We describe the statistical-mechanical theory of macromolecules which consist of N solute molecules in anisotropic solvents. The result of conformations is one-dimensional inextensible semiflexible rod, which interacts with the solvent anisotropy. To analyze the dynamics of a such chain we use path integral approach. Analytical study shows that the contribution of kinetic energy to the partition function is simulated by the power laws dependence and plays unique role in thermodynamic equilibrium.

1. Power-law contribution of the kinetic energy of
reversible self-assembly worm-like chain in
thermal bath.
PhD Student Andrii S. Lukianets
Head of department Pyotr M. Tomchuk
Theoretical Physics | Institute of Physics NASU
”
October 13, 2015

2. Statistical theory of polymer growth: predictions
𝐻 =
𝑛
𝑈𝑐𝑜𝑛𝑓 𝑟𝑛 +
𝑛
𝐾( 𝑣 𝑛) 2
𝑛
a. The product of aggregation is one-dimensional inextensible worm-like chain. b. Dependence of the total number of dye molecules in all n-mers on the length.
𝑐 𝑛
𝑍 = 𝑒𝑥𝑝
−𝐻
𝑘𝑇
98% of all boos that deals with Polymer Physics
1
Aggregation phenomena and chain conformation

3. c. The product of aggregation is a rigid rod. d. Partition function renormalization.
Statistical mechanics of aggregation in anisotropic solvents: kinetic energy of aggregates and universal power-law behavior far from criticality/ Pergamenshchik
𝑐 𝑛
𝑛
Kinetic energy contribution
Only Potential energy
𝐻 =
𝑛
𝐻 𝑛 =
𝑛
𝑈𝑐𝑜𝑛𝑓
𝑛
+ 𝐾𝑡𝑟
𝑛
+ 𝐾𝑟𝑜𝑡
𝑛
+ 𝐾𝑓𝑙𝑒𝑥
𝑛
+ 𝑖𝑛𝑡 𝑍 𝑎,𝑛 =
1
𝑁𝐶 𝑛 !
𝜈𝑛 𝑞 𝑒 𝑛−1 𝜀/𝑘𝑇 𝑁𝐶 𝑛
𝜀
43
The motion of a polymer in a solution
= 𝐵−1
𝑛 𝑞
𝑒−𝑛(𝜑−𝜀/𝑘𝑇)
Principled contribution of the kinetic energy
Self-intersection

4. Modeling and attempts of description:
Kinetic energy contribution to the partition function
e. System modeling
5
𝑑𝑝 𝑒𝑥𝑝 −
𝑝2
2𝑚
~𝑚
1
2~𝑛
1
2

5. Coupled pendulums
𝐾 =
𝑚𝑙2
2
𝑘=1
𝑁−1
𝑁 − 𝑘 𝝋 𝒌
2 +
𝑘=2
𝑁−1
2 𝑁 − 𝑘
𝑘=2
𝑘−1
𝝋𝒊 𝝋 𝒌 cos(𝝋𝒊 − 𝝋 𝒌) 7
𝐾 =
𝑚
2
𝑛=1
𝑁
𝑹 𝒏
2
𝑅 𝑛 = 𝑅1 +
𝑛=1
𝑁−1
𝑟𝑖 𝑟𝑖 = 𝑃𝑖, 𝑃𝑖+1
6
Attempt #1: Kinetic energy of freely jointed chain
𝑦
𝑥
𝑹 𝒏
𝑹 𝒏+𝟏
𝑹 𝑵
𝒓𝒊 𝝋 𝑵−𝟏
h. System consists
of N-beads
connected by N-1
links of fixed length
𝒑𝒊 =
𝜕𝐻
𝜕 𝝋𝒊
8

6. 𝜓(𝑅0, 𝑅𝑓, 𝑡𝑓) = 𝐴
𝑅0
𝑅 𝑓
𝒟𝑹 𝑡 𝑒𝑥𝑝 −
0
𝑡 𝑓
𝑹 𝒏
𝟐
𝑡
4𝐷
𝑑𝑡
𝜕𝜓
𝜕𝑡
= 𝐷2
𝜕𝜓2
𝜕𝑅2
𝜓 = 𝐶
𝑛=1
𝑁
𝑅0
𝑅 𝑓
𝒟𝑹 𝒏 𝑡 𝑒𝑥𝑝 −
𝑀
4𝑘𝑇𝜏𝐿
𝑛=1
𝑁
𝑎
0
𝑡 𝑓
𝑹 𝒏
𝟐
𝑡 𝑑𝑡
𝑛=2
𝑁
𝜹
𝑹 𝒏 𝑡 − 𝑹 𝒏−𝟏 𝑡 2
𝑎2 − 1
Dynamics of a three-dimensional inextensible chain/ Ferrari, Paturej, Vilgis
10
9
11
Brownian particles: constrains
𝑅0
𝑅𝑓
Attempt #2: Path integral approach to the dynamics of a polymer chain
g. Brownian particle in motion

7. 𝑍 𝑎 = 𝒟𝒓𝑖 𝒟𝒗𝒊 𝑒𝑥𝑝 −
𝑖
𝑚
2𝑘𝑇
𝒗𝑖
2
𝑹 𝒏
𝑹 𝒏+𝟏
𝒓𝒊
𝒗𝒊+𝟏
𝒗𝒊
12
Path integral form
13
𝑖=1
𝑁−1
𝜹 𝒓𝑖(𝝊𝑖+1 − 𝝊𝑖)
Attempt #3: Path integral approach to the motion of a polymer chain
g. Constrained trajectories
𝑍 𝑎,𝑣 = 𝐴
𝑖=1
𝑁−1
𝜋
𝛼𝑖
𝛼𝑖+1 = 1 +
𝑐𝑜𝑠2 𝜑𝑖,𝑖+1
4𝛼𝑖
𝛼1 = 1

8. 𝑹 𝒏
𝑹 𝒏+𝟏
𝒓𝒊
𝒗𝒊+𝟏
𝒗𝒊
𝑍 𝑎,𝑣 =
𝜋
2
1
𝑙
𝑁−1
2𝑘𝑇𝜋
𝑚
𝑁+1
2
𝜋
𝑁−1
2 1 +
1
4
𝑖=1
𝑁−2
𝑐𝑜𝑠2 𝜑𝑖,𝑖+1
−
1
2
14
Estimation:
15
Path integral approach to the motion of a polymer chain
𝑐𝑜𝑠2
𝜑𝑖,𝑖+1
4
≤
1
4

9. Thanks