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Study Guide - Practical 7
Polycondensation
Introduction
Step-growth polymerization refers to a type of polymerization process in which polymer chains are formed through successive reactions between dissimilar functional groups. A wide range of chemical transformations can be employed to produce polymeric materials via this mechanism, including esterification, transesterification, amidation, and the formation of urethanes, among others.
The polymerization typically involves reactions between two complementary functional groups, for example, hydroxyl and carboxyl groups in the synthesis of polyesters, or hydroxyl and isocyanate groups in the formation of polyurethanes.
Molecules capable of undergoing step-growth polymerization generally fall into two main cate- gories:
- Single monomers bearing two different reactive functional groups within the same molecule. A classic example is amino acids, which contain both an amine group (–NH 2 ) and a car- boxylic acid group (–COOH), allowing them to react intramolecularly to form polyamides.
– H 2 O
Figure 1: Self-polycondensation of 11-amino undecanoic acid to form Nylon 11
Reactions involving such monomers are commonly referred to as AB-type systems, where each monomer contains one functional group of type A and one of type B. Polymerization occurs through self-condensation: the A group of one monomer reacts with the B group of another, forming a linear chain. This mechanism allows polymer formation without the need for a second monomer.
n AaB ⇄ A(a)nB + (n − 1)AB
- Two different polyfunctional monomers, each containing only one type of reactive group. These are often bifunctional molecules (e.g., diacids and diols) but may also include tri- or higher-functional monomers depending on the desired polymer structure. An example is the reaction between dicarboxylic acids (e.g., terephthalic acid, containing two –COOH groups) and polyols (e.g., ethylene glycol, containing two –OH groups), which yields polyesters such as polyethylene terephthalate (PET).
– H 2 O
Figure 2: Polycondensation of Terephthalic Acid (TPA) and Ethylene Glycol (EG) to form Polyethylene Terephthalate (PET)
This class of reactions is typically described as AA–BB-type systems, in which monomers with two A-type functional groups react with monomers containing two B-type groups. Polymer growth occurs through alternating reactions between the two types of monomers, producing stepwise chain extension.
n AaA + n BbB ⇄ A(ab)nB + (2n − 1)AB
Functionality
The functionality of a monomer, commonly denoted as f , indicates the number of reactive func- tional groups per molecule. In step-growth polymerization, monomer functionalities typically range from 2 to 8.
The choice of functionality directly impacts the structure and properties of the resulting poly- mer:
- A functionality of f = 2 results in linear polymer chains
- A functionality of f > 2 leads to branched or crosslinked network structures
This structural distinction is critical when designing materials with specific mechanical, thermal, or chemical properties.
Polycondensation of AA-BB Systems
Step-growth polyreactions involve monomers with electronically independent or clearly defined functional groups, and they mainly refer to polycondensations. However, a few important examples of polyadditions also fall into this category.
Let’s now take a closer look at bifunctional polycondensations, meaning polyreactions that occur between bifunctional monomers to form linear polymers.
The first case, also the simplest, though not very common, is the polycondensation of het- erobifunctional monomers of the type AaB. This kind of polyreaction can be represented as follows: n AaB ⇄ A(a)nB + (n − 1)AB
The index n can take any integer value because, in step-growth reactions, all species, like monomers and oligomers, can react with each other.
A good example is the polycondensation of α–ω hydroxy acids to form polyesters, as shown Figure 3.
DPn =
Number of initial monomer molecules Number of polymer molecules
(n^0 A + n^0 B ) 2 (nA + nB ) 2
n^0 A + n^0 B nA + nB
In this expression, both monomer and polymer molecules are divided by 2 because nA and nB refer to functional groups, and each molecule contains two of them.
With a bit more algebra, we can arrive at the final expression for DPn:
DPn =
n^0 A + n^0 B nA + nB
n^0 A +
n^0 A r n^0 A (1 − p) + n^0 A
r
− p
1 + r 1 − r (2p − 1)
This equation reveals two important limiting cases: one when r = 1 and another when p = 1:
Case 1 – r = 1
DPn =
1 − p
Case 2 – p = 1
DPn =
1 + r 1 − r
Now that we’ve defined the degree of polymerization, calculating the number-average molecular weight becomes straightforward. It can be determined using the formula:
M (^) n = Meg + DPn · M 0
Here, Meg is the molecular weight of the end group, while M 0 is the average molecular weight of the repeating unit, calculated as:
- M 0 = Mru in AB systems
- M 0 =
Mru 2
in AA–BB systems
It’s also important to note that the purity of the reagents—in terms of the number of functional groups—affects the maximum achievable molecular weight. For instance, consider a reaction involving a mixture of bifunctional and monofunctional molecules with B-type reactive groups. The system is composed of AaA + BbB + Bb. In this case, the monofunctional molecules act as chain stoppers, limiting the attainable molecular weight.
To account for their presence, the stoichiometric ratio must be redefined as:
r =
n^0 A n^0 B + 2n^0 B,m
where n^0 B,m is the number of monofunctional B groups. The factor of 2 reflects the fact that one monofunctional molecule has the same effect on the reaction as an excess of one bifunctional molecule.
Effect of Monomer Functionality in Polycondensations
It is also important to take into account the structure of the reagents, especially in terms of their functionality. Functionality, commonly indicated by the symbol f , is defined as the number of reactive groups present on a single monomer molecule. For example, a difunctional monomer has f = 2, while a trifunctional monomer has f = 3, and so on. In Figure 5, polyols with different functionalities are shown:
f = 2 f = 3 f = 3 f = 6
Figure 5: Polyols with different functionalities
In real polymerization systems, especially industrial ones, it’s common to have a mixture of different monomers with varying functionalities. In such cases, the standard definition of the stoichiometric (or reactivity) ratio needs to be adjusted to reflect the total number of reactive groups of type A and B, not just the number of molecules.
The generalized definition of the reactivity ratio r becomes:
r =
P
i
fi · n^0 A,i P j
fj · n^0 B,j
In this expression:
- n^0 A,i and n^0 B,j are the initial numbers of moles of the i-th and j-th monomers carrying reactive groups of type A and B, respectively,
- fi and fj are their corresponding functionalities.
This approach ensures that the stoichiometric ratio r correctly represents the actual balance between reactive groups, rather than simply the number of molecules. Accounting for functionality is especially important when monomers with different numbers of reactive groups are involved, or when impurities such as monofunctional molecules (which act as chain stoppers) are present. Ignoring these contributions could lead to an overestimation of the achievable molecular weight and degree of polymerization.
Effect of the Equilibrium in Closed Systems
Polycondensation reactions are reversible and therefore subject to equilibrium constraints.
To drive the reaction toward the formation of higher molecular weight products, specific strate- gies can be adopted, such as applying a vacuum to the reaction system, to remove low-molecular- weight byproducts. In most polycondensations, these byproducts are typically water or alcohols.
Let’s now focus on closed systems, meaning systems where these strategies are not implemented and the byproducts accumulate over time. Under typical reaction conditions, if the mixture is allowed to reach thermodynamic equilibrium, the equilibrium constant is usually relatively low. As a result, it becomes impossible to achieve very high molecular weights.
- The number fraction (xi) is the fraction of molecules that have a specific molecular weight Mi. It is defined as: xi =
ni P^ ∞ i=
ni
ni N
Where ni is the number of chains with molecular weight Mi, and N =
P∞
i=
ni is the total number of chains.
- The weight fraction (wi) is the fraction of the total polymer mass contributed by molecules of weight Mi. It is defined as:
wi =
niMi P^ ∞ i=
niMi
ci P^ ∞ i=
ci
ci c
Where ci is the mass of chains with molecular weight Mi, and c =
P∞
i=
ci is the total mass of the polymer.
The concept of number fraction and weight fraction are very useful in defining the three key quantities that are typically used to describe MWD:
- Number Average Molecular Weight - M (^) n
M (^) n =
X^ ∞
i=
xiMi =
X^ ∞
i=
ni N
Mi =
P^ ∞
i=
niMi
P^ ∞ i=
ni
This average gives equal weight to all chains, regardless of their size, and is particularly sensitive to low-molecular-weight species.
- Weight Average Molecular Weight - M (^) n
M (^) w =
X^ ∞
i=
wiMi =
X^ ∞
i=
ci c
Mi =
P^ ∞
i=
ciMi
P^ ∞ i=
ci
This average places greater emphasis on high-molecular-weight chains, making it more reflective of properties such as viscosity and tensile strength.
- Polydispersity Index - P DI
P DI =
M (^) w M (^) n
The P DI quantifies the breadth of the molecular weight distribution. A perfectly monodis- perse polymer has P DI = 1, whereas most synthetic polymers exhibit P DI > 1 due to the stochastic nature of polymerization reactions.
Experimentally, the molecular weight distribution of polymers is commonly determined through chromatographic techniques, with Gel Permeation Chromatography (GPC), also referred to as Size Exclusion Chromatography (SEC), being the standard analytical method. GPC separates polymer chains based on their hydrodynamic volume, which depends on both the molecular weight and the chain conformation in solution.
In a typical GPC setup, the chromatographic column is packed with porous silica or polymeric beads. As the polymer solution passes through the column, smaller molecules are able to penetrate deeper into the pores, resulting in longer retention times. Conversely, larger molecules are excluded from most of the pore volume and leave the column earlier. Thus, the elution volume is inversely correlated with molecular size.
The instrument produces a chromatogram, from which key molecular weight parameters such as M (^) n, M (^) w, and the P DI can be calculated, typically using calibration curves based on polymer standards of known molecular weight.
Another technique employed for MWD characterization, particularly for low-molecular-weight species, is High-Performance Liquid Chromatography (HPLC). While HPLC offers superior resolution for small molecules and oligomers, its application in polymer analysis is generally limited to oligomer detection due to its reduced effectiveness in separating high-molecular- weight species.
Molecular Weight Distribution according to Flory-Schulz’s Theory
For step-growth polymerizations, the molecular weight distribution can be predicted using purely statistical considerations. From this point onward, we will consider the special case where the stoichiometric ratio r = 1, under the assumption that all terminal groups have equal reactivity.
Let’s assume we have a total of N 0 polymer molecules in the system. The probability of selecting a molecule with a degree of polymerization i corresponds to its mole fraction, defined as xi = NiN 0.
The probability that a molecule contains at least one monomeric unit is 1, since we’re only considering existing molecules. Statistically, the extent of reaction p can be interpreted as the probability that a functional group has reacted. So, the probability that a molecule has undergone one reaction (i.e., is at least a dimer) is given by 1 · p.
Similarly, the probability that a molecule is a trimer (i.e., has reacted twice) is 1 · p^2.
By generalizing this reasoning, the probability that a molecule has a length i, that is, it consists of i repeating units, is given by: xi = p(i−1)(1 − p)
Here, p(i−1)^ represents the probability of undergoing (i − 1) successive reactions, while (1 − p) accounts for the probability that the chain stops growing after reaching length i. In other words, it reflects the chance that no further reaction occurs at the terminal group of the chain.
It is possible to derive the expression for the mass fraction wi in a similar way:
wi =
Wi W 0
Ni Mi N 0 M 0
Ni M 0 i N 0 M 0
Ni i N 0
Given that: Ni = xi N = ni N 0 (1 − p)
