Corrections to scaling in multicomponent polymer solutions

0.2 MB·English

by  Andrea Pelissetto

Corrections to scaling in multicomponent polymer solutions
Andrea Pelissetto
Dipartimento di Fisica and INFN – Sezione di Roma I
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Universita` degli Studi di Roma “La Sapienza”
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Piazzale Moro 2, I-00185 Roma, Italy
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J e-mail: [email protected]

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Ettore Vicari
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o Dipartimento di Fisica and INFN – Sezione di Pisa
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at Universita` degli Studi di Pisa
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Largo Pontecorvo 2, I-56127 Pisa, Italy
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e-mail: [email protected]
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Abstract
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6
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1
0
6
0
/ We calculate the correction-to-scaling exponent ω that characterizes the
t T
a
m
approach to the scaling limit in multicomponent polymer solutions. A di-
-
d
rect Monte Carlo determination of ω in a system of interacting self-avoiding
n T
o
c walks gives ω = 0.415 ± 0.020. A field-theory analysis based on five- and
T
:
v
i six-loop perturbative series leads to ω = 0.41 ± 0.04. We also verify the
X T
r
renormalization-group predictions for the scaling behavior close to the ideal-
a
mixing point.
PACS: 61.25.Hq, 82.35.Lr, 05.10.Cc
I. INTRODUCTION
The behavior of dilute or semidilute solutions of long polymers have been investigated
at length by using the renormalization group,1–3 which has explained the scaling behavior
observed in these systems and has provided quantitative predictions that become exact when
the degree of polymerization becomes infinite. Most of the work has been devoted to binary
systems, i.e. to solutions of one polymer species in a solvent. The method, however, can
be extended to multicomponent polymer systems, i.e. to solutions of several chemically
different polymers. The general theory has been worked out in detail in Refs. 4–6. In the
good-solvent regime in which polymers are swollen, the scaling limit does not change. For
instance, the radius of gyration R increases as Nν, where7,8 ν ≈ 0.5876 and N is the length
g
of the polymer. However, the presence of chemically different polymers gives rise to new
scaling corrections in quantities that are related to the polymer-polymer interaction. In the
dilute regime one may consider, for instance, the second virial coefficient B between two
2
polymers of different species. Its scaling behavior is4–6
B = A(R R )3/2 1+a(R R )−ωT/2 +···+b(R R )−ω/2 +··· , (1.1)
2 g,1 g,2 g,1 g,2 g,1 g,2
(cid:2) (cid:3)
where R and R are the gyration radii of the two polymers, A, a, b are functions of
g,1 g,2
R /R , and ω , ω are correction-to-scaling exponents. Eq. (1.1) is valid for long polymers
g,1 g,2 T
in the good-solvent regime; more precisely, for N ,N → ∞ at fixed N /N (or, equivalently,
1 2 1 2
at fixed R /R ), where N and N are the lengths of the two polymers. The exponent
g,1 g,2 1 2
ω is the one that controls the scaling corrections in binary systems. The most accurate
estimate of the exponent so far yields7 ∆ = ων = 0.517±0.007+0.010, corresponding to ω =
−0.000
0.880±0.012+0.017. The exponent ω is a new exponent that characterizes multicomponent
−0.000 T
systems. Perturbative calculations9,6 indicate that ω is quite small, ω ≈ 0.4. Thus, scaling
T T
corrections decrease very slowly in multicomponent systems and can be quite relevant for
the values of N and N that can be attained in practice. Therefore, the determination of
1 2
the scaling behavior in multicomponent systems may require extrapolations in N and N ,
1 2
which, in turn, require a precise knowledge of the scaling exponents.
2
In this paper we improve the previous determinations9,6 of ω . First, we extend the
T
perturbative three-loop calculations of Ref. 6. We analyze the five-loop expansion of ω
T
in powers of d = 4 − ǫ, d being the space dimension, and the six-loop expansion of ω
T
in the fixed-dimension massive zero-momentum (MZM) scheme. Second, we compute ω
T
by numerical simulations. For this purpose we consider interacting self-avoiding walks and
compute the second virial coefficient. A careful analysis of its scaling behavior provides us
with a estimate of ω . Wealso consider the ideal-mixing point where the interaction between
T
the two different chemical species vanishes. A renormalization-groupanalysis of the behavior
close to this point was presented in Ref. 6. An extensive Monte Carlo simulation allows us
to verify the theoretical predictions.
The paper is organized as follows. In Sec. II we present our renormalization-group cal-
culations. In Sec. III we determine the correction-to-scaling exponent by means of a Monte
Carlo simulation. In Sec. IV we discuss the ideal-mixing point where the effective interaction
between the two chemically different species vanishes. Conclusions are presented in Sec. V.
II. PERTURBATIVE DETERMINATION OF ω
T
The starting point of the calculation is the Landau-Ginzburg-Wilson Hamiltonian6
H = ddx 1 (∂ φ )2 +(∂ φ )2 +r φ2 +r φ2 +
2 µ 1 µ 2 1 1 2 2
R (cid:8) (cid:2) (cid:3)
1[u φ4 +2w φ2φ2 +v φ4] , (2.1)
4! 0 1 0 1 2 0 2
(cid:9)
whereφ andφ aren-componentfields. Asusual, thepolymer theoryisobtainedinthelimit
1 2
n → 0. In this specific case, the fixed-point structure of the theory is particularly simple
and is explained in detail in Ref. 6. One finds that the β functions satisfy the following
properties: β (u,v,w) = β(u), β (u,v,w) = β(v), β (g,g,g) = β(g), where β(g) is the β
u v w
function in the vector O(n = 0) ϕ4 model and u,v,w are renormalized four-point couplings
normalized so that u ≈ Cu , v ≈ Cv , w ≈ Cw at tree level. The relevant fixed point is the
0 0 0
symmetric one u∗ = v∗ = w∗ = g∗, where g∗ is the zero of β(g). The exponent ω is given
T
3
by
∂β
w
ω = . (2.2)
T
∂w (cid:12)
(cid:12)u=v=w=g∗
(cid:12)
(cid:12)
Theexponent ω canbecomputeddirectlyintheO(n = 0)ϕ4 model. Indeed, asdiscussed in
T
Ref. 6, ω = −y , where y is the renormalization-group dimension of φ2φ2 in the symmetric
T 4 4 1 2
theory with u = v = w . It corresponds to an O(2n) vector theory and, for n → 0, one
0 0 0
is back to the O(n = 0) ϕ4 model. Using the results of Ref. 10 one can also show that, for
n → 0, φ2φ2 is a spin-4 perturbation of the O(2n) model and thus y is the renormalization-
1 2 4
group dimension of the cubic-symmetric perturbation ϕ4 of the O(n = 0) ϕ4 model.
a a
P
Thus, one can use the perturbative expansions reported in Refs. 11 and 12. The ǫ expansion
of ω is
T
1 19
ω = ǫ− ǫ2 +0.777867ǫ3−2.65211ǫ4+11.0225ǫ5 +O(ǫ6). (2.3)
T
2 64
At order ǫ3 it agrees with that given in Ref. 6. In the fixed-dimension MZM scheme we have
at six loops
3 185
ω = −1+ g − g2 +0.916668g3−1.22868g4+1.97599g5−3.59753g6+O(g7), (2.4)
T
2 216
where g is the four-point zero-momentum renormalized coupling normalized so that β(g) =
−g + g2 + O(g3), as used in, e.g., Ref. 13; the fixed point corresponds to14–18 g∗ = 1.40 ±
0.02. In order to obtain quantitative predictions, the perturbative series must be properly
resummed. We use here the conformal-mapping method13,19 that takes into account the
large-order behavior of the perturbative series.
From the standard ǫ expansion we obtain in three dimensions (ǫ = 1) ω = 0.42±0.04,
T
while in the fixed-dimension MZM scheme we find ω = 0.37 ± 0.04. Using the pseudo-ǫ
T
expansion, Ref. 20 obtained ω = 0.380 ± 0.018 in the MZM scheme. Though compati-
T
ble, the MZM result is lower than the ǫ-expansion one. This phenomenon also occurs for
other exponents and is probably related to the nonanalyticity of the renormalization-group
functions at the fixed point.14,21,15,16
4
A more precise estimate is obtained by considering ζ ≡ ω − ω/2. The perturbative
T
expansion of ζ has smaller coefficients than that of ω and thus ζ can be determined more
T
precisely. Its ǫ expansion is
1
ζ = ǫ2 −0.133936ǫ3 +0.490572ǫ4−2.41405ǫ5 +O(ǫ6) . (2.5)
32
The term proportional to ǫ is missing, while the other coefficients are smaller by a factor of
5-10 approximately. Similar cancellations occur in the MZM scheme:
1 1 85
ζ = − + g − g2+0.136823g3
2 2 432
−0.110394g4+0.074425g5+0.024718g6+O(g7). (2.6)
Resumming the perturbative series, we obtain
ζ = −0.006±0.009 (MZM),
ζ = −0.008±0.012 (ǫ exp). (2.7)
We can combine these estimates with those for ω. If we use the Monte Carlo result of Ref. 7
reported in the introduction, we obtain
ω = 0.433±0.016+0.008. (2.8)
T −0.000
If instead we use the field-theory estimates of ω reported in Ref. 17, we obtain
ω = 0.399±0.018 (MZM),
T
ω = 0.407±0.022 (ǫ exp). (2.9)
T
Collecting results, we estimate
ω = 0.41±0.04, (2.10)
T
where the error should be quite conservative.
5
III. MONTE CARLO RESULTS
In order to determine ω numerically, we consider lattice self-avoiding walks (SAWs)
T
with an attractive interaction −ǫ between nonbonded nearest-neighbor pairs. If β ≡ ǫ/kT is
the reduced inverse temperature, this model describes a polymer in a good solvent as long
as β < β , where β ≈ 0.269 corresponds to the collapse θ transition.22,23 We consider two
θ θ
walks with different interaction energies ǫ and ǫ , i.e. with different β and β . We assume
1 2 1 2
β ,β < β , so that both walks are in the good-solvent regime. Then, we consider the second
1 2 θ
virial coefficient
1
B2(N1,N2;β1,β2,β12) ≡ d3rh1−e−H(1,2)i0,r, (3.1)
2 Z
where the statistical average is over all pairs of SAWs such that the first one starts at the
origin, has N steps, and corresponds to an inverse reduced temperature β ; the second
1 1
one starts at r, has N steps, and corresponds to an inverse reduced temperature β . Here
2 2
H(1,2) is the reduced interaction energy: H(1,2) = +∞ if the two walks intersect each
other; otherwise, H(1,2) = −β N ,whereN isthenumber ofnearest-neighborcontacts
12 nnc nnc
between the two walks and β ≡ ǫ /kT is the reduced inverse temperature. In order to
12 12
generate the walks we use the pivot algorithm24–28 with a Metropolis test, while the second
virial coefficient is determined by using the hit-or-miss algorithm discussed in Ref. 29. We
study the invariant ratio
B (N ,N ;β ,β ,β )
2 1 2 1 2 12
A (N ,N ;β ,β ,β ) = , (3.2)
2 1 2 1 2 12
[R (N ;β )R (N ;β )]3/2
g 1 1 g 2 2
whereR (N;β)istheradiusofgyration. Aswehavealreadydiscussed, inthelimitN ,N →
g 1 2
∞ at R (N ;β )/R (N ;β ) fixed, A obeys a scaling law of the form6
g 2 2 g 1 1 2
R (N ;β )
g 2 2
A (N ,N ;β ,β ,β ) = f , (3.3)
2 1 2 1 2 12
(cid:18)R (N ;β )(cid:19)
g 1 1
where f(x) is universal. This applies for β ,β < β and, as we shall see, for β sufficiently
1 2 θ 12
small. Note that all the dependence on the inverse temperatures is encoded in a function
of a single variable. Moreover, f(x) is also the scaling function associated with a polymer
6
solution made of two different types of polymers that have the same chemical composition
(hence β = β = β ) but different lengths. In that case R (N ;β )/R (N ;β ) = (N /N )ν,
1 2 12 g 2 2 g 1 1 2 1
so that Eq. (3.3) implies that A (N ,N ;β,β,β) = g(N /N ), with g(x) = f(xν) universal.
2 1 2 1 2
The function f(x) satisfies the condition f(x) = f(1/x) and de Gennes’ relation30 f(x) ∼ xp,
p = 3/4−1/(2ν), for x → 0.
We will be interested here in the corrections to Eq. (3.3). In the scaling limit we can
write
a (β ,β ,β )
nm 1 2 12
A (N ,N ;β ,β ,β ) = f(ρ)+ f (ρ)+··· (3.4)
2 1 2 1 2 12 xnωT+mω nm
n+Xm≥1
where ρ ≡ R (N ;β )/R (N ;β ), x ≡ [R (N ;β )R (N ;β )]1/2, and we have neglected the
g 2 2 g 1 1 g 2 2 g 1 1
contributions of the additional correction-to-scaling operators with renormalization-group
dimensions −ω . They give rise to additional corrections proportional to x−p, p = nω +
i T
mω + n ω . Little is known about ω , though we expect them to satisfy ω > ω , ω > ω.
i i i i T i
P
In the following we will assume that all such exponents satisfy ω & 3ω ≈ ω + ω. The
i T T
scaling functions g (ρ) areuniversal once a specific normalizationhasbeen chosen. Instead,
nm
the coefficients a depend on the model and, in particular, on the specific values of the
nm
parameters β , β , and β .
1 2 12
We have simulated two SAWs with β = 0.05 and β = 0.15, two values that are well
1 2
within the good-solvent region. Then, we have computed A for 100 ≤ N = N ≤ 64000
2 1 2
and several values of β in the range 0 ≤ β ≤ 0.30. The results are plotted in Fig. 1. For
12 12
β < 0.25, as N = N = N increases, the estimates of A tend to become independent of
12 1 2 2
β although the convergence is very slow. The behavior changes for β & 0.25 and indeed
12 12
the data indicate that A = 0 for N → ∞ for some β = β slightly larger than 0.25. This
2 12 12,c
value corresponds to thecase inwhich the short-distance repulsion is exactly balanced by the
solvent-induced attraction proportional to β . In field-theoretical terms, this means that
12
the renormalization-group flow is no longer attracted by the symmetric fixed point discussed
in Sec. II, but rather by the unstable fixed point with w∗ = 0. Thus, at β = β there
12 12,c
is effectively no interaction between the chemically different polymers. For β > β , A
12 12,c 2
7
becomes negative signalling demixing. The behavior of A does not change significantly if β
2 1
and β are varied. In Fig. 2 we report results for shorter walks for different pairs of β and
2 1
β , and also for walks with N = 4N . Note that β depends very little on the parameters
2 1 2 12,c
β and β .
1 2
In order to determine ω , we have considered the data with β = 0,0.05,0.10,0.15 that
T 12
are sufficiently far from the critical value β . We performed a fit that is linear in ω
12,c T
c (β ) c (β )
2 12 3 12
ln[A (N;β = 0)−A (N;β )] = c (β )−ω lnx+ + , (3.5)
2 12 2 12 1 12 T
xωa xωb
whereβ = 0.05,0.10,0.15. Theexponentsω andω shouldtakeintoaccounttheadditional
12 a b
scaling corrections. Since 2ω ≈ ω, we should have ω ≈ ω ≈ ω−ω . Using the field-theory
T a T T
estimateofω andω ≈ 0.88±0.03(Ref.7),itshouldbesafetotake31 ω = 0.43±0.06. Asfor
T a
ω we should have ω ≈ 2ω ≈ ω. Moreover, there is also the possibility that there exists an
b b T
additional correction exponent ω not very much different from 3ω , which would contribute
1 T
a correction with exponent ω −ω . For this reason we have taken ω = 0.85±0.20. The
1 T b
error should be large enough to include all possibilities. The results are reported in Table I.
The systematic error reported there gives the variation of the estimate as ω and ω vary
a b
within the reported errors. The results with N = 250 and 500 are compatible within
min
errors and thus we can take the estimate that corresponds to N = 500 as our final result.
min
To be conservative, however, the error bar takes also into account the possibility that the
observed small trend is a real one. If the neglected corrections are of order x−3ωT we expect
the results to depend on N as N−3νωT ≈ N−0.7. This implies that the estimate of ω can
min min min T
decrease at most by 0.005 when N is further decreased. This leads to the result
min
ω = 0.415±0.020. (3.6)
T
As a check we perform a nonlinear fit of the form (fit 2)
a (β ) a (β ) a (β )
A (N;β ) = A∗ + 1 12 + 2 12 + 3 12 ; (3.7)
2 12 2 xωT x2ωT x3ωT
since, x ∼ Nν, we can also fit the data to (fit 3)
8
a (β ) a (β ) a (β )
A (N;β ) = A∗ + 1 12 + 2 12 + 3 12 , (3.8)
2 12 2 N∆T N2∆T N3∆T
where ∆ = ω ν. The results are reported in Table I. They agree with those obtained
T T
before and allow us to estimate the universal constant A∗ [A∗ = f(ρ) for ρ ≈ 1.24; f(ρ) is
2 2
defined in Eq. (3.3)]: A∗ = 5.495±0.020.
2
IV. IDEAL-MIXING POINT
In this section we consider the behavior close to the ideal-mixing point (IMP) β
12,c
where the effective interaction between the two chemically different species vanishes. The
renormalization-groupanalysis ispresented inRef.6. Thebehavior iscontrolledby anunsta-
ble fixed point characterized by an unstable direction with renormalization-group dimension
y = 2/ν − 3 and by a stable direction with exponent −ω, where ω ≈ 0.88 is the usual
I
correction-to-scaling exponent. These results imply that close to the IMP a renormalization-
group invariant quantity R (for instance, the invariant ratio A introduced above) scales
2
as
R(β ) = f [(β −β )(R R )yI/2,R /R ]
12 R 12 12,c g,1 g,2 g,1 g,2
1
+ g [(β −β )(R R )yI/2,R /R ]. (4.1)
(R R )ω/2 R 12 12,c g,1 g,2 g,1 g,2
g,1 g,2
In this section we wish to verify this scaling behavior for the second virial coefficient. For
this purpose we have made simulations for six different pairs of β and β with N = N
1 2 1 2
and 0.25 ≤ β ≤ 0.28. Since N = N = N and R (N;β) ≈ a(β)Nν, we can rewrite the
12 1 2 g
previous equation as
1
ˆ
R(N;β ,β ,β ) = f (b,ρ)+ gˆ (b,ρ),
1 2 12 R N∆ R
b ≡ (β −β )Nφ,
12 12,c
ρ ≡ R (N;β )/R (N;β ), (4.2)
g 1 g 2
where β also depends on β and β , and
12,c 1 2
9
φ ≡ νy = 2−3ν = 0.2372±0.0003. (4.3)
I
The critical value β can be characterized by requiring A (N → ∞;β ,β ,β ) = 0. We
12,c 2 1 2 12,c
can also define a finite-N IMP as the value of β where A (N;β ,β ,β ) vanishes (this is
12 2 1 2 12
the analogous of the Boyle point in θ solutions): we define βeff (N) such that
12,c
A (N;β ,β ,βeff (N)) = 0. (4.4)
2 1 2 12,c
Inserting in Eq. (4.2) we obtain for N → ∞ the behavior
a
βeff (N) = β + . (4.5)
12,c 12,c Nω+φ
Finally, we can replace β with βeff (N) in Eq. (4.2) obtaining the equivalent form
12,c 12,c
1
R(N;β ,β ,β ) = fˆ [(β −βeff )Nφ,ρ]+ g¯ [(β −βeff )Nφ,ρ], (4.6)
1 2 12 R 12 12,c N∆ R 12 12,c
where g¯ (b,ρ) vanishes at the IMP b = 0. Eq. (4.6) is more suitable for a numerical check
R
close to the IMP than Eq. (4.2), since scaling corrections vanish at the IMP and are therefore
small close to it. In the following we verify numerically predictions (4.5) and (4.6).
In Fig. 3 we show βeff (N) vs N−ω−φ for three different pairs of β and β . The data
12,c 1 2
show a quite good linear behavior: only the two points corresponding to N = 2000 and 4000
are in some cases off the linear fit, probably because our sampling is not yet adequate for
these large values of N. These results allow us to obtain β :
12,c
β = 0.2574±0.0006 β = 0.05, β = 0.10,
12,c 1 2
β = 0.2588±0.0007 β = 0.05, β = 0.15,
12,c 1 2
β = 0.2609±0.0003 β = 0.05, β = 0.20,
12,c 1 2
β = 0.2596±0.0004 β = 0.10, β = 0.15,
12,c 1 2
β = 0.2609±0.0009 β = 0.10, β = 0.20,
12,c 1 2
β = 0.2626±0.0009 β = 0.15, β = 0.20.
12,c 1 2
Note that the dependence on β and β is tiny.
1 2
10