AMTF Reading Session

Rubber Elasticity (Part 2)

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Time: 2017, 12, 1 (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.      Energetic contribution to rubber elasticity

2.      Limitations and modifications of the statistical theory

3.      Affine deformation

4.      Effect of degree of crosslinking

5.      Strain-induced crystallization

6.      The phenomenological treatment

7.      Swelling of network

 

 

Discussion:

        Although the rubber elasticity is primarily entropic in origin, there is still a small contribution from the energy. From the thermodynamic equation of elasticity, the energetic contribution is given by:

圖片

        Then consider a single chain whose end-to-end distance is r, the conformational entropy of this chain is given by:

 圖片

        Next, we use the expression of the retractive force:

圖片

So,

圖片

        In a real network, defects may exist, and hence the number of network chains contributing to the elasticity (effective chains) is less than the total number of chains. The elasticity equation should be modified as:

圖片

Where ne is the number of moles of the effective network chains per unit volume.

Now we consider a network with tetrafunctional crosslinks. Even if all crosslinks are normal, there must be two terminal chains on the crosslinks in the end of the network architecture:

圖片

        So the number of effective network chains is Ne=N-2NP, where N is he total number of the network chains, and NP is the total number of the polymer chains (a polymer chain is defined as the chain formed from one end to the other).

        Note that n=(ρ/Mc), thus


圖片

        Finally we consider the situation when a network is mixed with a solvent. Since the chains in the network are all tied by the crosslinks, the rubber will only “swell” but not get dissolved in the solvent:

圖片

        The free energy change, ΔG, associated with the swelling of a network can be separated into two parts:

(a) the free energy of mixing involving the mixing of pure solvent with the initial unstrained network, ΔGm

(b) the elastic energy involving the subsequent expansion of the network,ΔGel.

        After further calculation, we obtain:

圖片

        The plot of lnσ vs. lnqm should thus give a straight line with the slope of -5/3 according to the theoretical prediction. The experimental results have shown the value of the negative slope is somewhat greater than 5/3.

Feedback:

        It is helpful to know the theoretical calculation and prediction of the viscoelastic behavior of polymer. Besides, we also know the mechanism of the elasticity, swelling, and expansion behavior from this part. We can do further research base on these knowledge.

Future works:

        This is the final part of our discussion. We will keep reading this book to get more knowledge.

 


蕭舒云 / 2017-12-12

Rubber Elasticity (Part 1)

 Time: 2017, 11, 24 (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.

Bradley Mansel, Chemical Engineering, NTHU

Detail:

1.      Rubber Elasticity

2.      Free energy state

3.      Energetic in origin

4.      Entropic in origin

5.      Crosslinking

6.      Equation of state

7.      Thermoelastic inversion

8.      Three essential conditions for rubber elasticity

9.      Structure-dependent term and volume-dependent term

10. Helmhotz free energy

11. Elongation ratio

12. Statistical Theory of Rubber Elasticity

13. Ideal rubber

Discussion:

A rubber band can be stretched to several times its original length, and when the force is released, the rubber band will return to its original length. This elastic behavior of rubbers is called “rubber elasticity”

A spontaneous process always goes toward the lower free energy state. The free energy of unstretched state is lower than of the stretched state. Gu < Gs

Because when we stretch a metal, the distance of separation between the atoms will be increased, and the interaction energy is increased. The elasticity of metals is “energetic in origin”. The elasticity of polymer chains is “entropic in origin”.

 The three essential conditions for rubber elasticity

圖片

(a) the chain between two crosslinkages (i.e. the network chain) should be long enough

(b) the chains can neither be crystalline nor glassy

(c) permanent network architecture

 

Use the classical thermodynamic treatment of rubber elasticity:

圖片

dE = dq – dw dE = TdS – dw

dw = Pdv - fdl

G = H – TS

Then we can write f = (dH/dt)T,P+T(df/dT)l,P , This equation may be regarded as a thermodynamic “equation of state”.

Then f = (dE/dl)T,V + T(df/dT)l,V , fe = (dE/dl)T,V , fs = -T(dS/dl)T,V

Thus f = fe + fs

Because (dS/dl)T,V ≈ (df/dT)P,α , f = (dE/dl)T,V +T(df/dT)l,V  ≈ (dE/dl)T,V + T(df/dT)P,α

Where α is the elongation ratio, α=l/l0.

 

l0 will vary with temperature due to the effect of thermal expansion. So we will have to adjust the length of the rubber.  α= l(T2)/l0(T2) = l(T1)/l0(T1)

圖片

Because (dE/dl)T,V ≈ 0, the rubber elasticity is primarily due to the decrease in comformational entropy upon stretching. We can define an “ideal rubber” as the rubber which  (dE/dl)T,V = 0. Therefore, for an ideal rubber f = -T(dS/dl)T,V .

 In the statistical theory of rubber elasticity, 圖片

   This is the elasticity equation of a rubber. The assumptions made are:

(a) perfect network structure

(b) affine deformation

(c) uniaxial stretching

(d) constant volume

 

The stress can be in terms of moles of chains per unit volume, n0

圖片

it results that at high elongation, an abrupt upturn is observed experimentally.

Feedback:

  Rubber elasticity is very interesting section. In our life, more things

contain their elasticity. This section let me learn the rubber motion and what are their free energy state. Then I can know it’s elasticity can be affected by temperature. Also can know it’s elasticity primarily due to the decrease in comformational entropy upon stretching. I will look forward the next time discussion about energetic contribution to rubber elasticity.

Future works:

 

We will discuss the energetic contribution to rubber elasticity in the next discussion time. We will learn more about the theories of rubber elasticity.


蕭舒云 / 2017-12-07

Thermodynamics of Polymer Solutions (Part 1)

 Time: 2017, 11, 10 (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: Yi-Chun Liao

Guidance introduction:

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

Bradley Mansel, Chemical Engineering, NTHU

Detail:

1.      Thermodynamic Function

2.      Chemical Potential

3.      Osmotic Pressure

4.      Properties of Mixing of Ideal Solution

5.      Statistical Thermodynamics of an Ideal Solution

6.      Conditions of Miscibility

7.      Problems Associated with the Application of Classical Ideal Solution Theory to Polymer Solution

8.      Entropy of Mixing in Flory-Huggings Theory

9.      Athermal Solution in Flory-Huggins Theory

10. Heat of Mixing in Flory-Huggins Theory

11. The Solubility Parameter

Discussion:

  At first we recall some thermodynamic function include Internal Energy, Heat (q), Work (w), Entropy (S), Helmholtz free energy (A) and Gibbs free energy (G).

Work (w): dE=dq-dw

Entropy (S): dS=dqrev/T

Helmholtz free energy (A): A=E-TS

Gibbs free energy (G): G=H-TS

 

  Then we focus on Entropy(S) and Gibbs free energy(G) to derive Chemical potential. In the muticomponent system, G is the function of temperature, pressure and numer of moles, i.e. G=G(T, P, n1, n2, ...)

So dG=(dG/dT)P,nidT+(dG/dP)T,nidP + Σi(dG/dni)T,P,ni≠jdnj

We can define chemical potential μi=(dG/dni)T,P,ni≠j

        For a two phase solution at equilibrium under constant T and P

dG=0=Σμidniμidni=μ1Adn1A+μ2Adn2A+μ1Bdn1B+μ2Bdn2B

In closed system, n1=n1A+n1B and dn1=0 → dn1A= -n1B

We can derive, μ1A=μ1B μ2A=μ2B

We can know if two phase are in equilibrium at constant T and P, the chemical potentials of the components in each phase must be identical.

 

By the chemical potential we can calculate the osometer pressure (π)

π=(μ10-μ1)/V1μ1=μ10+RTlnx1 → π=-RTlnx1/V1

x1=mole fraction of the solvent in the solution

x2=mole fraction of the solute in the solution

The Gibbs free energy of mixing ideal solution △Gm=RT(n1lnx1+n2lnx2)

The Entropy of mixing ideal solution △Sm=-R(n1lnx1+n2lnx2)

If we use statistical thermodynamic to calculate entropy S=kBlnP

P=CΩ, Ω=conformational probability or statistical weight Ω=N!/Πini!

And we will use a calculate skill “Stirling approximation” lnR!=RlnR-R

 

Therefore, the entropy of mixing of an ideal solution

△Sm=NS0-N1S0-N2S0-kB(N1lnx1+N2lnx2)=-kB(N1lnx1+N2lnx2)=-R(n1lnx1+n2lnx2)

 

In order to have good miscibility, we hope our system has two characteristics:

1. △Gm<0

2. (d2△Gm/dx12)>0

Flory-Huggins Theory

 

  Flory-Huggins Theory is a lattice theory for polymer solution. This theory takes into account that solute (polymer) and solvent molecules may be of very different sizes.

圖片

The concept of Flory-Huggins Theory

 

  It suppose solution has N lattices, and N1 is occupied by solvent and N2 is occupied by polymer.

N=N1+xN2, x is the degree of polymerization, x=(molar volume of the polymer)/(molar volume of the solvent)

The conformational probability of the solution is Ω=(y/N)N2(x-1)(N!/N1!N2!)

y = availble coordinate number for polymer segment

 

And we can expree entropy with Flory-Huggins Theory

△Sm=-kB(N1lnΦ1+N2l2)=-R(n1lnΦ1+n2lnΦ2)

Φ1=N1/N, Φ2=xN2/N

By Flory-Huggins Theory, we consider the nearest neighbor interactions, the internal energy of the solution is given by

E12=(N11ε11+N22ε22+N12ε12)

And we can calculate △ε12=ZNav(ε12-(ε11-ε22)/2)

Then we define Flory-Huggins Interaction Parameter χ=△ε12/RT

 

Now we can write down the complete expression for the Gibbs free energy of mixing based on Flory-Huggins Theory:

△Gm=RT(n1lnΦ1+n2lnΦ2+χn1lnΦ2)

Feedback:

  Thermodynamic is a very important part in my research. In order to more understand Thermodynamic, I study not only Polymer Phycis but also Statistical Thermodynamic. This part of the polymer physcis is combind two subjects I learn this semester, so I think it is a good chance for me to reorganize what I learn this semester.

Future works:

 

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


蕭舒云 / 2017-11-24

Thermodynamics of Polymer Solutions (Part 2)

 Time: 2017,11,17 (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: Thermodynamic function & Flory-Huggins parameter

2.          Relationship between χ and polymer solutions

3.          Flory-Krigbaum Theory

4.          Phase Diagram of Polymer solutions

5.          Application

Discussion:

With osmotic method, we can obtain the π/c2 vs.c2 plot and distinguish the relationship between solvents and polymers, as shown below.

圖片

 

However, the figure shown above is based on Flory-Huggins theory. If we want to discuss the behavior in dilute solution, Flory-Krigbaum theory should be introduced. Different from F-H theory, F-K theory considers excluded volume effect and no interaction between polymer coils and finally we can obtain:

圖片

 

Another issue in polymer solution is its phase diagram. Different volume fraction with various temperatures can build the phase diagram as shown below:

圖片

To realize the stability of these phase, we can calculate △Gm with various Φ2. From calculating the 1st and 2nd order derivation we can obtain the critical points in phase diagram.

With the background knowledge shown below, we can do experiments to obtain M.W, θ temperature and phase diagram.

Feedback:

We have learned some basic knowledge about polymer but don’t know how to measure these properties. Today’s reading session gives us an initial understanding to approach the polymer properties.

Future works:

 

Next week, we will discuss another special property of polymer, called elasticity. Different from last chapter, mechanical properties are the key point in this chapter.


蕭舒云 / 2017-11-24

Conformation Statistics of a Single Chain (Part 4)

 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:

 

圖片

 

After further calculation, we finally get:

圖片

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:

圖片

圖片

After further work, we obtain:

圖片

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:

圖片

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:

圖片

 

Then we get the probability of a given chain conformation:

圖片

In reality, it is impossible to have a single chain in vacuum situation, so <r2> of the molecules may be effected by inter- and intra-molecular interactions.

 

Now we disscuss the three following situations:

圖片

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

 

Then we have <r2> and <Rg2> in “perturbed dimension.”

圖片

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.


蕭舒云 / 2017-11-13

Conformation Statistics of a Single Chain (Part 2)

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).

圖片…………………….…… (1)

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

圖片……………….…… (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.

 

 圖片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.


蕭舒云 / 2017-11-02

Conformation Statistics of a Single Chain (Part 3)

 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 r0 (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.

圖片

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.

  圖片

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

圖片

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).

圖片

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.

圖片

圖片

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.


蕭舒云 / 2017-11-02

Conformation Statistics of a Single Chain (Part 1)

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 Rg)”. 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 Rg 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.


蕭舒云 / 2017-10-23

Some Basic Concepts of Macromolecules

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.


蕭舒云 / 2017-10-19
Rubber Elasticity (Part 2) 2017-12-12 Rubber Elasticity (Part 1) 2017-12-07 Thermodynamics of Polymer Solutions (Part 1) 2017-11-24 Thermodynamics of Polymer Solutions (Part 2) 2017-11-24 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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