Question

The critical point of a Flory-Huggins solution. For the mixing of a polymer \(A\left(N_{A}=10^{4}\right)\) with a small molecule \(B\) ( \(N_{B}=1\) ), compute the critical volume fraction \(\phi_{c}\) and the critical exchange parameter \(\chi_{c}\).

Short answer

\( \phi_{c} = 0.01 \) and \( \chi_{c} = 50.5 \).

Step-by-step solution

- Understanding the given variables

Identify the values given in the problem: The degree of polymerization for polymer A is given by \(N_A = 10^4\). For the small molecule B, the degree of polymerization is \(N_B = 1\).

- Using the critical point conditions

For the Flory-Huggins solution theory, the critical volume fraction \(\phi_{c}\) and critical exchange parameter \(\chi_{c}\) are needed. The critical point conditions are given by the equations: \[ \phi_{c} = \sqrt{ \frac{ N_B }{ N_A } } \] and \[ \chi_{c} = \frac{1}{2}(1 + \sqrt{ \frac{N_A}{N_B} }) \].

- Calculating the critical volume fraction

Substitute \(N_A\) and \(N_B\) into the formula for \(\phi_{c}\): \[ \phi_{c} = \sqrt{ \frac{ 1 }{ 10^4 } } = \sqrt{ 10^{-4} } = 10^{-2} = 0.01 \].

- Calculating the critical exchange parameter

Substitute \(N_A\) and \(N_B\) into the formula for \(\chi_{c}\): \[ \chi_{c} = \frac{1}{2}(1 + \sqrt{ \frac{10^4}{1} }) = \frac{1}{2}(1 + 10^2) = \frac{1}{2}(1 + 100) = \frac{101}{2} = 50.5 \].

Key concepts

critical volume fraction

The critical volume fraction, \(\phi_{c}\), is a critical concept in Flory-Huggins solution theory. It represents the fraction of the polymer volume at which the solution is at the threshold of phase separation. In simpler terms, it tells us the point where a polymer solution stops being uniform and starts to separate into distinct phases. For a polymer A mixed with a small molecule B, this can be calculated using the formula:
\[ \phi_{c} = \sqrt{ \frac{N_B}{N_A} } \]
Here, \(N_A\) and \(N_B\) represent the degree of polymerization of polymer A and the small molecule B, respectively.
To calculate \(\phi_{c}\) for a polymer A where \(N_A = 10^4 \) and a small molecule B where \(N_B = 1\), you substitute these values into the equation:
\[ \phi_{c} = \sqrt{ \frac{1}{10000} } = \sqrt{ 10^{-4} } = 0.01 \]
Therefore, the critical volume fraction for this specific scenario is 0.01 or 1%.

critical exchange parameter

The critical exchange parameter, \(\chi_{c}\), is another vital concept in Flory-Huggins solution theory. It measures the interaction between the polymer and solvent molecules. Essentially, it determines the compatibility of the polymer with the solvent. A higher \(\chi_{c}\) indicates poorer compatibility, leading to phase separation.
The critical exchange parameter is calculated using:
\[ \chi_{c} = \frac{1}{2}(1 + \sqrt{ \frac{N_A}{N_B} }) \]
Using the given \(N_A = 10^4\) and \(N_B = 1\), we substitute these values:
\[ \chi_{c} = \frac{1}{2}(1 + \sqrt{ 10000 }) = \frac{1}{2}(1 + 100) = \frac{101}{2} = 50.5 \]
Hence, the critical exchange parameter in this case is 50.5. This number tells us that the interaction is such that phase separation will likely occur when this threshold is reached.

degree of polymerization

The degree of polymerization, denoted as \(N\), is a crucial parameter in polymer science. It describes the number of repeating units in a polymer molecule. Think of it as the length of the polymer chain. A higher degree of polymerization means a longer chain and typically higher molecular weight.
In the Flory-Huggins solution theory, the degree of polymerization plays a significant role in determining both the critical volume fraction and the critical exchange parameter.
For example, in our problem, \(N_A = 10^4\) for polymer A, which means it has 10,000 repeating units. On the other hand, \(N_B = 1\) for the small molecule B, indicating it has only one repeating unit and is not a polymer but a monomer.
Understanding the degree of polymerization helps in predicting the behavior of polymer solutions, including their phase separation tendencies and interactions with solvents.