P iNa t; R
ð Þ¼
X n ia
m¼1
m!
m m À 1
ð
Þ!
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
! m
1 À
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
! m
&
' 2=2
t > t
0
À
Á
ð11Þ
They are fitted for the normalized conditions:
ð
P iNE t; R
ð ÞdR ¼ N iE
ð
P iNa t; R
ð ÞdR ¼ N ia
ð12Þ
N T ¼
X
i¼1
N ia þ
X
i¼1
N iE ¼ N Ta þ N TE
ð13Þ
ξ T ¼
X
i¼1
X
i¼1
À ξ iE N iNE þ ξ ia N iNa
Á
ð14Þ
where, N T is the total number of polymeric chains in the system, ξ T is the total
number of effective elastic chains in the system, N TE and N Ta are the total numbers
of entangled and adsorbed chains in the system, respectively; Γ(mE) and Γ(ma) are
Euler Gamma functions of m E and m a , respectively; ξ iE and ξ ia are the numbers of
effective elastic chains on the i kind of polymeric chains, respectively. In Eqs. (10)
and (11),
1
Γ m E
ð Þ t À t
0
À
Á m E À1 1 À P iE τ; Λ Ei
À
Á
À
Á
e
Àμ E tÀt
0
ð Þ P iE τ; Λ Ei
À
Á
h
i
and
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
h
i
represent the probabilities
of every single E and A-constituent chain to be reorganized, respectively,
therefore,
1 À
1
Γ m E
ð Þ t À t
0
À
Á m E À1 1 À P iE τ; Λ Ei
À
Á
À
Á
e
Àμ E tÀt
0
ð Þ P iE τ; Λ Ei
À
Á
h
i m
n
o
and
1 À
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
h
i m
n
o
represent the
probabilities of the tail segments lying on the both ends of testing polymeric
chain. When P iNa < 1, P iNE < 1, and summation of Eqs. (10) and (11) by geometric
series, they can be simplified to be Eqs. (15) and (16):
P iNE t; R
ð Þ ¼
1
Γ m E
ð Þ
t À t
0
m E À1
e
Àμ E tÀt
0
ð Þ p iE τ; Λ Ei
À
Á
ð15Þ
P iNa t; R
ð Þ ¼
1
Γ ma
ð Þ
t À t
0
m a À1
e
Àμa tÀt
0
ð Þ pa τ; Λ ai
À
Á
ð16Þ
For a transient double-network at deformation, the distribution functions of
statistical conformation for two kinds of chains are obtained after introducing a
condition of affine deformation, which is shown as follow:
168
Y. Chen and C. Xu
ð Þ¼
X n ia
m¼1
m!
m m À 1
ð
Þ!
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
! m
1 À
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
! m
&
' 2=2
t > t
0
À
Á
ð11Þ
They are fitted for the normalized conditions:
ð
P iNE t; R
ð ÞdR ¼ N iE
ð
P iNa t; R
ð ÞdR ¼ N ia
ð12Þ
N T ¼
X
i¼1
N ia þ
X
i¼1
N iE ¼ N Ta þ N TE
ð13Þ
ξ T ¼
X
i¼1
X
i¼1
À ξ iE N iNE þ ξ ia N iNa
Á
ð14Þ
where, N T is the total number of polymeric chains in the system, ξ T is the total
number of effective elastic chains in the system, N TE and N Ta are the total numbers
of entangled and adsorbed chains in the system, respectively; Γ(mE) and Γ(ma) are
Euler Gamma functions of m E and m a , respectively; ξ iE and ξ ia are the numbers of
effective elastic chains on the i kind of polymeric chains, respectively. In Eqs. (10)
and (11),
1
Γ m E
ð Þ t À t
0
À
Á m E À1 1 À P iE τ; Λ Ei
À
Á
À
Á
e
Àμ E tÀt
0
ð Þ P iE τ; Λ Ei
À
Á
h
i
and
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
h
i
represent the probabilities
of every single E and A-constituent chain to be reorganized, respectively,
therefore,
1 À
1
Γ m E
ð Þ t À t
0
À
Á m E À1 1 À P iE τ; Λ Ei
À
Á
À
Á
e
Àμ E tÀt
0
ð Þ P iE τ; Λ Ei
À
Á
h
i m
n
o
and
1 À
1
Γ ma
ð Þ t À t
0
À
Á m a À1 1 À P ia τ; Λ ai
À
Á
À
Á
e
Àμ a tÀt
0
ð Þ P ia τ; Λ ai
À
Á
h
i m
n
o
represent the
probabilities of the tail segments lying on the both ends of testing polymeric
chain. When P iNa < 1, P iNE < 1, and summation of Eqs. (10) and (11) by geometric
series, they can be simplified to be Eqs. (15) and (16):
P iNE t; R
ð Þ ¼
1
Γ m E
ð Þ
t À t
0
m E À1
e
Àμ E tÀt
0
ð Þ p iE τ; Λ Ei
À
Á
ð15Þ
P iNa t; R
ð Þ ¼
1
Γ ma
ð Þ
t À t
0
m a À1
e
Àμa tÀt
0
ð Þ pa τ; Λ ai
À
Á
ð16Þ
For a transient double-network at deformation, the distribution functions of
statistical conformation for two kinds of chains are obtained after introducing a
condition of affine deformation, which is shown as follow:
168
Y. Chen and C. Xu