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Monte Carlo test of the self-consistent field theory of a polymer brush
Pik-Yin Lai, E. Zhulina
To cite this version:
Pik-Yin Lai, E. Zhulina. Monte Carlo test of the self-consistent field theory of a polymer brush.
Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.547-560. �10.1051/jp2:1992148�. �jpa-00247651�
Classification Physics Abstracts
36.20 61A0K
Monte Carlo test of the self-consi§tent field theory
of a polymer brush
Pik-Yin Lai and E. B. Zhul1~la
Institut fir Physik, Johannes-Gutenberg Universit£t-Mainz, Postfach 3980, D-6500 Mainz, Germany
(RecHved 29 August 1991, accepted in final form 8 November 1991)
Abstract The analytic predictions from the Self-Consistent Field(SCF) theory for grafted polymer layers are compared in detail with the recent Monte Carlo simulations using the bond- fluctuation model. Quantities describing the equilibrium structure of the brush are derived from the SCF theory and compared with the Monte Carlo data with no free parameter. In most
cases the results are in agreement with the SCF predictions. Causes for discrepancies are also
discussed.
1 Introduction.
Because of their wide applications in polymer technology and their importance in understand-
ing the fundamental problems of polymeric materials, grafted polymer layers 11,2] have been
a subject of recent interest both theoretically [3 24] and experirnentauy [25 33]. Theoreti- cally, analytic Self-Consistent Field (SCF) theory [7,10 11,14] gives a very detailed solution of the equilibrium structural properties of the polymer brush while experiments usually can
only provide more global information like the thickness, concentration profile and the force
profile between two interacting brushes. On the other hand, detailed information like the poly-
mer chain trajectory, fluctuations etc. can conveniently be obtained in a computer simulation and compared with the SCF theory. Furthermore, in a simulation the system is precisely char-
acterised and approximations used in the theories are avoided. In this paper,
we shall compare the recent Monte Carlo(MC) simulation results [34] of the polymer brush in a good solvent with the analytic SCF predictions in detail. The bond-fluctuation model of macromolecular
chains [35 38] was used in the simulations. In the next section, all the predictions that will be tested by the MC simulations are listed or derived using analytic SCF theory with Gaussian
chain stretching and binary interactions between monomers. The excluded volume parameter (*) Alexander von Humboldt Fellow. Permanent address: Institute ofmacromolecular Compounds, Academy of Science, Leningrad 199004, U-S-S-R-
for the bond-fluctuation model is then calculated using results from other independent stud- ies. Then the MC results are compared with the SCF predictions and possible causes for the
observed deviations are discussed.
2. Analytic results from SCF.
The idea of SCF theory for polymer brushes is based on the assumption that monomer-
monomer interactions can be represented by a position dependent potential U(z) where z
is the distance from the grafting surface. This potential win determine the monomer volume fraction #(z) which in turn gives rise to U(z). The equations determining #(z) thus have to be solved self-consistently- One exploits the observation [12] that grafted chains are stretched and the number of polymer chain configurations is greatly reduced in which
case the partition
function is dominated by the chain configuration that minimizes the total free energy. Fluc- tuations about this minimal chain path can be neglected and the problem is reduced to a one
dimensional classical mechanics problem and U(z) is found to be parabolic [7,10 ill- The
analytic results of the SCF theory are summarized below, the details of this theory can be found in references [7] and ill].
Monodispersed polymer chains of N monomers with one end grafted to a flat impenetrable
surface with grafting density I/S are considered (S-area per chain). If f[$(z)] is the density
of the free energy volume interactions in a layer with concentration $(z) (number of monomer per unit volume), then one has ill]
~fjj( )j ~ 2 2
&I(( " ~~~~ " ~°~~~~~~ 8iiiN ~~~
where (R() is the average square end-to end distance of an unperturbed, Gaussian chain.
Assuming binary interaction between monomers, iii]
= vi~ where v is the second virial coefficient of monomer interactions, one obtains
I(z)
= )U(z) (2)
and hence the concentration profile is also parabolic [7,10 ii]. If a~ is the volume of a monomer, then the volume fraction of polymer units, #(z) = a~$(z). Introducing surface coverage a = a~/S, we obtain
~(~) ~~2/3(~~~~)
~~~
((~ (~)2j (~)
2 8pv h
where p
= (R() /(Na~) is the stiffness parameter of a chain. The values of p and v for the three dimensional bond-fluctuation model are calculated in the appendix, h is the brush height and is given by
h = (~)l"ii'3N (4)
The "trajectory" of the grafted chain, which is the position of the ith monomer along the
chain, is given by [7]
~~
' ~~ ~~
2N ~~~
The probability distribution of the free end is given by [7,10 iii
PN(~) " $@~ (6)
These major SCF predictions will be compared with our MC simulation data. We now use the above SCF results to derive the other quantities that are measured in the MC simulations.
Using (3), it is trivial to obtain the first moment of #(z) which is
a measure of the brush
thickness,
j~j ~~ ~(~l~v~jl/3p/ (~)
8 8 x2
The z-component of the square of the radius of gyration of the chains is defined as
iRizi + ij fiz; ( L z;i~i
181
Using (5), (6) and N » I, one gets,
jj~2 j~2jj~ ~ j ~j~ ~ )j~P~~)2/3p/2 jgj
gz N ~ ,2 5 ~ ,2 ,2
The orientation of the ith bond is characterised by (cos R;) which is defined as
ICOS R;) +i ljl'Bl'~jjl = ~~)i/~llz;I lz;-i)1
(1°1
where (£~)Q~ is the bond length between successive monomers along a chain. Using (5) and N » I, after some algebra, one obtains
ICOS °;I = ~j(jij~l«~vp«)~'~
CDS ) (11)
The probability distribution of the ith monomer, p;(z) can also be obtained
" Z "
~
z ~
ix
~ ~~ ~ ~~~ @)PN(ZN)dZ~ (h 32
sin jj3 (h sin ~~ )2 ~~
Finally, the relative mean square displacement of the ith monomer can be measured by iAzii
=
izii
_~ j~~j
iz;i~ iz;i~
Using (5) and (6), one gets
)l'~~
=
)($)~ i
= o.1528 1i4l
which is a constant independent of I, N and a.
3. Monte Carlo results.
The MC data were taken from the simulation studies in reference [34]. The bond fluctuation model [35 38] was used in these MC simulations. The details of the simulation work can be found in reference [34]. The surface coverage a ranged from 0.025 to 0.2 and chain length
from N = 10 to 80. Here we will briefly mention some features of the model that is relevant in
analysing the data. Each monomer occupies a cube of 8 lattice sites in a simple cubic lattice
and no two monomers can share a common site. Thus when comparing the MC results with
the SCF predictions, one should take a = 2. We consider a polymer brush in a good solvent and there is no monomer-monomer interaction other than self- avoidance. The bond length
between successive monomers along a chain ranges from 2 to @ and is characterised by the root-mean-square of the bond length, (£~)~/~ For the range of concentration in the simulations, (£~)~/~ ci 2.71~ 0.02. To allow a precise quantitative comparison to the SCF theory, the values of the excluded volume parameter v and stiffness parameter p for the 3-dimensional bond- fluctuation model are essential. v and p can be obtained independently using the results from other studies [37, 38] of the 3-dimensional bond-fluctuation model as given in the appendix.
We get v ci 11.55 and p ci 2.7 ~ 0.2 and these values will be used in the subsequent analysis
of the data. The product up can also be obtained from the results of the present study by a linear least square fit of (z) as a function of Na~/~
as implied in equation (7). As shown in
figure I (upper part), the data do lie nicely on a straight line as predicted in (7). Linear least square fit (dashed line) gives up ci 33 ~ 2 which is in good agreement with the independent estimate, ci 31.2 (see Appendix).
50
~4-
40
,' ,'
,' 30 ,,'
~,
,, 20
Q ~gA~'
~ ~
~ 10
~q~
f,
,~4 0
Fig. I. First moment of the density profile (z) (upper set of data) and (R(z)~'~ vs. Na~/~. The
dashed line is a least square fit for the (z) data. The solid line is the SCF result from equation (8).
Symbols: a
= 0.025(o), o.05(n), o.075(6), o.lo(o), o.15(V)> o.20(*).
The brush thickness can be characterised by (R(z)Q~ whose MC results are plotted against Na~/~ in figure I (lower part). The data fall nicely on a straight line verifying the scaling
behavior. Smau systematic deviations for low values of Na~'~
occur as expected. The solid
line is calculated from the SCF results in (9) which show almost perfect agreement.
2
1. o
B m
, cq
~ 6
h4
~ 0.2
0
~
can be estimated from #(h) = 0 in figure 2 and extrapolation to the large N limit. We get h ci (2.8~ 0.2)Na~/~ which is consistent with equation (4). The tail of #(z) has been predicted
theoretically [2] to decay as
#(z) c< exp(-constant N~Q~aQ~(z h)~'~) for z > h (is)
This is checked by plotting £n#(z) corms N~~'~aQ~(z-h)~'~ in figure 3. Here h is estimated from #(h) = 0 for large N from the MC data. The data show straight lines consistent with
a decay
~- exp(-constant (z h)~/~). However the data do not collapse to a single curve as
would be implied by equation (15). Thus there appears to be corrections in the prefactor to the power law dependences in a and N as given by equation (15).
The average position of the ith monomer along a chain is shown in figure 4. The data fall
approximately on a single curve implying (z;) is a function of ilN only. Furthermore the data agree reasonably well with the SCF result given by equation (5) (solid curve). More significant discrepancy occurs for the monomers near the free end. This discrepancy does not decrease with increase in N for the range of N values investigated. Thus the deviation from the SCF theory, though smau in this case, is not due to finite N but probably arises from stronger chain configuration fluctuations around the most probable path near the brush end and correction to the mean-field behavior.
The probability distribution of monomers along a chain are displayed in figure 5a for a
= 0.05
and N
= 30. The SCF result in equation (12) suggests a scaling plot of p;(z)h sin (ix/2N)
corms z/(h sin (ix/2N)) which is shown in figure 5b. The data collapse roughly verifying the
dependence of sin (ix/2N) in p;(z). Again deviation from the SCF prediction (solid curve)
is largest for the end monomer, as expected. The scaling dependence of N and a is tested by plotting Na~'~pN(z) corms z/Na~/~ in figure 6. In this case, the data do not scale very well. Furthermore data for higher values of a show a much sharper distribution than predicted by SCF. A similar sharpening in the distribution at higher coverage has been obtained in
analytic SCF calculations taking into account of higher order interactions [10,16] and the finite extensibility of the chains [16] However the locations of the peaks in figure 6 agree quite closely with the SCF result.
The SCF prediction for the average orientation induced by grafting in equation (11 is tested by plotting (cos 9;)a~~/~ versw il(N I) in figure 7. The data show a rough couapse except
for monomers near the wall. This is expected since SCF theory does not take into account for
the shrot range order effects near the wall. Such a disagreement with the SCF theory near the wall has also been observed in the molecular dynamics simulation in reference [21]. The SCF prediction (solid curve) agrees with the major trend of the data. The root-mean-square bond length is taken to be (£~)~/~ = 2.71 in equation ill). Again for larger a the discrepancy
between the data and theory gets larger. The MC data indicate the brush is less stretched but the stretching is more uniform over the entire brush.
The discrepancy with the SCF prediction is mostly manifested in the relative mean square
displacement of the monomer positions as displayed in figure 8 a and b for different values of N and a respectively. While SCF predicts (Eq. (14)) a constant independent of I, N and a, our MC data show a minimum for I cS N/2 and a systematic decrease for increasing a (N fixed) and increasing N (a fixed). These are in disagreement with the analytic SCF theory
which assumes pair interactions and Gaussian stretching energy. As « increases, the system gets denser and monomers are more confined and it is not surprised that the mean-square
displacement decreases since one expects as a - I, Az; - 0. This is also reflected in p;(z)
which gets narrower for larger coverage but the peak stays more or less the same. We believe this deviation from the analytic SCF prediction mainly arises from higher order terms in the
-2
-3
-4
- Jov
t~
_~ ~ ~ ~~
~ ~
° $~$+~6d
o
~ o ~ +~ ~
o ~ ~v ~
° ~ Qk
° $ vK©
-7 o +w p
o~ ~~ a
-B
0 2 3 4 5
J/-1/2 ~l/6(~ /~)3/2
Fig. 3. in #(z) vs. N~~'~a~'~(z h)~'~ near the end of the brush. Symbols: (N, a)
= (60, o.05) (o), (40,0.05) (n), (30,0.05) (6), (20,0.05) (o), (20,0,I) (V)> (30,0,1) (*), (40,0.1) (+), (60,0,1) (~).
2. 0
5
= 0
fi
5
0.
0 2 4 6 B 1.0 1.2
ilN
Fig. 4. Average position of the ith monomer in a chain as a function of the normalized position ilN along the chain. Solid curve is the SCF result from equation (5). Symbols: (N, a)
= (80, o-I) (o), (60, 0.2) (n), (60, 0,1) (6), (40,0,I) (o), (40,0.05) (V)> (30, o.2) (*), (30, 0,1) (+), (30, 0.05) (~).
o.15
o-to
0fl51
0
0 5 lo 15 20 25 30 35 40 45
z a)
2. o
a
~
_ °O°
~ l.5 '~°
C4
~
~
V '~
w-£
~
-
~4 ~
- V
'M OV~
9yO
w 0.
0
hi
Fig. 5. a) Probabiity distribution p;(z) for chain length N
= 30 and surface coverage a
= 0.05.
The curves are just guides to the eye. Symbols: ilN
= 0.2(o), o.4(n), o.6(/£), 0.8(o), 1.0(V). b) Scaling plot for the data in a). Solid curve is the SCF result from equation (12).
equation of state and correction to Gaussian elastic free energy due to finite extensibility of the chains. On the other hand, the decrease in relative fluctuation with increasing N in figure
8b is probably due to other reasons since the system in that case is not too dense (a = 0.05).
Since the deviations from the SCF prediction (solid line in Fig. 8b) get larger for longer chains,
we believe the cause for the discrepancy is not due to finite chain length, but probably arises
from corrections to the mean-field picture.
a
_
6 h4
~~
m
~~
2~
0.
0
Fig. 6. Scaling plot of the chain end probabiity distribution pN(z). Solid curve is the SCF result in equation (6). Symbols: (N,a) = (80,0,I) (o), (80,o.05) (n), (60, 0.1) (6), (60,o,05) (o), (40, 0.1) (V)> (40, o,05) (*), (30,o.1) (+), (30,o.05) (~).
1. 5
f I-D
~ >.
b fi
Q3$
° 5
0.
0 2 4 .6 B 1.0 1.2
il(N 1)
Fig. 7. Scaring plot of the projection (cos 9;) of the local orientation vector of bond connecting
monomer I -1 and I vs, the normalized position of the ith bond along the chain. Solid curve is the
SCF result from equation (11). Symbols: (N, a) = (80, 0.1) (o), (60, 0.2) (n), (60,o-1) (/£), (40,o.1) (o), (40,0,05) (V)> (30,0.2) (*), (30,0.1) (+), (30, 0.075) (~), (30, 0.05) (~).
25
»
20 cq
fi
~
~ h4
~
- *
V O
~ ~
- 15
c~pw ~ V ~
~ a ~ i
a
- 6
° a a o
lo o
~
o
05
0 2 4 .6 B 1.0 1.2
ilN a)
25
o 20
m ~
~
- f
o ~ ~ ~
~
~__
15 Q ~
'4 » j ~ g 6 j
~ ©
~ ~
$
10
05
0 2 4 .6 B 0 1.2
ilN b)
Fig. 8. a) Plot of ((bz;)~)/(z;)~
os, ilN with N
= 30. Solid curve is the SCF result from equation (14). Symbols: a
= 0.2(o), 0.15(n), o.10(/£), 0.075(o), o.05(V), 0.025(*). b) Same
as a) but for various chain length at a
= o,05. Symbols: N
= 20(o), 30(a), 40(/£), 60(o), 80(T7).
4. Discussions and conclusion.
The plausible causes for the discrepancies between the SCF theory and the MC results
are (I)
finite chain length in the simulations (it) correction to the Gaussian elastic flee energy due to the finite extensibility of the cllain and higher order terms in the equation of state [10,16,17].
(iii) fluctuations of a chain about the minimal path are important and (iv) corrections to the mean-field description due to strong spatial fluctuations in concentration.
Since most of our data showed the predicted scaling quite well and in many cases the
