Brownian dynamics simulation of diffusion-limited polymerization of rodlike
molecules: Isotropic translational diffusion
J. Srinivasalu Gupta and D. V. Khakhar
Citation: J. Chem. Phys. 107, 3289 (1997); doi: 10.1063/1.474679
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Brownian dynamics simulation of diffusion-limited polymerization
of rodlike molecules: Isotropic translational diffusion
J. Srinivasalu Gupta and D. V. Khakhar
Department of Chemical Engineering, Indian Institute of Technology—Bombay, Powai,
Bombay 400076, India
~Received 7 March 1997; accepted 8 May 1997!
Rigid rodlike polymers have considerable technological importance due to their excellent
mechanical properties. The polymerization kinetics of such condensation polymers are qualitatively
different from flexible polymers, and exhibit significant slowing at the later stages of the reaction.
This is due to the slow rotational diffusion of the molecules to an appropriate configuration for
reaction. In this work we have carried out Brownian dynamics ~BD! simulations to obtain the
effective rate constant for reaction between rodlike molecules in the presence of diffusional
limitations. The theory of Northrup et al. @J. Chem. Phys. 80, 1517 ~1984!# for pairwise BD
simulation of reactions is extended to the case of rodlike molecules assuming isotropic translational
diffusion. The computed results are compared to exact analytical predictions. Good agreement
between computation and theory is obtained over a wide range of parameter values. © 1997
American Institute of Physics. @S0021-9606~97!53030-3#
I. INTRODUCTION
Rigid rodlike polymers have gained considerable technological importance in the production of high-modulus and
high-strength fibers and films due to their excellent mechanical properties.1,2 The polymerization kinetics of these polymers, which are primarily made by condensation polymerization, are qualitatively different from those of flexible
polymers. At the molecular level, apart from the requirement
of the functional groups at the ends of the molecules approaching close enough, the rods also must have near parallel relative alignment, for the reaction to take place. The
latter requires rotation of the entire molecule ~determined by
the rotational diffusivity about its center of mass, D r ! which
becomes very slow once the rodlike molecule becomes long
enough.3 Recent theoretical studies4,5 have shown the role of
rotational and translational diffusion in limiting the rate of
polymerization under simplified conditions. Several experimental studies have also shown severe slowing of the reaction at the late stages due to diffusion control.6–9 Such slowing of reactions due to diffusional limitations has an effect
not only on the molecular weight distribution but also on the
final molecular weight obtained in the presence of side
reactions.8,9 An understanding of the polymerization kinetics
is important to predict the reaction time and the molecular
weight distribution, which ultimately determine the reactor
design and the processing parameters to obtain a product
with the required properties.
The basic method for analysis of the diffusion controlled
reactions originates in the classical work of Smoluchowski10
and has been applied to a number of systems ~see Refs. 11
and 12 for comprehensive reviews!. In processes involving
various degrees of complexity, such as angular constraints
for reaction, charge effects, anisotropic translational diffusion, and flow, it is quite difficult to evaluate the effective
rate constant analytically and several numerical techniques
have been proposed.13–17 In this work we use the novel pairJ. Chem. Phys. 107 (8), 22 August 1997
wise Brownian dynamics ~PWBD! simulation method to obtain the effective rate constant for reaction between rodlike
molecules. The method was proposed by Northrup et al.,18
and involves the computation of the rate constant from the
statistics of the Brownian diffusion of a single molecule in
the neighborhood of a second reactive molecule. The main
advantages of this method over other numerical techniques
are: ~i! it is computationally less intensive since the rate constants are obtained directly without first computing the concentration field;19 and ~ii! mathematical formulation of problems involving complex boundary conditions, flow fields,
and charged molecules is simple. The PWBD approach has
been continuously refined, and some of the major developments are the use of variable time steps,20 efficient inclusion
of charge effects,21 and incorporation of finite rates of
reaction.22 More recently, the reactions between complex
molecules involving local rotation and translation have been
studied;23–27 the far-field diffusion in these cases is, however, isotropic. The case of rodlike molecules has not been
considered previously.
In this work we focus on the diffusion limited reaction
between rodlike molecules with reactive functional groups at
their ends @e.g., formation of poly~p-phenylene
terephthalamide!#,9 in dilute solutions ~C 0 L 3 !1, where C 0 is
the number concentration and L is the length of the rods!.
The main objective of the work is to adapt the pairwise
Brownian dynamics method for analysis of the above class
of reactions. The ratio of the translational diffusivity parallel
to the rod axis (D i ) to the diffusivity perpendicular to the
rod axis (D' ) in dilute solutions is approximately 2.3 Here
we consider, for simplicity, the case of isotropic translational
diffusion, for which an analytical solution is available,4 since
the purpose is mainly to illustrate and validate the approach.
Extension of the approach to take into account anisotropic
translational diffusion is straightforward, and computations
for dilute solutions (D i /D' 52) show that the approxima-
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3289
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J. Srinivasalu Gupta and D. V. Khakhar: Simulation of polymerization
rod axis, respectively. Expressions for diffusivities of rodlike
molecules are reported in Ref. 29. The above equation is
written with respect to a coordinate system fixed on the nondiffusing reactive site. If the test rod is also diffusing, D t and
D r are the sum of the individual rod diffusivities, and Eq. ~1!
will contain an additional translational diffusion term due to
rotation of the rod on which the coordinate system is fixed.30
The far-field boundary conditions and the symmetry boundary conditions for the problem are
C5C ` ,
~2a!
r→`,
]C
50,
]u
u 50,p /2,
~2b!
where C ` is the bulk concentration. For small u c , the reaction zone based on the center of mass position of the diffusing rod can be approximated to be spherical4 and boundary
conditions at the reactive site are ~Fig. 1!
Dt
]C
5k 8 C at r5R for u , u c
]r
~3!
and
FIG. 1. Schematic diagram showing the criteria for reaction of rodlike molecules. Molecules can react only if proximity (r,R) and orientation ( u
, u c ) requirements are satisfied. Initiation (r5b) and truncation (r5q)
boundaries used in the Brownian dynamics simulations are shown.
tion of isotropic translational diffusion is reasonable.28 In
Sec. II we define the problem under study and in Sec. III we
present the theoretical framework of the PWBD method for
calculating the effective rate constant for rodlike molecules
for the case of dilute solutions, for which the translational
diffusion is nearly isotropic. The computational procedure is
given in Sec. IV, and the numerical results and comparison
to previous analytical results4 are given in Sec. V. Conclusions of the work are given in Sec. VI.
II. PROBLEM FORMULATION
We consider the diffusion controlled reaction between
rodlike molecules with the following constraints for reaction:
~i! The rod ends must be closer than a distance R, and ~ii! the
angle between the two rod axes must be less than a critical
angle ( u c ). Following Smoluchowski,10 the effective rate
constant is obtained from the diffusive flux to the reactive
site at the end of a rod ~Fig. 1!. Assuming the translational
diffusion of the rods to be nearly isotropic, a reasonable assumption for dilute solutions, the governing equation for diffusion of the rods reduces to4
S
D
S
D
] 2C 2 ] C
1 ]
]C
Dt
1
1D r
sin u
50,
]r2 r ]r
sin u ] u
]u
]C
50 at r5R for u > u c ,
]r
where k 8 is the surface reaction rate constant. The effective
second order rate constant is then
k eff5
4pR2
C`
where C(r, u ) is the number concentration of the molecules
at position r with orientation angle u ~Fig. 1!, and D t 5(D i
12D' )/33 is the effective translational diffusivity, with D i
and D' being the diffusivity parallel and perpendicular to the
E
uc
0
Dt
]C
sin u d u .
]r
~5!
The relationship between the surface reaction rate constant
(k 8 ) and the homogeneous rate constant (k R ), i.e., the rate
constant in the absence of diffusional resistance, is
k 85
kR
.
4 p R ~ 12cos u c !
~6!
2
The above equation is obtained by putting C'C ` in Eq. ~3!,
in the limit of rapid diffusion, and substituting for the flux in
Eq. ~5!.
We next cast the above equations into a dimensionless
form, rescaling the radial distance as j 5r/R and the concentration as C̄5C/C ` . The diffusion equation is then
s
S
D
S
D
] 2 C̄ 2 ] C̄
] C̄
1 ]
1
1
sin u
50,
]j 2 j ]j
sin u ] u
]u
~7!
where s5D t /R 2 D r is a dimensionless parameter which
gives a measure of the rotational diffusional resistance. The
boundary conditions in dimensionless form are
C̄51,
~1!
~4!
j →`,
] C̄
50,
]u
] C̄
5 a C̄
]j
~8a!
u 50,p /2,
at j 51
~8b!
for u , u c ,
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~8c!
J. Srinivasalu Gupta and D. V. Khakhar: Simulation of polymerization
] C̄
50
]j
at j 51
for u > u c ,
~8d!
where a 5k 8 R/D t is the dimensionless surface reaction rate
constant. The dimensionless effective rate constant is given
by
k̄ eff5
E ]]j U
uc
C̄
0
sin u d u ,
5q; Fig. 1!. We next consider the relationship between ^ b q &
and ^ b ` & . The first visit flux for a finite domain is given by
j*
05
D tC qq
~ q2b ! b
C50,
j 51
C5C q ,
The flux normal to any surface enclosing the reactive
site is equal to the reaction flux for the case of steady diffusion, due to continuity. The effective rate constant for steady
diffusion can thus be written as
k eff5
4pb2
C`
E
p /2
0
j ~ b, u ! sin u d u 5
4pb2
^ j&,
C`
~10!
where j(r, u ) is the radial component of the translational
diffusion flux, r5b is any spherical surface enclosing the
sink ~Fig. 1!, and ^•& denotes an average taken over all orientations. The basic idea of the PWBD approach is to write
the radial flux as
j ~ b, u ! 5 j 0 ~ b ! b ` ~ u ! ,
~11!
where j 0 is the first visit flux ~i.e., the flux of particles starting from r5` that visit the surface r5b for the first time!
and b ` ( u ) is the reaction probability ~i.e., the fraction of
particles starting from r5b with orientation angle u that react rather than escape to r5`!. The first visit flux is obtained as
j 05
D tC `
b
~12!
r5b,
r5q.
~15a!
~15b!
We take C q such that the reaction flux for the finite domain
case is equal to that for the infinite domain, hence
^ j & 5 j *0 ^ b q & 5 j 0 ^ b ` & .
~16!
The average escape fluxes in the case of the finite and infinite
domain cases are then
^ j bq & 5 j *0 2 j *0 ^ b q &
~17!
^ j b` & 5 j 0 2 j 0 ^ b ` & ,
~18!
and
respectively. Defining V5 ^ j b` & / ^ j bq & we finally obtain
^ b `& 5
III. THEORY
~14!
using Eq. ~1! and the boundary conditions
~9!
where k̄ eff5keff /kD and k D 54 p D t R is the Smoluchowski
rate constant for a uniformly reactive sphere with instantaneous reaction at the surface.
The dimensionless parameters of the process are u c , s,
and a. For rigid rodlike polymers, u c is expected to be
small30 although experimental values for particular systems
have not been reported. Large values of s correspond to slow
rotational diffusion and small values of a represent intrinsically slow reactions or large translational diffusivity, which
result in a kinetically controlled reaction with a negligible
influence of rotational diffusion parameter s. The effects of
rotational constraints thus become apparent for large values
of both a and s.
3291
^ b q&
.
b
1
~
^ q & 12 ^ b q & ! V
~19!
When the diffusion for r.b is isotropic ~flux depends only
on radial distance, r!, Northrup et al.18 obtained
V512b/q.
~20!
Here we generalize the above result to the case when the
diffusive flux may not be isotropic for r.b as
V512a ~ b/q ! 1••• .
~21!
Note that V→1 as b/q→0 as required. Thus if ^ b q & is computed for two or more values of b/q, ^ b ` & can be calculated.
The dimensionless rate constant is then
k̄ eff5b̄ ^ b ` &
~22!
and the ratio of escape fluxes in dimensionless form is V
512a(b̄/q̄), where b̄5b/R and q̄5q/R. The above result
is similar to that of Northrup et al.,18 however, the averaging
over orientations and the relaxing of the necessity of isotropic diffusion for j .b̄ facilitates computations. The derivation is somewhat more rigorous than the branching diagram
approach of Northrup et al.18 since variation of the probabilities with initial orientation are explicitly taken into account.
The simplicity of the above analysis allows for generalization to more complex cases ~e.g., anisotropic translational
diffusion!.
on solving the diffusion equation @Eq. ~1!# along with the
boundary conditions
C50,
C5C ` ,
r5b,
r5`.
~13a!
~13b!
Numerical simulations can be carried out only for a finite domain, hence it is possible to calculate only b q ( u ) ~i.e.,
the fraction of trajectories that react rather than escape to r
IV. COMPUTATIONAL PROCEDURE
The numerical method essentially involves the generation of Brownian trajectories of molecules starting from r
5b, and terminating the trajectories when they visit the surface r5q ~Fig. 1!. The reactive surface is taken to be perfectly reflecting. Each Brownian step comprises random
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3292
J. Srinivasalu Gupta and D. V. Khakhar: Simulation of polymerization
density in the presence of a reflective boundary at r50. A
sufficiently small time step (Dt) is used in the computations
so that v ik '1 except very near the target surface and that
the surface can be considered flat ~i.e., curvature effects can
be neglected!. Analytical expressions for P rxn and P refl can
be obtained in this limit. For isotropic translational diffusion
of the rods the expressions reduce to the results of Lamm and
Schulten32 for spheres when the region in phase space near
u 5 u c is neglected.28 These results are used in the computations.
A variable time step is used in the computations with the
dimensionless time step given by
Dt5
FIG. 2. Variation of reaction probability ( b q ) with the initial orientation
angle of the molecules ~u! for different values of the rotational diffusion
resistance (s). The upper dotted line gives the theoretically obtained reaction probability for a uniformly reactive surface @Eq. ~27!# and the vertical
line indicates the critical orientation angle ( u c ). Error bars give the standard
deviation for five iterations. ~a! s5104 ; ~b! s5108 .
Gaussian translational displacements in the three coordinate
directions with variance A4D t Dt ~Ref. 31! and a random
rotation step with probability density
P r ~ u ,Dt u u 0 ! 5
1
4 p D r Dt
H
3exp 2
J
u 2 1 u 20 22 uu 0 cos D f
. ~23!
4D r Dt
for u 0 ,0.01 rad, where u 0 is the orientation before the time
step, and a random Gaussian rotation step ~Du! with variance
A4D r Dt if u 0 .0.01 rad. The average reaction probability is
calculated for sufficiently large number of trajectories (N)
initiated at each orientation u as22
1
b q ~ u ! 512
N
N
( )k v ik ,
i51
~24!
where v ik is the survival probability for a single Brownian
step given by
P rxn~ r,Dt u r8 !
v ik 5
.
P refl~ r,Dt u r8 !
~25!
P rxn(r,Dt u r8 ) is the probability density of finding a molecule, which was initially at r8 , at r after a time Dt, in the
presence of a reactive surface at r50 with a surface reaction
rate constant k 8 . Similarly, P refl(r,Dt u r8 ) is the probability
H
t fac~ j 2r̄ c ! 2 ~ q̄2 j ! 2 1 t min,
t min,
j .r̄ c
j >r̄ c ,
~26!
where D t 5D t Dt/R 2 and t fac is a constant in the range 0.2–
0.001. The survival probability is computed only if the particle is within certain critical radius r̄ c . We take r̄ c 51.05
and t min51024 so that the reaction probability ratio, @ 1
2 v ik (r̄ c ) # / @ 12 v ik (0) # , is less than 10210, following Allison et al.22 Further, t fac is chosen so that the reaction probability ratio is less than 10210 for all j .r̄ c .
Since the critical orientation angle and the rotational diffusivity are very small ~u c !1, s@1! only those molecules
initially oriented close to u c have a nonzero reaction probability @ b q ( u ) # . Hence, molecules are initiated with orientations in the range u P(0,u c 1 d ), where d is chosen such that
b q ( u c 1 d )/ b q (0)<1023 . Scaling analysis of the Eq. ~1!
gives d ;Aq̄/ As and the criterion for the reaction probability
at u 5( u c 1 d ) gives A.1.5. The reaction probability b q ( u )
is computed at equally placed intervals in the above range of
u, and then integrated using a numerical technique ~Simpson’s method! to obtain ^ b q & which is of interest. For very
large values of s (s.106 ) we have d , u c , and thus b q ( u ) is
calculated only in the range u P( u c 2 d , u c 1 d ). In this case
the reaction probability for molecules with initial orientations in the range u P(0,u c 2 d ) is
b q5
a ~ q̄2b̄ !
b̄ ~ q̄ ~ 11 a ! 2 a !
~27!
and corresponds to the case of a uniformly reactive surface
~i.e., reaction is possible for all orientations u!. A sufficiently
large number of Brownian trajectories (N) are used so as to
get small standard deviations. Computations are carried out
for 100 trajectories at each of the 20 u values for cases in
which d , u c and for 500 trajectories at each of the 20 u
values for cases when d . u c . In all cases the average of 5
such computations are reported.
V. RESULTS AND DISCUSSION
The variation of the reaction probability ( b q ) with the
initial orientation angle of the rod ~u! is presented for different values of the rotational diffusion resistance (s) in Fig. 2.
The molecules initially oriented at u 50 have the highest
value of the reaction probability, and the reaction probability
levels off to zero at relatively large initial orientation angles.
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J. Srinivasalu Gupta and D. V. Khakhar: Simulation of polymerization
3293
FIG. 4. Variation of b̄ ^ b q & with b̄/q̄ for a 51.0, u c 50.01 rad, and b̄
51.1 for different values of the rotational diffusion resistance (s). Error
bars give the standard deviation of the computed values and the solid lines
are obtained from a least squares fit. !: s5104 , n: s5106 , 3: s5108 , h:
s51010, and 1: s51012.
k̄ eff5
FIG. 3. Variation of b̄ ^ b q & with b̄/q̄ for a 51.0, u c 50.01 rad, and different
values of the initiation radius (b̄) for two values of the rotational diffusion
resistance (s). Error bars give the standard deviation of the computed values
and the lines are obtained by a least squares fit. ~a! s5104 ; ~b! s5108 , 1:
b̄51.05, 3: b̄51.1, and n: b̄52.0.
The reaction probability for a uniformly reactive surface @Eq.
~27!# is also shown in both graphs ~dotted lines!. For very
large values of s @Fig. 2~b!# the reaction probability at low
orientations is equal to the theoretical value as required.
However, for smaller values of s @Fig. 2~a!# the maximum
value of b q is lower than the theoretical value. The computations thus verify the assumptions made for the range of
initial orientations of the molecules.
For small values of ^ b q & , as obtained here, Eq. ~22!
simplifies to
b̄ ^ b q &
12a ~ b̄/q̄ !
.
~28!
Figure 3 shows the variation of b̄ ^ b q & with b̄/q̄ for different
values of the initiation radius (b̄), and the rotational diffusion resistance (s). The error bars give the standard deviation of the computed values over five iterations. In all the
cases considered, straight lines are obtained in accordance
with Eq. ~28!. The fitted values for k̄ eff and a are given in
Table I. The values of the rate constants for all the three
values of b̄ are the same within the obtained standard deviations. Furthermore, the values of the constant a obtained for
very slow rotational diffusion (s5108 ) are significantly less
than 1, indicating a deviation from isotropic flux in the region j .b̄. The constant a approaches unity as b̄ increases
for this case. The constant a is close to unity for faster rotational diffusion (s5104 ). Both of the above trends are consistent with an increase in isotropy for the region j .b̄.
Typical computational times ~on an IBM PC, Pentium 120!,
also given in Table I, show a decreasing trend with the ini-
TABLE I. Table showing the values of the constants of the lines fitted to the
data in Fig. 3 for different values of b̄ and two different values of s, along
with the computational time consumed.
s
b̄
k̄ eff3105
a
CPU ~h!
104
1.05
1.10
2.00
3.7760.15
3.7360.10
4.0060.25
0.995
0.992
1.083
1.28
1.25
0.83
108
1.05
1.10
2.00
2.6260.02
2.6260.02
2.6060.04
0.744
0.767
0.885
0.31
0.30
0.20
FIG. 5. Effect of rotational diffusional resistance (s) on the dimensionless
effective rate constant (k̄ eff) for a 51.0, u c 50.01 rad. Symbols denote the
computed results and the solid line gives the analytical result ~Ref. 4!.
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3294
J. Srinivasalu Gupta and D. V. Khakhar: Simulation of polymerization
The effective rate constant for reaction between rodlike
molecules computed using pairwise Brownian dynamics is
equal ~within computational error! to the value obtained
from theory,4 thus validating the computational method. The
method can easily be extended to more complex situations,
for example, diffusion controlled reaction with anisotropic
translational diffusion. The numerical procedures are more
complicated in this case as compared to isotropic translational diffusion, and the computations are significantly more
intensive particularly for low values of D' /D i . Results of
these computations will be reported separately.33
FIG. 6. Effect of critical orientation angle ( u c ) on the dimensionless effective rate constant (k̄ eff) for a 51.0. Symbols denote the computed results
and the solid line gives the analytical result ~Ref. 4!.
tiation radius (b̄) for a fixed number of trajectories. The
errors are small at small b̄ and increase with increasing values of the initiation radius (b̄). Considering a balance between magnitude of errors and computational times, an intermediate value of the initiation radius b̄51.1 seems to be
most suitable for the computations.
Figure 4 shows the variations of b̄ ^ b q & with b̄/q̄ for
different values of the rotational diffusion resistance (s) for
b̄51.1. Again, straight lines are obtained for all cases, in
agreement with Eq. ~28!. The value of the intercept ~which is
equal to the rate constant! decreases with increase in the
rotational diffusion resistance (s) as expected. The variation
of the effective rate constant (k̄ eff) with the rotational diffusion resistance (s) is compared with the analytically obtained values4 in Fig. 5. There is excellent agreement between the theory and the computed values within the
obtained standard deviations. Figure 6 shows a graph of the
variation of the effective rate constant (k eff) with the critical
orientation angle ( u c ) for s5106 and a 51.0. Again, these
values show a good agreement with the analytical results.
VI. CONCLUSIONS
The pairwise Brownian dynamics method of Northrup
et al.18 for the calculation of effective rate constants of diffusion controlled reactions is extended to the case of rodlike
molecules. A new approach is presented for calculation of
the finite domain correction for the reaction probability. The
approach is somewhat more rigorous as compared to that of
previous works since variation of probabilities with the initial orientation of the molecules are explicitly taken into account. Other improvements that facilitate computations include numerical integration of the reaction probability over a
limited range of initial molecular orientations, and relaxation
of the requirement of isotropic diffusion for j .b̄ which allows for using smaller values of the initiation radius (b̄). All
of the above would be useful for pairwise Brownian dynamics simulations of diffusion controlled reactions for systems
other than the one considered here as well.
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