5 [ 3 ]
𝑐𝑚
1
𝛿Ethylene (𝑑𝑖𝑙𝑢𝑒𝑛𝑡)
𝑐𝑎𝑙 2
= 8 [ 3]
𝑐𝑚
1
𝛿𝑉𝐴
𝑐𝑎𝑙 2
= 9.6 [ 3 ]
𝑐𝑚
Comment on the results:
The system will be miscibility in areas where there are no red dots. A mixture of two polymers with close solubility
parameters can give us a wider range of miscible areas, for example VPh-co-BR (2.5) compared to VPh-co-ST (1.1) and
MMA-co-ST (0.5) compared to E-co-VA (1.6).
Additionally, when polymers with low solubility parameter are combined with polymers with high solubility parameter,
a higher percentage of polymer with high solubility parameter will be added, so the solubility parameter of the
copolymer will be closer to that of the polymer with high solubility parameter.
Example of calculation:
VPh-co-BR/ MMA-co- ST
AD - MMA-co- ST
BC - VPh-co-BR
VPh-co- ST/E-co-VA
AD - E-co-VA
BC - VPh-co- ST