**Time:** 2017, 11, 3 (Fri) 12:00~14:00

**Location: **CHE R701

**Guidance: **Chia-Hui Chen, Chih-Hsuan Lin

**Member:** Yi-Chun Liao, Shu-Yun Hsiao, Yu-Hsuan Hsieh, Yu-Fan Chang, Kai Chen, Chih-Mei Young

**Recorder: **Kai Chen

**Guidance introduction: **

Chia-Hui Chen, Institution of Polymer Science and Engineering, NTU.

Chih-Hsuan Lin, Chemical Engineering, NTHU

**Detail:**

1. Independent bond rotations

2. Interdependent bond rotations

3. An ensemble of polymer chains

**Discussion:**

We consider a simple case called “independent bond rotation” which means that the rotation of each bond is independent, that is, the conformation of a bond is independent of the conformation of its preceding bond.

The chain partition function for independent bond rotation is given by:

After further calculation, we finally get:

This equation is derived by assuming:

(a) independent bond rotation

(b) all bond lengths and bond angles are the same

To make this equation even simpler and more convenient to be used, we do:

After further work, we obtain:

With three assumption:

(a) independent bond rotation

(b) symmetric potential

(c) the chain is volumeless

Now we discuss “interdependent bond rotations,” the conformational state of a bond depends on the conformations of its neighboring bonds. For example, consider the three conformations of pentane:

This figure shows the “pentane effect,” that the distance between the methyl groups at the end with g^{+}g^{+} is longer than that with g^{+}g^{-}.

We consider the nearest-neighbor correlation, the energy of this conformation is:

Then we get the probability of a given chain conformation:

In reality, it is impossible to have a single chain in vacuum situation, so <r^{2}> of the molecules may be effected by inter- and intra-molecular interactions.

Now we disscuss the three following situations:

The short-range and long-range interaction are different in these situations.

Then we have <r^{2}> and <R_{g}^{2}> in “perturbed dimension.”

**Feedback:**

After this disscussion, we know how to calculate the end-to-end distances. And we started to discuss the phenomena in reality that we actually concern about. It is very useful to know the properties of polymer in further research.

**Future works:**

We will disscuss thermodynamics of polymer solutions which is important in polymer physics and polymer industry.

**Time:** 2017, 10, 20 (Fri) 12:00~14:00

**Location: **CHE R701

**Guidance: **Chia-Hui Chen, Chih Hsuan Lin

**Member:** Yi-Chun Liao, Shu-Yun Hsiao, Yu-Hsuan Hsieh, Yu-Fan Chang, Kai Chen, Chih-Mei Young

**Recorder: **Yu-Hsuan Hsieh

**Guidance Introduction: **

Chia-Hui Chen, Institution of Polymer Science and Engineering, NTU.

Chih Hsuan Lin, Chemical Engineering, NTHU.

**Detail:**

1. Review: The Definition of End-to-End Distance and Radius of Gyration

2. How to Describe Polymer Chain Distribution in Mathematics

3. Statistical Distribution of End-to-End Distance of a Freely Jointed Chain

i. Random-Walk Statistics: Gaussian Distribution

ii. Radial Distribution

**Discussion:**

We discussed the topic of end-to-end distance same as the contents mentioned last week. In addition to the value of radius of gyration, the distribution (probability) of end-to-end distance is important. The concept is simple and same as we learned in Physical Chemistry (a course in Department of Chemical Engineering), so the entropy of the chain would be introduced.

To derive the distribution of polymer chain, we should determine the entropy of the chain first. It is determined by calculating all probabilities of the position of the polymer chain will go. This concept is like a drunk guy is restricted to walk on a line and walks randomly, so called “random walk”. By many calculating steps, we obtain the statistical distribution of the 1-D random walk: Gaussian distribution in (1).

…………………….…… (1)

Furthermore, the same concept can help us obtain the 3-D freely jointed chain distribution, as shown in (2). (calculating steps is complicated.)

……………….…… (2)

Finally, A typical question “What is the probability of finding the other end that is r away from origin?”. We can find the answer by plotting W(r)dr vs. r.

**Feedback:**

In this session, we can not only obtain the useful knowledge about polymer physics but review some concepts we learned before. Solving the problem all together is a good experience for us because teamwork is important in Lab.

**Future works:**

Next time we will discuss the elasticity of a single chain in vacuum. From the knowledge about the distribution of end-to-end distance, chain length and bond rotation, we can obtain more information in polymer physics.

**Time:** 2017, 10, 27 (Fri) 12:00~14:00

**Location: **CHE R701

**Guidance: **Chia-Hui Chen, Bradley Mansel

**Member:** Yi-Chun Liao, Shu-Yun Hsiao, Yu-Hsuan Hsieh, Yu-Fan Chang, Kai Chen, Chih-Mei Young

**Recorder: **Yu-Fan Chang

**Guidance introduction: **

Chia-Hui Chen, Institution of Polymer Science and Engineering, NTU.

Bradly Mansel, Chemical Engineering, NTHU.

**Detail:**

1. What is elasticity of a single chain in vacuum?

2. Entropic origin

3. End-to-end distance

4. Trans and gauche states

5. Temperature coefficient

6. Persistent length

7. Transformation matrix

8. Conformation probability

**Discussion:**

The elasticity can be described by the classical thermodynamics. We can consider an unstretched polymer chain whose end-to-end distance is r_{0 }(Figure 1). We learned that trans and gauche states have lower energies, we can assume that in a polymer chain, the bond conformation is either in the trans state or in the gauche state.

**Figure 1.** polymer chain behavers.

Then we can conclude the probability of the gauche conformation is proportional to the energy difference between the trans and gauche state.

When the temperature is low enough, there are many trans states in a chain. Then the chain extends like a rod (Figure 2).

**Figure 2.** The conformation of polymer when temperature is low enough.

When the temperature is high enough, there are many gauche states in a chain. Then the chain contracts (Figure 3).

**Figure 3.** The conformation of polymer when temperature is high enough.

Finally we can use the coordination transformation method to set up the conformation of polymer. Then we can get the result relationship about probability.

**Feedback:**

I haven't learned the class of physical chemical of polymer before. But in this section discussed many mechanism about elasticity of a single chain in vacuum. This section is so interesting. We can easily know how the polymer chain will behave in high temperature or low temperature. I think this is helpful for our research.

**Future works:**

I think next time we can discuss more detail about the mechanism of polymer and find some example to do. We can also practice our mathematical ability to calculate the problem of the polymer behavers. Then we can use these results to solve our experiment problems.

**Time:** 2017, 10, 13 (Fri) 12:00~14:00

**Location: **CHE R701

**Guidance: **Chia-Hui Chen, Bradly Mansel

**Member:** Yi-Chun Liao, Shu-Yun Hsiao, Yu-Hsuan Hsieh, Yu-Fan Chang, Kai Chen, Chih-Mei Young

**Recorder: **Shu-Yun Hsiao

**Guidance introduction: **

Chia-Hui Chen, Institution of Polymer Science and Engineering, NTU.

Bradly Mansel, Chemical Engineering, NTHU.

**Detail:**

1. What is Conformation?

2. End-to-End Distance and Radius of Gyration.

3. Freely Jointed Chain and Freely Rotating Chain in Vacuum.

4. Kuhn’s Concept of Equivalent Statistical Element.

5. Non-linear Polymer Chains.

**Discussion:**

Conformation means the structure, shape, or status of something. Three types of bond conformations (trans-, Gauche-, and cis-states) results from rotation of two atoms about a single bond. Different bond conformations have different energies. Normally, bonds in molecules would like to stay in lower energy conformations, like trans-state or Gauche-state, rather than cis-state.

For polymer, which contains many molecule bonds, it is hard to distinguish the conformations between different atoms. Therefore, chemist used to express the conformation of a polymer chain as “end-to-end distance (notated as r)” or “radius of gyration (notated as R_{g})”. End-to-end distance refers to the distance between one end and the other of a polymer. Radius of gyration, however, means the average distance of each atoms to the mass center of whole polymer chains.

The ideal polymer chain model is the “freely jointed chain” in vacuum. Any bonds in freely jointed chain could be randomly placed in space, so r in this kind of model only depends on the total number of chemical bonds and the length of these bonds. Another more complex model in vacuum is called “freely rotating chain”. The angles between chemical bonds take into consideration, so r depends on not only the sum and the lengths of bonds, but also the rotation angles.

The assumption in freely rotation chain model in vacuum (all rotation angle of chains are equal) is far from the reality. Kuhn proposed a concept for the “real” polymer chain in vacuum. The main concept is that the unit to characterize polymer chain replaced from single bond to the statistical element (conformed by m bonds).

For non-linear polymer chains, r is hard to define and characterize. Therefore, we then resort to the expression of R_{g} in the characterization of conformation of non-linear polymer chains.

**Feedback:**

Physical chemistry of polymers is the most basic subject for my graduate study and research. Group discussion is always more effective than individually study. Members could share different thoughts to help each other to learn a concept, and the guidance could give us some feedback or correct our thoughts immediately. However, the reading content is too broad that falls out of our imagination. It is a little pity that after the content discussion, we cannot have enough time to talk about our research or give feedback to each other.

**Future works:**

Beside the reading discussion, maybe we can prepare some interesting topic that related to every-week discussed content and share to others.

**Time:** 2017, 10, 06 (Fri.) 12:00~14:00

**Location: **CHE R701

**Guidance: **Chia-Hui Chen, Chih-Hsuan Lin

**Member:**

Yi-Chun Liao, Shu-Yun Hsiao, Yu-Hsuan Hsieh, Yu-Fan Chang, Kai Chen, Chih-Mei Young

**Recorder: **Yi-Chun Liao

**Guidance introduction: **

Chia-Hui Chen, Institution of Polymer Science and Engineering, NTU.

Chih Hsuan Lin, Chemical Engineering, NTHU.

**Detail:**

1. What are polymers?

2. Monomer

3. Linear, Branched, and Crosslinked Polymer Chains

4. Classification of Polymers According to their Chemical Structure

5. Copolymers

6. Thermoplastic polymers

7. Configuration and Conformation

8. Average Molecular Weight of Polymers

**Discussion:**

Polymer means marcomolecules; it combines by lot of monomers so it must has high molecular weight. Polymers have many classifications depending on different method of monomer combination, and it will influence physics properties of polymers. Therefore, how to control it such as blending, condensation, quenching and annealing is that we are interested in. Different polymer has different processing, so there are a lot of knowledge we have to learn and practice to use the learning as the basis of our research.

**Feedback:**

I have taken some polymer course at college, so this section is not too hard for me. This part is the basic knowledge of polymer; though I have learned it before, I would still learn it again and do my best to discuss with our members.

**Future works:**

I think next time we cannot only discuss the textbook but also combine the content of the textbook with our practical experiments.

Conformation Statistics of a Single Chain (Part 4) 2017-11-13
Conformation Statistics of a Single Chain (Part 2) 2017-11-02
Conformation Statistics of a Single Chain (Part 3) 2017-11-02
Conformation Statistics of a Single Chain (Part 1) 2017-10-23
Some Basic Concepts of Macromolecules 2017-10-19

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