SUPPLEMENTARY INFORMATION SECTION
A.1 Select Experimental Detail
A.1.1 Determination of Acid-Content of Functionalized Polymer
The extent of MPA incorporation was determined upon analysis of
1
H NMR spectra
by integration of backbone and side-group peaks, with the following complication. For
PB functionalization with MPA, we e
xpect two peaks between 2.85 and 2.55 ppm
corresponding to HO
2
CC
H
2
CH
2
S- protons and HO
2
CCH
2
C
H
2
S-, with one or more
additional peaks (depending on whether cyclic structures
are formed or not) around 2.55
ppm, corresponding to HO
2
CCH
2
CH
2
SC
H
2
- protons. Experimental results consistently
showed two large, partially overlapping
peaks between 2.825 and 2.575 ppm, and a much
smaller, broad overlapping shoulder below
2.575 ppm, extending in some cases down to
2.4 ppm (for instance, for 510kPB4.7A polymer th
at shoulder peak was about 1/3 of the
size of the first two). These observations indicate that the signal for HO
2
CCH
2
CH
2
SC
H
2
-
is broad, with significant overlap with the
middle peak, and that for simplicity the 2.85-
2.4 ppm range should be integrated as a
whole, accounting for 6 protons for each
functionalized monomer. The
complication is that the inte
gration of the small shoulder
region between 2.55 and 2.4 ppm was found to be
completely unreliable (due both to its
small size at the very low extents of functi
onalization investigated,
and apparently to the
presence of the very large neighboring PB b
ackbone peak): in many cases, the software-
computed integral was a physically nonsensical
negative number. As a result, extents of
MPA functionalization were approximated inst
ead in all cases by integrating between
2.825 and 2.575 ppm and estimating that that integral accounted for 5 protons for each
functionalized monomer.
A.1.2 Extensional Viscosity Results
Figure A.1 below reports apparent extens
ional viscosity for the solutions whose
capillary breakup behavior is shown in Figure 13.
A-1
2
3
4
5
6
7
8
9
1
2
3
4
5
Apparent Extentiona
l Viscosity (Pa.s)
10
9
8
7
6
5
Hencky Strain
1250kPB
1250kPB0.3A
1250kPB0.6A
Figure A.1
Effect of pairwise, self-associations
on apparent extentional viscosity during
capillary breakup rheology (refer to Figur
e 13 for experimental details).
A.2 Numerical Approach for Chain Stat
istics of Self-Associating Chains at
Infinite Dilution in
θ
-Solvent
A.2.1 Model Description
Our objective is to determine
the size of a linear chain of
N
monomers,
f
of which are
modified to act as stickers capable of form
ing pair-wise only, physical associations. The
stickers are assumed to be equidistantly spaced
l
monomers apart along the chain, and the
energy of association is
ε
kT
. We will assume Gaussian chain statistics for any segment
of the chain whose configuration is unre
strained by reversible crosslinks.
To calculate the size of the chain in th
e very dilute regime (all associations are
intramolecular), we define a semi-Markov process X(
t
),
t
> 0 such that each state
i
is fully
specified by identifying which pa
irs of stickers form bonds. (Note here that a given state
has an infinite number of chain configurations.) The chain goes from one state to the
next by either breaking or forming
a bond, as illustrated in Figure A.2.
A-2
Stickers are indexed from one end of the chain to the other:
3
5
6
1
4
2
7
(a)
Assume the chain is in the following state at time t:
Say the next state occurs by
breaking a bond, e.g. (2,7):
And the following state occurs by
forming a bond, e.g. (6,7):
Free:
2
6
7
Bonded:
(1,3)
(4,5)
(6,7)
Free:
2
Bonded:
(1,3)
(4,5)
Bonded:
(1,3)
(4,5)
(2,7)
Free:
6
l
monomers
(b)
(c)
(d)
Figure A.2
Schematic illustration of the transition from one state of the chain to another
by bond breaking and bond forming, for a chain with
f
= 7 stickers
This semi-Markov process is complete
ly specified if we know both (a) the
distribution of times T
i
the chain spends in any given state
i
and (b) the probabilities P
ij
that once it leaves state
i
it next enters state
j
. Thus, if we can determine the distributions
A-3
of T
i
, compute the transition probabilities P
ij
, and calculate relevant properties of the
chain in any state
i
, then we can estimate the long-run av
erage of chain properties such as
size by simply running the Markov process
for a sufficiently long time. Fortunately,
although the total number of states is extremely large for
f
as small as 20, the number of
states that are
accessible in one step
from any given state is much more manageable, that
is, P
ij
= 0 for most values of
j
for a given state
i
.
Assume the polymer chain enters state
i
at time
t
. Clearly, the state it enters next is
determined by which bond is broken or formed
first; and the time spent in the present
state is the time it took for that bond-breakin
g or bond-forming event to take place.
Because the times for bond breaking and bond forming are random variables, in order to
solve the problem we need to determine th
e distribution of the breaking time and the
forming time of all the possible bonds, for any state
i
.
Consider the breaking of a single bond. A
bond that has been “alive” for any given
time
s
is just as strong as a bond which has just formed; in other words the expected time
to break a bond given that is has been “alive” for time
s
is independent of
s
. Accordingly,
bond breaking is a memoryless (hence exponential)
process, and the tim
e to break a bond
is given by the exponential density function:
b
T
1
f()= exp-
bb
t
t
τ
τ
⎛⎞
⎜
⎝⎠
⎟
t
≥
0
where
τ
b
is the expected time to
break a bond, and has the sa
me value for all bonds.
Bond forming can also be argued to be an expo
nential process, as discussed below, with
expected time
τ
f
dependent on the number of monomers
L
qq’
in the shortest connected
path
between the two free stickers
q
and
q’
. Observe that th
e shortest path
L
qq’
between
stickers depends on the pair of stickers
q
and
q’
and on the current state of the chain; for
example, the shortest distance between sticke
rs 1 and 6 in state (b) of Figure A.1 is
L
1,6
=
2
l.
Let’s now see if we can determine expressions for
τ
b
and
τ
f
. Rubinstein and Semenov
1
give
τ
b
=
τ
0
exp(
ε
+
ε
a
), where
τ
0
is the monomer relaxation time and
ε
a
kT
is a potential
barrier for bond breaking and
also the activation energy for bond forming. I get
τ
f
≅
A-4
τ
0
/
p
c
p
s
to a first approximation (as discussed below), where
p
c
= (6/
π
L
3
)
1/2
is the contact
probability that the two stickers separated by
L
monomers be within distance
b
of each
other (
b
is the Kuhn length, assumed to be the
maximum distance over which the stickers
can associate), and
p
s
is the sticking probability that a bond is formed at any given “visit.”
If
p
s
≈
(
V
b
/
b
3
) exp(-
ε
a
) where
V
b
is the bond volume, then we get
τ
f
≈
α τ
0
exp(
ε
a
)
L
3/2
where
α
= (
b
3
/
V
b
)(
π
/6)
1/2
.
Let {
pp’
}
i
be the set of all pairs of s
tickers that are bonded in state
i
, and
{T
b,pp’
}
i
be
the set of random variables
corresponding to
their breaking time. Among the unpaired
stickers in state
i
, let {
qq’
}
i
be the set of all possible pa
irings for formation of a new
bond, and
{T
f,qq’
}
i
be the set of random variables co
rresponding to their expected bond-
forming time. For a given state
i
, then, we have independent, exponential random
variables {T
b,pp’
}
i
and {T
f,qq’
}
i
, with expected values E[T
b,pp’
] =
τ
b
and E[T
f,qq’
] =
α
τ
0
exp(
ε
a
) (
L
qq’
[
i
]
)
3/2
, where
L
qq’
[
i
]
is the number of monomer in
the shortest connected path
between the two free stickers
p
and
p’
in state
i.
The probability that the next state is
achieved by breaking a specific bond
pp’
, or by forming a specific bond
qq’
is the
probability that T
b,pp’
or T
f,qq’
is the shortest of all the br
eaking and forming times. For
independent exponential random va
riables this probability is (the rate of the given
exponential variable)/(the sum of
all the rates). Thus, the probab
ility that the
next state is
obtained from forming a bond between any two free stickers
q
and
q’
is:
[]
[]
-1
-3/2
-1
-3/2
e
P=
+e
qq' i
qq' i
ε
ε
qq'
α
L
Q
α
L
⋅
∑
where
Q
is the number of bonds in the present state
i
and the sum is over all the possible
pairs of unpaired stickers in
the present state. The probability that the next state is
obtained from breaking a given bond is:
[]
-1
-3/2
1
P=
+e
qq' i
ε
qq'
Q
α
L
⋅
∑
.
Note that these probabilities are independent
of the activation energy. Given that all the
breaking times and forming times are independe
nt exponential random variables, the time
A-5
T
i
to transition from any state
i
to the next is also exponen
tially distributed
, with rate
equal to 1/(sum of all the rates), so with mean:
[]
-1
-3/2
'
=
+e
qq' i
b
i
ε
qq
τ
τ
Q
α
L
⋅
∑
.
In this case (exponentially
distributed transition times), the semi-Markov process is a
continuous-time Markov chain. Continuous-tim
e Markov chains are characterized by the
Markovian property that,
given the present state, the future
is independent of the past.
This result is intuitive for the bond-breaking and bond-forming processes which
determine the time-evolution of
our chains: given the present
state, the next state and the
time to transition into the next state are bot
h independent of what states were visited
previously or how long the chai
n has been in the present state already. As a result of the
memoryless property of a continuous-tim
e Markov chain, the amount of time T
i
spent in
state
i
, and the next state visited are
independent
random variables. The reader is referred
to a text by Ross
2
for an excellent description of
Markov chains, semi-Markov chains,
and continuous-time Markov chains.
A.2.2 Distribution Function
for the Time to Form a Bond
Consider a strand of
L
monomers with 2 stickers at its ends. What is the probability
density function of the random variable T
f
, the time for the sticke
rs to form a bond (given
that they are not bonded at
the present time)?
Clearly there is an infinite number of
configurations for the strand, where a
configuration is specified by
specifying the position vectors
r
1
,
r
2
,...,
r
L
of all the
monomers.
However, if we break up space into a
3D lattice with arbitrarily small but
finite volume elements, then there is now a fin
ite number of strand configurations. If we
further define a macroscopic time
τ
0
over which the polymer configuration does not
change (and renormalize time in units of
τ
0
), then the configurations the strand takes over
time constitute an irreducible (all the st
ates communicate), positive recurrent (the
expected time to return to the present state is finite for all the states) Markov chain.
A-6
Therefore, there exist stationary probabilities
independent of initial state
u
for all states
v
.
n
n
π
=limP
v
→∞
uv
Given a probability density function for the
initial polymer configuration, there exists
a distribution function for T
f
, the time it will take for the stickers to bond
given
that they
are not bonded at the present time
t
= 0. If the probability de
nsity of the initial chain
configuration is the stationa
ry probability density, and
given
that the sticker ends have not
bonded after time
s
, the probability density to find the ch
ain in any given configuration at
time
s
is still equal to the sta
tionary probability density. Therefore, nothing has changed
after time
s
, so that the remaining time it will take to form a bond is independent of
s.
Accordingly, bond forming is a memoryless (h
ence exponential) proce
ss, and the time to
break a bond is given by:
b
T
1
f()= exp-
bb
t
t
τ
τ
⎛⎞
⎜⎟
⎝⎠
where
τ
f
is the expected value of T
f
. To determine
τ
f
for a strand of
L
monomers with
stickers at its ends, we need to know the time it takes for the stickers to come within close
enough distance of each other to associate. Assume stickers form a bond with sticking
probability
p
s
if they come within distance
b
(= Kuhn length) of each other. Consider
another semi-Markov process with
the following two states only:
the strand is in state 1 if
the stickers are within distance
b
of each other, and in state 2 otherwise. Let
μ
1
and
μ
2
be
the mean times spent in states 1 and 2, respectively (we do not need to know the
distribution of transition times). For
ε
= 0 (corresponding to a non-associating strand of
L
monomers), the long-run fraction
of time spent in state 1 is:
1/2
1
2
1
3
11
22
12
6
==
+
π
c
μ
p
μμ
L
⎛⎞
⎜⎟
⋅
⎝⎠
where
p
c
is the contact probability for the chain ends to be within distance b of each
other, and the only assumption is that of Gaussian statistics. But the mean time spent in
state 1 is
μ
1
≈
τ
0
, from which we get
μ
2
≈
τ
0
(1-
p
c
)/
p
c
. Note here that the above
expression for
μ
1
is also valid when
ε
≠
0 if stickers fail to st
ick while the strand is in
A-7
state 1, and that the above expression for
μ
2
is likewise also valid when
ε
≠
0. By
conditioning on the present state of the chai
n, the expected time to form a bond is:
()
()
12
2
2
ET =ET1P+ET 2P
=E T 1
+
μ
+E T 1 1-
=
μ
(1-
) + E T 1 .
ff
f
f
cf
cf
c
p
p
p
⎡⎤ ⎡ ⎤ ⎡ ⎤
⎣⎦ ⎣ ⎦ ⎣ ⎦
⎡⎤ ⎡⎤
⎣⎦ ⎣⎦
⎡⎤
⎣⎦
We find the expected time to bond give
n that the strand is
in state 1, E[T
f
|1], by
conditioning on whether the sticke
rs stick the first time around:
(
)
12
ET1
+ + +ET1 (1- )
f
s0
f
s
p
τμμ
p
⎡⎤
⎡⎤
≈
⎣⎦
⎣⎦
where p
s
≈
(V
b
/b
3
) exp(-
ε
a
) is the probability that the stic
kers form a bond while the chain
segment is in state 1. So after rearranging:
12
2
s
+
ET1
-
f
μμ
μ
p
⎡⎤
≈
⎣⎦
.
Substituting, we get:
00
τ
(1-
)(1-
)
τ
ET
ccs
f
cs
cs
ppp
p
pp
⎡⎤
≈≈
⎣⎦
p
.
A.2.3 Simplifying Assumptions of the Model
We made two simplifying assumptions in our
construction of the model. First, we
have assumed Gaussian statistics for any strand whose configuration is unrestrained by
reversible crosslinks. This assumption is reasonable in
θ
-solvents only to the extent that
congestion is not an issue. In reality there is a limit to the number of monomers that can
be collapsed into a given volume, and fo
r a high-enough number density of paired
stickers and long-enough chains, congesti
on is expected to become important.
Second, in our derivation of the time for
bond formation we assumed that the initial
configuration of the strand between the sticke
rs was chosen according to the stationary
probabilities
π
u
, where a state
u
corresponds to a specific conf
iguration of the strand.
Justification for this assumption is derived from
the fact that the probability density of the
strand configuration
will
reach the stationary probability
density, for any arbitrary initial
probability density, after a sufficiently long time during which the stickers do not pair up.
A-8
A-9
The assumption however presents
limitations for a pair of stic
kers right afte
r their bond is
broken: due to spatial proxi
mity, these may reassociate be
fore the strand between them
reaches its stationary probability density.
In other words, the memoryless Markovian
property may be violated to that extent in that the future is not completely independent of
the past.
A.2.4 References and Notes
1.
Rubinstein, M.; Semenov, A. N., Dynami
cs of entangled solutions of associating
polymers.
Macromolecules
2001,
34, (4), 1058–1068.
2.
Ross, S. M.,
Introduction to Probability Models
. 8th ed.; Academic Press: 2003.
.