Monomer concentration profile in the depletion region
Hisao Nakanishi (1,2) and Sang Bub Lee (1,**)
(1) Department of Physics, Purdue University, W. Lafayette, IN 47907 U.S.A. (*)
(2) Collège de France, Physique de la Matière Condensée, Unité associée au CNRS (UA N° 792)
11 Place Marcelin-Berthelot, 75731 Paris Cedex 05, France
(Reçu le 11 janvier 1988, accepté le 21 mars 1988)
Résumé. 2014 On étudie par des simulations de Monte Carlo le profil de concentration, près d’une paroi impénétrable mais non répulsive, des monomères de polymères linéaires en solution diluée. On traite seulement le cas de polymères en bon solvant, soit flexibles, soit semi-flexibles. Les résultats obtenus sur un
réseau cubique simple donnent un profil défini par un exposant 5/3 en accord avec les théories d’échelle
proposées par Joanny et al. et par de Gennes. D’autre part, les résultats pour les chaînes semi-flexibles
suggèrent que l’image de blobs proposée par Ausserré et al. est assez correcte à condition de décaler la position
effective de la surface.
Abstract. 2014 We study by Monte Carlo simulation the monomer concentration profile of the dilute solution of linear chain polymers in the depletion region near an impenetrable but otherwise non-interacting wall. Only
the case of a good solvent is treated here but we allow both for flexible and semi-flexible chains. Our results on
the simple cubic lattice indicate a profile exponent of 5/3 in agreement with the scaling theories of Joanny et al.
and of de Gennes. Moreover, the results for the semi-flexible case suggest that the simple blob picture proposed by Ausserré and coworkers is reasonably accurate if a modification is made in the offset of the effective location of the surface.
Classification
Physics Abstracts
05.40 - 61.41 - 81.40
1. Introduction.
The depletion region problem [1] we treat here is the
problem of monomer density profile of a dilute
solution of linear polymers in good solvent in the
vicinity of an impenetrable wall which does not
otherwise interact with the solution. Therefore the non-uniform concentration profile, specifically, a
decrease in concentration near the wall, is entirely
due to the fact that the configurational entropy decreases for the chains near the wall, thus favoring
the chains to be farther away from it. An excellent review of the problem, together with recent exper- imental results, is presented in the thesis of Ausserre
[2] where many previous references are also given.
This problem is probably the simplest among those of a non-uniform profile for a solution of polymers
near a wall and certainly one of the most amenable (*) Present and permanent address.
(**) Present address : Department of Mechanical and
Aerospace Engineering, North Carolina State University, Raleigh, NC 27695 U.S.A.
for direct numerical studies even for the semi-flex- ible polymers.
For a semi-dilute solution of linear chains in good solvent, there are at least two scaling theories due to
Joanny et al. [3] and to de Gennes [4] that lead to the
monomer concentration profile near the wall, at
distance z from it, of the form :
asymptotically for long chains. Here, v - 3/5 is the
end-to-end distance exponent of a single chain in good solvent , § is the correlation length in the semi- dilute solution, and cb is the bulk monomer concen- tration. These results should also apply to the dilute
case with the radius of gyration RG replacing
§ [5]. Thus, the scaling prediction for the dilute case
is simply
We note that, although the asymptotic bulk concen-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070129500
tration for a dilute (i.e., non-overlapping) solution is
zero, the z-dependence of the concentration after
dividing through by cb remains non-trivial. Since the chain can be located anywhere in a semi-infinite space, the profile c(z) as discussed here is slightly
different from that treated in Eisenriegler et al. [5]
(cf. Fig. 11 of that reference), where they give
Monte Carlo results for a single chain with one end
fixed at the surface [6]. They, however, give exten-
sive discussions of scaling for more general (but only flexible) chains.
Exact solutions for the analogous problems for the
solution of hard spheres (see, e.g., Ausserre’s thesis
[2]) and for the random (non-excluded volume)
chains in the Gaussian approximation [5, 7] are well
known. In both of these cases, the concentration
profile is of the form of equation (2) with the
exponent replaced by 2. For the Gaussian chain, this
can be understood as the special case of equation (2)
with v = 1/2, the classical value of the exponent
v. In fact, it is easy to show (see appendix) that even
for a chain with intrachain concentration correlation
(as appropriate for excluded volume, swollen
chains), we again obtain the exponent of 2 as long as
the angular fluctuations are averaged over and the
chain is regarded as a hard sphere (of radially non-
uniform distribution of monomers). This implies
that the non-classical exponent is due to the con- formational fluctuations.
Ausserre et al. [8] considered a solution of semi- flexible chains in the depletion region. This was prompted by the experimental consequence that the concentration of monomers in the extreme vicinity
of the surface is greater for the less flexible chains,
thus providing greater signal in their Evanescent Wave Fluorescence Spectroscopy experiments [9]
for the concentration profile. They presented [8] an extremely simple and appealing blob picture for the
semi-flexible case, arguing that a semi-flexible chain
can be renormalized to a flexible one by grouping
monomers within a length scale of D into a blob
(D being the Kuhn length, see, e.g., Yamakawa
[10]). This picture is sketched in figure 1 and as depicted there, it also entails the offset of the origin
Fig. 1. - The blob picture of semi-flexible polymer chain
in the depletion region proposed by Ausserr£ et al. [2, 8] is
sketched.
as compared to the flexible case. This argument yields the concentration profile of semi-flexible . chains in terms of that for the flexible ones. Since
both z and RG are scaled by D, we obtain
where f is a scaling function and the scaling variable
x is just
Using the Joanny-de Gennes prediction for the
function f (x ), the modified blob picture would then predict, very near the surface,
The stiffness is contained entirely in the offset C and RG. If the blob picture is taken seriously, the
most natural candidate for C is the radius of the
blob, or D/2, although Ausserre et al. [8] proposed using just D. However, the precise relation to
D may turn out to depend on the details of the
model, and in any event, we are unable to calculate the Kuhn length for fully excluded-volume chains.
Thus, at this point we will leave the value of C undetermined.
In what follows, we first treat the case of random (non-excluded volume) flexible chains in section 2,
then excluded volume flexible chains in section 3,
and finally, the semi-flexible excluded volume chains in section 4. Our Monte Carlo results will be shown to be generally in good agreement with the scaling
theories [3, 4] for the flexible case and with the blob
picture [8] for the semi-flexible case with a modifi- cation in the size of the offset.
2. Non-excluded volume chains.
As a first step toward treating excluded volume
chains, we can look at the non-excluded volume
case. Thus our first Monte Carlo simulation deals with flexible random walks of N steps on the simple
cubic lattice. Note that even for completely ideal
chains the depletion region must exist because of the decrease of the configurational entropy near the surface. Surprisingly, the Gaussian solution [5, 7]
and previous simulations [5, 11] similar to the one performed here were in disagreement : the Gaussian, continuum chain gives the concentration profile as in equation (2) with the exponent 2, while the simu- lation result shown in a figure of reference [11]
indicated a finite, non-zero slope of the profile as it approaches z = 0 (surface of the wall). The result of
[5] for a non-reversal random walk did imply a zero slope but failed to give the exponent 2, which they
attributed to finite size effects. In this work we
quantitatively take account of the finite size effects
and show that our simulation agrees with the Gaus- sian solution precisely.
We grow random walks on a simple cubic lattice which is oriented so that the surface of the wall is
always to the left of the chain and parallel to one of
the faces of the unit cell ([100] surface). We do not
put in the surface explicitly into the simulation algorithm as was done in [5] but rather we follow [11] so that the effect of the wall appears only in the
last stage when the computation of the profile c (z ) is made. Thus, if a particular configuration of a freely grown chain has a profile of the number of
monomers, g(n), where n labels the transverse coordinate in such a way that n = 0 corresponds to
the left-most monomer and, say, n = ni to the
rightmost monomer, then, the contribution of this
configuration to the average profile c(z) is pro-
min (z, nl )
portional to Y g (n ). This corresponds to the
n=0
fact that, although a configuration can occur any- where in the half space, non-zero contributions to
c (z ) occur only if some monomers of the chain are at
distance z from the wall. For more detailed illus- tration, we refer to [11]. Thus,
where (...) is the simple average over all random walks of N steps generated without constraint.
Our numerical results for N = 1 000, 2 000, 3 000,
and 4 000 are sketched in figure 2. (Some of the
parameters of this and other simulations in this work
are given in Tab. I.) Note that the abscissa of
figure 2 is a scaled variable x - (z + l p/2 )/RG where 1 p (= 1.2 lattice constants here) corresponds to the
average distance between two successive gauche
turns as discussed in section 4. For these long chains (compared, e.g., with N = 100 for non-reversal random walks used in [5]), evidently finite size effects are so small that the concentration profiles
are already very close to the Gaussian prediction (shown as a solid line).
The remaining small deviation can then be re-
moved by a proper extrapolation of these results to N -> oo. If the N-dependence of the profile ON (x ) at
fixed x is of power-law nature, then we write
where 0394 is an appropriate correction exponent. In this case, we find that A = 1/2 provides a good
numerical fit and yields points lying on the dashed
line in figure 2, which is so close to the solid line
(Gaussian result) that the difference is only visible
for x > 0.6. When we follow this procedure for much
shorter chains, A = 1/2 still yields a final profile in
Fig. 2. - The concentration profile for free random walks obtained by simulations near an impenetrable wall which
is the [100] surface of a simple cubic lattice. All distances
are in units of lattice spacing and l p -1 /p = 1.2 in this
case and different symbols correspond to different chain
lengths N. The solid line is the Gaussian solution and the broken line is the result of interpolating through the points
obtained by extrapolation of the form of equation (7) with
4 = 0.5. The simulation data are grouped into 5 batches for each value of N, and the standard deviation for the batch averages varies depending on z and N but in all cases
less than 3 % or so and thus smaller than the symbols used.
Table I. - The parameters used in the Monte Carlo simulations presented in figures 2, 3 and 4.
good agreement with that obtained from the long
chains used in figure 2.
Similar simulations in two and four dimensions also agree quantitatively with that for the simple ’
cubic lattice when rescaling of the abscissa is made in accordance with the Gaussian predictions. There-
fore, it seems clear that our simulation, which is essentially the same as that of [11], is nonetheless
capable of reproducing correct behaviour which is
predicted by the Gaussian solution.
3. Excluded volume chains.
For the flexible linear chains with full excluded
volume, we performed Monte Carlo simulations on
the simple cubic lattice with chains of length up to N = 100 steps. This length is similar to that reported
in reference [5], but again as for the ideal chains, we
allow the chain ends to be anywhere in the semi- infinite space and perform an extrapolation to
N -> 00 of the form of equation (7).
Our raw results plotted in figure 3 appear to indicate an initial non-zero slope, implying that the exponent in equation (2) should be replaced by 1, just as it appeared in a figure of reference [11] for
the ideal chains. However, a closer inspection reveals
definite signs that the chain length N is insufficient
to attain asymptotic limits as was found in [5].
However, as for the ideal chains, we postulate that
the extrapolation can be done at fixed
x =- (z + l p/2 )/RG according to the form of equation
(7), provided that we use the appropriate correction-
to-scaling exponent [12] 03941. (Also, lp here is equal to
1.25 lattice spacing.) When we use a value [13]
obtained from the analogous magnetic n-vector
model in the n - 0 limit [14],
Fig. 3. - The concentration profile for flexible excluded- volume random walks obtained by simulation near the
[100] surface of a simple cubic lattice. Similarly to figure 2,
different symbols correspond to different chain lengths
N and the broken line is an interpolation through the points obtained by extrapolation of equation (7) with
d = 0.47. The data are grouped into 5 batches and the standard error among the batch averages is smaller than the data for figure 2 and also smaller than the symbols. In
the inset, the broken line shows the same extrapolated
data in a double logarithmic plot. The solid straight line corresponds to 0.7 X 5/3.
we obtain points that lie on the broken curve of
figure 3 (which is a simple cubic spline of the extrapolated points). Clearly, this extrapolation has
turned the original profiles for N -- 100 that ap-
peared linear near the origin, into an asymptotic one
that fits reasonably well with the power law with the exponent close to l I v - 5/3 (see inset of Fig. 3).
If we leave the exponent dl as a free parameter to
see its effect on the extrapolated profile, we find that
there is a range of L11, approximately between 0.4
and 0.6, which gives rise to a profile that can be
fitted with a power law over a region of x comparable
to that of figure 3. This power is rather close to 5/3.
On the other hand, 41 beyond this range yields profiles that cannot be fitted with any power law
over a reasonable range of data points. Since an
error in the identification of the offset of the origin
(lpl2 = 5/8 in this case) could also induce a correc-
tion corresponding to a power of v - 3/5, our
numerical results cannot rule out such a possibility ; however, it seems consistent and reasonable that the
procedure employed here is the correct one.
4. Stmi-flexible excluded volume chains.
As discussed in section 1, Ausserre et al. [3] studied
the concentration profile of a dilute solution of semi- flexible chains in terms of an appealing blob picture
as shown is figure 1. For such a picture to be sensible, we must have the number of blobs
N/g > 1 where g is the number of monomers in each blob. The diameter of such a blob is the Kuhn
length D and it should equal the size of a monomer times g if this picture is taken seriously.
In this study, we use a model called Biased Self- Avoiding Walk (BSAW), introduced in [15], in
which the stiffness of a chain is reflected algorithmi- cally in the gauche step probability p in the process of numerically generating a self-avoiding random
walk on a lattice. Thus, a self-avoiding walk already generated up to n steps has a probability 1- p of having the (n + 1 )-st step in the direction of the n-th step and probability p /4 (on the simple cubic lattice)
of taking the next step in one of the four perpendicu-
lar directions. Of course, if the (n + 1 )-st step results in a self-intersection, the walk is discarded from the sample pool. BSAWs are generated on the simple cubic lattice by simple sampling precisely as
described above, with a [100] surface as in sections 2 and 3. Clearly, the choice of the lattice and the surface affects in which way the stiffness must be
implemented ; our particular choice ensures that a
very stiff chain can still adsorb completely onto the
surface.
Now, because of the excluded volume constraint,
the Kuhn length D (blob diameter) is not known for
the BSAW model. However, for the non-excluded volume analog, i.e., for a stiff, random chain (with
the parameter p playing the same role), the Kuhn length was shown [16] to reduce to 2 l p = 2/p for
small p in units of lattice spacing using the solution for (R2). It is reasonable that D oc lip even for the
excluded volume case since the expected number of gauche turns should be proportional to p at least for small p. However, D is a non-trivial function of p for larger values of p even for the non-excluded volume, stiff chain, and in particular it is not known
for the excluded volume BSAWs. For this reason,
we turn to a simpler choice (l p/2 = 1/(2 p)) as our
empirical choice of the offset C of equation (5) ; the
extra factor of 2 is motivated by the approximate
relation that the average distance between two
successive gauche turns should be about lp itself, but
the choice is ultimately arbitrary.
p
We check, by simulation of the BSAW model, the scaling predictions of equations (3)-(5), and in particular, the specific power-law form of equation (7). The parameters used in this work are
summarized in table I, and the main results are
shown in figure 4 which corresponds to figure 3 given earlier for the flexible, excluded volume chains. For a fixed value of p and for large N, the scaling predictions are reasonably well obeyed
as far as the exponent appearing in equation (7), although the functional form of f (x ) of equation (3)
seems to be slightly different from the flexible case
shown in figure 3. These deviations are likely to be suggesting the ultimate inadequacies of the simple
Fig. 4. - The concentration profile for stiff excluded- volume random walks obtained by simulation near the
[100] surface of a simple cubic lattice. The BSAW stiffness parameter here is p = 0.1. The chain lengths N varies
from 200 to 1 000 here, and again the broken line is the
interpolation of the extrapolated data with 4 = 0.47. The data are grouped into 5 batches and the standard error
among the batch averages is similar to the data in figure 3
and smaller than the symbols. In the inset, the same extrapolated data are plotted in a double logarithmic plot by the broken line. The solid line corresponds to
0.62 X 5/3.
blob picture to predict the entire functional depen-
dence of f.
In the extremely stiff limit of very small p, one may also consider a different scaling regime with the following form [17] for (R2) :
for p -+ 0, N - ao such that Np > 1. The corre- sponding scaling form for c (z ) would be
Although the asymptotic limits intended for
equation (10) and those for equations (3) and (4) are distinct, to the extent that the same data are going to
be tested for these scaling forms, there may be some
region of p and N for which both forms are
approximately satisfied. In fact, if the data satisfy equation (9) approximately then the form of c (z ) of equation (10) for those data would be consistent with the scaling form of equations (3) and (4).
Indeed such is the case for the data presented in figure 5 in the form of equation (10).
Fig. 5. - The scaling of 0 (z ) __ c (z )/cb in the very stiff
regime as in equation (10) is demonstrated for two values of Np, namely 10 and 20. For each Np, three different values of p are used (0.05, 0.1, and 0.2) and, consequently,
also three different chain lengths. For these data, the
number of walks used varies from 7 000 for p = 0.1 and
N = 200 to more than 14 000 for p = 0.2 and N = 50.
5. Summary.
In summary, we have used Monte Carlo technique
to study the so-called depletion region problem for a
dilute solution of linear polymers. All simulations
are done for various random walks on simple cubic
lattice with the surface oriented as the [100] plane.
Our warm-up project for the ideal chain shows
clearly that the present method is capable of showing
the Gaussian behaviour for the concentration profile
