Abbreviations
c
Concentration of polymer in g mL
À1
K
Constant dependent upon the polymer
K
Equilibrium rate constant
k À1
Rate coefficient for two ends diffusing apart
k 2
Rate coefficient for the chemical reaction between the two chain ends
k c1
Rate coefficient for diffusion of chain ends from different chains within
the capture volume
k l1
Rate coefficient for diffusion of two ends of the same chain within the
capture volume
l
Length of a covalent bond
M e
Molecular weight between entanglements
M w
Weight-average molar mass
N
Total number of polymer molecules in total volume V
N A
Avogadro’s number
P c
Probability that the two ends of the same chain are within the capture
volume
P L
Probability that chain ends from different chains are within the capture
volume
>
Mean square end-to-end distance of the chain
>
1/2
Root mean square end-to-end distance
> θ
Root mean square radius of gyration in a θ-solvent
T g
1
Glass transition temperature at infinite molecular weight
T g
Glass transition temperature
v s
Capture volume
W(r)
Possibility distribution function
ρ
Density of the polymer
1 Introduction
Cyclic polymers have one of the simplest topologies, yet they have some of the
most intriguing properties, many of which remain poorly understood. The physical properties of linear polymers in their melt state can be predicted using the
theory of reptation [1, 2]. Linear polymers diffuse within the constraints of
adjacent polymer chains, in which the chain ends play a most important role
due to their ability to explore a much greater volume than the interior of the
chain [1, 3]. Cyclic polymers have no chain ends and, at first sight, one would
assume that they diffuse in a very different way and at a much slower rate.
Diffusion experiments show the contrary; cyclic polymers have a diffusion rate
coefficient approximately twice as fast as linear polymers of the same molecular
weight [4]. This is postulated to be due to an amoebae-like motion for the cyclic
296
Z. Jia and M.J. Monteiro
c
Concentration of polymer in g mL
À1
K
Constant dependent upon the polymer
K
Equilibrium rate constant
k À1
Rate coefficient for two ends diffusing apart
k 2
Rate coefficient for the chemical reaction between the two chain ends
k c1
Rate coefficient for diffusion of chain ends from different chains within
the capture volume
k l1
Rate coefficient for diffusion of two ends of the same chain within the
capture volume
l
Length of a covalent bond
M e
Molecular weight between entanglements
M w
Weight-average molar mass
N
Total number of polymer molecules in total volume V
N A
Avogadro’s number
P c
Probability that the two ends of the same chain are within the capture
volume
P L
Probability that chain ends from different chains are within the capture
volume
>
Mean square end-to-end distance of the chain
>
1/2
Root mean square end-to-end distance
> θ
Root mean square radius of gyration in a θ-solvent
T g
1
Glass transition temperature at infinite molecular weight
T g
Glass transition temperature
v s
Capture volume
W(r)
Possibility distribution function
ρ
Density of the polymer
1 Introduction
Cyclic polymers have one of the simplest topologies, yet they have some of the
most intriguing properties, many of which remain poorly understood. The physical properties of linear polymers in their melt state can be predicted using the
theory of reptation [1, 2]. Linear polymers diffuse within the constraints of
adjacent polymer chains, in which the chain ends play a most important role
due to their ability to explore a much greater volume than the interior of the
chain [1, 3]. Cyclic polymers have no chain ends and, at first sight, one would
assume that they diffuse in a very different way and at a much slower rate.
Diffusion experiments show the contrary; cyclic polymers have a diffusion rate
coefficient approximately twice as fast as linear polymers of the same molecular
weight [4]. This is postulated to be due to an amoebae-like motion for the cyclic
296
Z. Jia and M.J. Monteiro