Free energy formalism for polymer adsorption: Self-consistent field
theory for weak adsorption
Blokhuis, E.M.; Skau, K.I.; Avalos, J.B.
Citation
Blokhuis, E. M., Skau, K. I., & Avalos, J. B. (2003). Free energy formalism for polymer
adsorption: Self-consistent field theory for weak adsorption. Journal Of Chemical Physics,
119(6), 3483-3494. doi:10.1063/1.1588998
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J. Chem. Phys. 119, 3483 (2003); https://doi.org/10.1063/1.1588998 119, 3483 © 2003 American Institute of Physics.
Free energy formalism for polymer
adsorption: Self-consistent field theory for
weak adsorption
Cite as: J. Chem. Phys. 119, 3483 (2003); https://doi.org/10.1063/1.1588998
Submitted: 10 March 2003 . Accepted: 13 May 2003 . Published Online: 24 July 2003 Edgar M. Blokhuis, Karl Isak Skau, and Josep B. Avalos
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Free energy formalism for polymer adsorption: Self-consistent field theory
for weak adsorption
Edgar M. Blokhuis and Karl Isak Skau
Colloid and Interface Science, Leiden Institute of Chemistry, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands
Josep B. Avalos
University Rovira y Virgili, ETSEQ, Department of Chemical Engineering, Avinguda dels Paı¨sos Catalans 26, Tarragona 43007, Spain
共Received 10 March 2003; accepted 13 May 2003兲
Polymer adsorption has been widely investigated in the context of self-consistent mean-field theories. As a further simplification, the ‘‘ground state dominance approximation’’ is often made, treating the polymer chains as infinitely long. For short polymers, or not so concentrated polymer solutions, corrections to ground state dominance may be important, however. In this work, we discuss analytical solutions to the full self-consistent field equations, valid for any chain length, in the limit of weak adsorption. We show how the resulting equations may be put into a free energy functional formalism, in analogy to the de Gennes–Lifshitz free energy for infinitely long polymer chains. Analytical expressions are derived for polymer density profiles, surface tension and the interaction potential between two planar, polymer-adsorbing surfaces. Particular attention is paid to the distal ordering of the polymer coils that shows up as oscillations in the polymer density profile and interaction potential at the scale of the polymer’s radius of gyration. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1588998兴
I. INTRODUCTION
Polymer surfaces have attracted considerable attention in recent years both from an experimental and a theoretical point of view. Interest in these systems is driven by practical applications, but also because it serves as a theoretical ex-ample of a confined polymer system.1–3More recently, inter-est has surged in the study of the interplay between colloidal particles and polymers in solution.4 –9Understanding the in-teraction between colloidal particles, as mediated through the polymer solution, supplies means to understand the mecha-nisms behind polymer induced colloidal stabilization and crystallization. This may be helpful in formulating conditions under which colloidal particles or globular proteins may crystallize.
Theoretically, the interplay of polymers and surfaces has been extensively studied. Various models for the interaction of the polymer and the 共colloidal兲 surface have been dis-cussed; irreversibly adsorbed or grafted polymers,10 revers-ibly adsorbed polymers,1,2 and polymers depleted from the surface.3,7–9 In the latter case, the polymer density is as-sumed to be zero at the surface so that a depletion layer exists around each particle. In these calculations, notably by Eisenriegler3,7 and by others,9,11 analytical solutions of the Edwards equations in the context of the so-called self-consistent field theory could be obtained.
Surfaces with enhanced polymer adsorption have first been studied by de Gennes1,12in self-consistent field theory, and by Scheutjens and Fleer in a lattice model for polymers.13 In the de Gennes model, the so-called ground state dominance approximation1,14,15 is made in which the polymer chain length is essentially set to infinity. A result of
the de Gennes model is that the polymer segment density profile is a monotonically decaying function and that the interaction between two planar surfaces is attractive at all separations. Various models have extended the de Gennes model to determine finite chain length corrections to, e.g., surface tension, polymer segment density profile, surface– surface interactions, etc.16 –20 Such calculations are of par-ticular interest since the polymer chain length is an important parameter in experiments,21 computer simulations,22–24 and numerical solutions of lattice models.13Variation of the chain length thus provides a more stringent testing of theoretical models. Moreover, it was expected,25,26 and later verified,16 –20 that in certain situations, the ‘‘tails’’ of the polymer chain become important, leading to qualitatively different behavior.
Extensions to the ground state dominance model were first investigated by Semenov et al.16 The presence of tails was taken into account by including a second order param-eter related to the end segment density. Good agreement was obtained for the loop and tail distribution of adsorbed poly-mer when the theoretical predictions are compared with nu-merical solutions of the lattice self-consistent mean-field theories.17Furthermore, it was shown that the interaction be-tween two planar surfaces becomes weakly repulsive at larger distances.18
It was subsequently shown by Semenov that the two-order parameter model could be cast into a free energy for-malism of a single order parameter19 for which the Euler– Lagrange equations are the Edwards equations, just as de Gennes had done previously in the context of the ground state dominance approximation.12 A free energy functional
JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 6 8 AUGUST 2003
3483
was proposed19 to describe the free energy correctly to
O(1/N2). This means that the three leading contributions in an expansion in 1/N to the free energy are captured: 共1兲 the ground state dominance contribution;共2兲 the leading correc-tion to ground state dominance of O(1/N), which gives a description on the same level as the two-order parameter model;27 共3兲 terms of O(1/N3/2). These last terms were shown to be responsible for a weak, oscillatory interaction potential at large distances between two planar surfaces.19 Similar oscillations were also observed in the segment den-sity profile in analytical work by Semenov et al. and in nu-merical solutions of the Scheutjens and Fleer lattice model for polymers.28
In this article, we investigate the surface tension, seg-ment density profiles, and interaction potential for 共planar兲 polymer-adsorbing surfaces. Analytical expressions, valid for any polymer length, are derived in the limit of weakly ad-sorbing polymers. Our expression for the free energy is con-sistent with the linear response model for bulk polymers1and the Semenov single order parameter free energy.19 We pay particular attention to the segment density profile at large distances from the surface and the interaction between sur-faces at large separations.
The outline of the paper is as follows: In the next section we discuss the self-consistent field theory in the context of which our calculations are made. In Secs. III and IV we provide a detailed derivation of the free energy from the Edwards equations in the case of weak external fields共Sec. III兲 and weak polymer adsorption 共Sec. IV兲. In Secs. V and VI, the free energy derived is used to determine surface ten-sion, segment density profiles, and interaction potential for polymer adsorbing surfaces. We end with a discussion of results.
II. SELF-CONSISTENT FIELD THEORY
The Green function G(r,r
⬘
,N) describes the statistical weight of a single polymer chain of length N with one end atr and the other end at r
⬘
. The Green function can be deter-mined by solving the Edwards equation, nG共r,r
⬘
,n兲⫽ a2 6 “ 2G共r,r⬘
,n兲⫺U共r兲 kBT G共r,r⬘
,n兲, 共2.1兲 with initial conditionlim
n→0
G共r,r
⬘
,n兲⫽␦共r⫺r⬘
兲, 共2.2兲 where a is the polymer segment length and U(r) is an as yet unspecified external potential.In terms of the Green function, one may construct the average segment density,
共r兲⫽Np 兰0 N dn兰dr
⬘
兰dr⬙
G共r⬘
,r,n兲 G共r,r⬙
,N⫺n兲 兰dr⬘
兰dr⬙
G共r⬘
,r⬙
,N兲 , 共2.3兲 where Np is the total number of polymer chains. Thispref-actor determines the scale of the Green function. As will
become apparent later, it is chosen such that the density be-comes equal to the uniform bulk density, (r)⫽NpN/V ⬅b, for a homogeneous system.
The general solution of the Edwards equation may be formulated in the form of an eigenfunction expansion,
G共r,r
⬘
,n兲⫽兺
k k共r兲k
*共r
⬘
兲 e⫺n Ek/kBT, 共2.4兲 with the eigenfunctions k and eigenvalues Ek determinedby a2 6 “ 2 k共r兲⫺ U共r兲 kBT k共r兲⫽⫺ Ek kBTk共r兲. 共2.5兲 In the self-consistent field model, the external potential is interpreted as being induced by the presence of the other polymer segments,14
U共r兲⫽kBTv共r兲, 共2.6兲
wherev is the so-called excluded volume parameter. In this
way, the set of Eqs. 共2.1兲–共2.3兲 becomes self-consistently closed. This approach has formed the basis of many theoret-ical treatments in the literature.1–3
For very long chains—a more precise criterion is formu-lated later—only the ground state contribution (k⫽0) in the eigenfunction expansion remains and the summation may be limited to the first term only,
G共r,r
⬘
,n兲⬇0共r兲0*共r⬘
兲 e⫺n E0/kBT. 共2.7兲
This is the so-called ground state dominance approximation.1 The Edwards equation, Eq. 共2.1兲, and seg-ment density, Eq. 共2.3兲, can then be written as
a2 6 “
2共r兲⫽v 兩共r兲兩2共r兲⫺ E0
kBT共r兲,
共r兲⫽兩共r兲兩2, 共2.8兲
where we have defined (r)⬅(NpN)1/20(r). It may now be noted1that the same equation follows from the minimi-zation of the following free energy functional:
F关兴 kBT ⫽
冕
dr冋
a2 6 兩“共r兲兩 2⫹v 2兩共r兲兩 4⫺ E0 kBT兩共r兲兩 2册
. 共2.9兲 The last term is added to fix the total number of segments,NpN⫽
冕
dr兩共r兲兩2. 共2.10兲The free energy can be rewritten in terms of the segment density,(r)⫽兩(r)兩2. One then has
F关兴 kBT ⫽
冕
dr冋
a2 24 兩“共r兲兩2 共r兲 ⫹ v 2共共r兲⫺b兲 2册
, 共2.11兲 where the uniform bulk density, b⬅NpN/V, is related toE0/kBT⫽vb. Furthermore, we subtracted the 共constant兲
bulk free energy in order for the above free energy to be the excess free energy.
en-ergy density functional may be formulated without making use of the ground state dominance approximation. In the present article we construct such a free energy without mak-ing the ground state dominance approximation.
III. INHOMOGENEOUS SYSTEM
Ultimately, we are interested in the case of polymer ad-sorption onto a solid surface, but for now we start by con-sidering an infinite system, without the presence of a surface, in which the segment density profile only depends on one coordinate z, (r)⫽(z). Furthermore, often we are only interested in the departure of the density profile with respect to the bulk density at z⫽⫾⬁ and introduce ␦(z)⬅(z) ⫺b, with limz→⫾⬁(z)⫽b.
A further reduction that we make, but could have made quite generally, is to integrate the Green function over one of the chain’s ends. This defines the statistical weight of a poly-mer chain of length n with one end fixed at position r,
G共r,n兲⬅
冕
dr⬘
G共r,r⬘
,n兲. 共3.1兲 In the bulk region G(r,n)→Gb(n), which is independent ofr. By solving the Edwards equation with the initial condition
关Eq. 共2.2兲兴, one finds that in the bulk, Gb(n)⫽e⫺vbn. It is
now convenient to redefine the statistical weight such so as to absorb this trivial n-dependence,
Z共r,n兲⬅evbn
冕
dr⬘
G共r,r⬘
,n兲. 共3.2兲 In the geometry under consideration, we have that Z(r,n) ⫽Z(z,n). The Edwards equation 关Eq. 共2.1兲兴 and segment density 关Eq. 共2.3兲兴 in the self-consistent field approximation 关Eq. 共2.6兲兴 are then given by nZ共z,n兲⫽ a2 6 2 z2Z共z,n兲⫺v共共z兲⫺b兲 Z共z,n兲, 共3.3兲 共z兲⫽Nb
冕
0 N dn Z共z,n兲 Z共z,N⫺n兲, 共3.4兲 with the initial condition,lim
n→0
Z共z,n兲⫽1. 共3.5兲
An exact solution of the above coupled differential equations can only be found when certain further approximations are made. In the following we consider weak fields leading to small deviations from bulk behavior.
Weak fields: Linear response theory: The assumption
of a weak field implies that ␦(z)⬅(z)⫺bⰆb and
␦Z(z,n)⬅Z(z,n)⫺1Ⰶ1. Linearization of Eqs. 共3.3兲 and
共3.4兲 then leads to n␦Z共z,n兲⫽ a2 6 2 z2␦Z共z,n兲⫺v␦共z兲, 共3.6兲 ␦共z兲⫽2Nb
冕
0 N dn␦Z共z,n兲, 共3.7兲with the initial condition,
lim
n→0
␦Z共z,n兲⫽0. 共3.8兲
We now rewrite these differential equations in terms of a single differential equation for the density ␦(z) alone. From the first integral we then proceed to construct the free energy functional. Only later do we explicitly solve the re-sulting differential equation for ␦(z) and determine the corresponding excess free energy.
The linearized differential equations can be solved by separating␦Z(z,n) into one part that no longer depends on n and one that still does depend on n: ␦Z(z,n)⬅(z) ⫹a(z,n). The differential equation in Eq. 共3.6兲 is then split up as na共z,n兲⫽ a2 6 2 z2a共z,n兲, 共3.9兲 a2 6 2 z2共z兲⫽v␦共z兲, 共3.10兲
with the segment density关Eq. 共3.7兲兴 now given by
␦共z兲⫽2b共z兲⫹
2b
N
冕
0 Ndn a共z,n兲, 共3.11兲 and the initial condition 关Eq. 共3.8兲兴,
lim
n→0
a共z,n兲⫽⫺共z兲. 共3.12兲
In this section, we obtain a solution to these differential equations for the infinite system with no boundaries present, i.e.,⫺⬁⬍z⬍⬁, and leave the more complicated situation of a semi-infinite system in the presence of a solid surface to the next section.
For the infinite system, the solution to the differential equation for a(z,n) in Eq. 共3.9兲, with the initial condition Eq. 共3.12兲, reads a共z,n兲⫽⫺
冉
3 2n a2冊
1/2冕
⫺⬁ ⬁ dz⬘
共z⬘
兲 exp冉
⫺3共z⬘
⫺z兲 2 2 n a2冊
. 共3.13兲 Inserting the above expression for a(z,n) into Eq.共3.11兲, we now have for the density profile in terms of (z),␦共z兲⫽2b共z兲⫺ b RG
冕
⫺⬁ ⬁ dz⬘
共z⬘
兲␣冉
z⬘
⫺z 2RG冊
, 共3.14兲 where the radius of gyration RG⬅冑
Na2/6, and where wehave defined
␣共x兲⬅
冑
2 exp共⫺x2兲⫺2 兩x兩 erfc共兩x兩兲. 共3.15兲 The Fourier transform of Eq. 共3.14兲 gives␦˜共k兲⫽2
b˜共k兲⫺2b˜␣共2kRG兲˜共k兲, 共3.16兲
where the tilde refers to the Fourier transform of the function, ˜ (k)⫽兰dz e⫺ikz(z) and (z)⫽(2)⫺1 ⫻兰dk eikz˜ (k). Explicitly, we have ␣˜ (x)⫽(4/x2)关1 ⫺exp(⫺x2/4)兴. Solving Eq. 共3.16兲 for˜ (k) then gives
˜共k兲⫽ 1 2b
␦˜共k兲
1⫺␣˜共2kRG兲. 共3.17兲
Next, we Fourier transform the original differential equation for (z) in Eq.共3.10兲,
⫺a 2
6 k
2˜共k兲⫽v␦˜共k兲. 共3.18兲
With the expression for ˜ (k) in Eq. 共3.17兲 inserted into it, this is written as ⫺1 bN ␦˜共k兲 fD共k2RG 2兲⫽v␦˜共k兲, 共3.19兲
where the Debye function is defined as fD(x)⬅(2/x2) ⫻关exp(⫺x)⫺1⫹x兴. The back Fourier transform of Eq. 共3.19兲 then finally yields the equation for␦(z) sought after ⫺1 bN
冕
⫺⬁ ⬁ dz⬘
␦共z⬘
兲冕
⫺⬁ ⬁ dk 2 eik(z⫺z⬘) fD共k2R G 2兲⫽v␦共z兲. 共3.20兲 We may now construct the free energy for which the Euler– Lagrange equation is given by the equation above. We findF关␦兴 A kBT ⫽ 1 2bN
冕
⫺⬁ ⬁ dz冕
⫺⬁ ⬁ dz⬘
␦共z兲␦共z⬘
兲 ⫻冕
⫺⬁ ⬁ dk 2 eik(z⫺z⬘) fD共k2RG 2兲⫹ v 2冕
⫺⬁ ⬁ dz关␦共z兲兴2. 共3.21兲 This expression for the free energy, with the corresponding Euler–Lagrange equation in Eq.共3.20兲, is the main result of this section.The free energy constructed from Eq. 共3.20兲 is only uniquely defined up to a constant, but we do know that the free energy in Eq. 共3.21兲 should reduce to the expression obtained in the ground state dominance approximation, Eq. 共2.11兲, in the limit N→⬁. That this is indeed the case may be verified by using that fD(k2RG
2
)⫺1⬇21k
2R G 2
→⫺ (N a2/12)␦
⬙
(z⫺z⬘
), and integrating by parts yieldingF关␦兴 A kBT ⫽
冕
⫺⬁ ⬁ dz冋
a 2 24b关␦⬘
共z兲兴 2⫹v 2关␦共z兲兴 2册
. 共3.22兲 One may verify that this expression indeed corresponds to the free energy in Eq. 共2.11兲 with the density expanded around the bulk density. Note that the condition kRG→⬁,implies that the description is strictly valid for distances smaller than the polymer’s radius of gyration, z⬀1/kⰆRG.
The first term in the expression for the free energy in Eq. 共3.21兲 could also have been derived from a linear response analysis,
␦F⫽12kBT
冕
dr冕
dr⬘
Cb共r,r⬘
兲␦共r兲␦共r⬘
兲, 共3.23兲in which Cb(r,r
⬘
)⫽Cb(兩r⫺r⬘
兩) is the bulk polymer directcorrelation function. For Gaussian chains in solution, the
Fourier transform of the direct correlation function is given by the Debye function,
C ˜ b共k兲⫺1⫽bN fD共k 2R G 2兲. 共3.24兲
A further important connection with previous work forms the expression for the free energy proposed by Semenov.19It has the following form:
FSem关兴 A kBT ⫽
冕
⫺⬁ ⬁ dz冋
a 2 24 ⬘
共z兲2 共z兲 ⫹ v 2共z兲 2⫺2 Nb 1/2共z兲1/2 ⫹ 共z兲册
⫹1 bN冕
⫺⬁ ⬁ dz冕
⫺⬁ ⬁ dz⬘
关b1/2共z兲1/2 ⫺共z兲兴关b 1/2共z⬘
兲1/2⫺共z⬘
兲兴 ⫻冕
⫺⬁ ⬁ dk 2e ik(z⫺z⬘)冋
1⫺e ⫺k2R G 2 ⫺k2R G 2 e⫺k2RG2 e⫺k2RG 2 ⫺1⫹k2R G 2册
. 共3.25兲 This form for the free energy is derived by Semenov19 in what is termed the ‘‘main approximation’’ which captures the free energy toO(1/N2). It is pointed out by Semenov19that the free energy reduces to the free energy in Eq.共3.21兲, when(z) is expanded around the bulk densityb. The first term
in Eq.共3.25兲 is the leading correction in 1/N to ground state dominance and gives a description on the same level as the two-order parameter model.27
Of course, without the existence of adsorption potentials responsible for the existence of an inhomogeneity in the sys-tem, the solution of the Euler–Lagrange equation in Eq. 共3.19兲 for ␦˜ (k) yields the trivial result ␦˜ (k)⫽0 and, therefore, ␦(z)⫽0. Since the Edwards equation is the Euler–Lagrange equation that minimizes the free energy in Eq. 共3.21兲, this result indicates that, from an energetic point of view, the most profitable situation is when (z)⫽b,
everywhere. Of more interest is therefore the semi-infinite system (0⭐z⬍⬁), considered next, in which the value of the density at the surface differs from the bulk value due the presence of an interaction with a solid surface.
IV. POLYMER ADSORPTION
We now turn to the situation of a polymer solution in contact with a solid surface共wall兲, located at z⫽0. The solid surface is considered to be impenetrable to the polymer so that the density is defined only for z⬎0 关see the density profile in Fig. 1共a兲兴. An attractive interaction with the wall is considered at z⫽0, Uwall共z兲 kBT ⫽⫺ a2 6 1 d␦共z兲. 共4.1兲
The inverse of the extrapolation length, d, is a measure of the surface interaction strength and is responsible for an en-hanced polymer adsorption, (0)⬎b, when d⬎0.
关(0)⫽0兴. Such a description is therefore expected to be valid only for enhanced polymer adsorption and for distances beyond a certain microscopic distance (⬀a) away from the surface.1
Eisenriegler and co-workers3,7,8 and others11 have ad-dressed the related problem of polymer depletion, in which there is no additional interaction with the wall considered, assuming the共Dirichlet兲 boundary condition,(0)⫽0. They showed that under these circumstances analytic solutions for the polymer partition function could be obtained. In the present situation, such an analytic solution cannot be ob-tained in general. However, when we assume that the exter-nal fields are weak, i.e., when we consider the case of weak adsorption, an analytic solution can indeed be formulated.
Weak adsorption: Before turning to the formulation of
the appropriate boundary conditions in the case of polymer adsorption, we come back to the expression for the free en-ergy in the ground state dominance approximation. The free energy for weak adsorption is then given by Eq. 共3.22兲 to which a term containing the interaction with the solid surface is added,1 F关␦兴 A kBT ⫽
冕
0 ⬁ dz冋
a 2 24b关␦⬘
共z兲兴 2⫹v 2关␦共z兲兴 2册
⫺a 2 6 1 d共0兲. 共4.2兲The minimization of the above free energy leads to the fol-lowing Euler–Lagrange equation and boundary condition for
␦(z):
a2
12b␦
⬙
共z兲⫽v␦共z兲,␦
⬘
共0兲⫽⫺2db. 共4.3兲We turn next to the evaluation of the full free energy, which we expect to reduce to Eq.共4.2兲 in the limit N→⬁.
Since the presence of the solid surface precludes the polymer for z⬍0, the partition function Z(z,n) is defined only for z⭓0. It turns out to be mathematically convenient to extend Z(z,n) also for z⬍0. Here we choose to extend sym-metrically 关see Fig. 1共b兲兴 and define a partition function Ze(z,n) defined for all z as
Ze共z,n兲⫽Z共⫺z,n兲⌰共⫺z兲⫹Z共z,n兲⌰共z兲, 共4.4兲
where⌰(x) is the Heaviside-function.
The presence of the potential induced by the surface modifies the external field in Eq.共2.6兲. The result is that the Edwards equation in Eq.共3.3兲 now becomes
nZe共z,n兲⫽ a2 6 2 z2Ze共z,n兲⫺v共e共z兲⫺b兲Ze共z,n兲 ⫹a 2 3 1 d␦共z兲Ze共z,n兲. 共4.5兲 Since the term containing the second derivative with respect to z is the only term that varies rapidly as a function of z near
z⫽0, the above differential equation reduces to the usual Edwards equation in Eq.共3.3兲 with the additional term lead-ing to the boundary condition,
zZ共z,n兲
冏
z⫽0 ⫽⫺1dZ共z,n兲冏
z⫽0 . 共4.6兲Again, we introduce ␦Z(z,n)⫽Z(z,n)⫺1⫽a(z,n)⫹(z), and the symmetrically extended functions ae(z,n) and
e(z), and expand in␦Z(z,n)Ⰶ1. The differential equation
for ae(z,n) then becomes
nae共z,n兲⫽ a2 6 2 z2ae共z,n兲. 共4.7兲
One may, again, verify that this expression reduces to the equation obtained in Eq. 共3.9兲, now complemented by the boundary condition,
za共z,n兲
冏
z⫽0
⫽0. 共4.8兲
The differential equation for e(z) reads
a2 6 2 z2e共z兲⫽v␦e共z兲⫺ a2 3 1 d␦共z兲. 共4.9兲
This expression reduces to the equation obtained in Eq. 共3.10兲 with the additional term leading to the boundary condition,
z共z兲
冏
z⫽0⫽⫺ 1d. 共4.10兲
We may now solve the differential equation for a(z,n) in Eq. 共3.9兲 taking into account the boundary condition in Eq. 共4.8兲. One may show that
FIG. 1. Sketches of the polymer density profile as a function of the distance to a solid surface.共a兲␦(z) is the polymer density profile near a single surface at z⫽0; 共b兲␦e(z) is the same density profile as in共a兲,
symmetri-cally extended to z⬍0; 共c兲␦e(z;h) is the polymer density profile between
two surfaces at z⫽0 and z⫽h, periodically extended to whole space.
a共z,n兲⫽⫺
冉
3 2n a2冊
1/2冕
0 ⬁ dz⬘
共z⬘
兲冋
exp冉
⫺3共z⬘
⫹z兲 2 2 n a2冊
⫹exp冉
⫺3共z⬘
⫺z兲 2 2 n a2冊册
共z⬎0兲. 共4.11兲 The segment density profile is obtained by inserting the above expression for a(z,n) into Eq. 共3.11兲. We then arrive at ␦共z兲⫽2b共z兲⫺ b RG冕
0 ⬁ dz⬘
共z⬘
兲 ⫻冋
␣冉
z2R⬘
⫹z G冊
⫹␣冉
z2R⬘
⫺z G冊册
共z⬎0兲, 共4.12兲 where ␣(x) is defined earlier 关Eq. 共3.15兲兴. This expression for ␦(z) is symmetrically extended for z⬍0,␦e共z兲⫽2be共z兲⫺ b RG
冕
⫺⬁ ⬁ dz⬘
e共z⬘
兲␣冉
z⬘
⫺z 2RG冊
. 共4.13兲 One may note that the above expression for the symmetri-cally extended density profile is the same as the expression for ␦(z) in the previous section, Eq.共3.14兲.In order to proceed as before, we need to determine the Fourier transform of Eq.共4.13兲 and the Edwards equation in Eq. 共4.9兲. One finds
␦˜ e共k兲⫽2b˜e共k兲⫺2b˜␣共2kRG兲˜e共k兲, 共4.14兲 ⫺a 2 6 k 2˜ e共k兲⫽v␦˜e共k兲⫺ a2 3 1 d. 共4.15兲
Substituting the expression for ˜e(k) in Eq.共4.15兲 into Eq. 共4.14兲 gives ⫺ a 2 12b k2 1⫺␣˜共2kRG兲␦˜e共k兲⫽v␦˜e共k兲⫺ a2 3 1 d. 共4.16兲 The back Fourier transform of this equation gives
1 4bN
冕
⫺⬁ ⬁ dz⬘
␦e⬙
共z⬘
兲␣D冉
z⫺z⬘
2RG冊
⫽v␦e共z兲⫺ a2 3 1 d␦共z兲, 共4.17兲where we have defined
␣D共x兲⬅
冕
⫺⬁ ⬁ dk 2 eikx 1⫺␣˜共k兲. 共4.18兲Again, we may construct the free energy for which the Euler–Lagrange equation is equal to the Edwards equation in Eq. 共4.17兲. One finds
Fe关␦e兴 A kBT ⫽ ⫺1 8bN
冕
⫺⬁ ⬁ dz冕
⫺⬁ ⬁ dz⬘
␦e共z兲␦e⬙
共z⬘
兲 ⫻␣D冉
z⫺z⬘
2RG冊
⫹v 2冕
⫺⬁ ⬁ dz关␦e共z兲兴2 ⫺a 2 3 1 de共0兲, 共4.19兲where e(0)⫽b⫹␦e(0). From the symmetrically ex-tended free energy, we may now go back to the free energy in terms of(z) instead ofe(z). Keeping in mind that the free energy of the symmetrically extended system is twice the free energy of the semi-infinite system, i.e., Fe⫽2 F, we
find F关␦兴 A kBT ⫽ 1 8bN
冕
0 ⬁ dz冕
0 ⬁ dz⬘
␦⬘
共z兲␦⬘
共z⬘
兲 ⫻冋
␣D冉
z⫺z⬘
2RG冊
⫺␣D冉
z⫹z⬘
2RG冊册
⫹v2冕
0 ⬁ dz关␦共z兲兴2⫺a 2 6 1 d共0兲. 共4.20兲 In this way we have constructed the free energy functional for weak polymer adsorption. We should verify that this ex-pression for the free energy reduces to the free energy given in Eq. 共4.2兲 for ground state dominance when N→⬁. One may show that this is indeed the case by using that in this limit ␣D((z⫾z⬘
)/2RG)→(N a2/3)␦(z⫾z⬘
).V. POLYMER DENSITY PROFILE AND SURFACE TENSION
In this section we turn our attention to obtaining explicit solutions for the segment density profile and surface tension.
A. Polymer density profile
An explicit solution for the polymer segment density profile is obtained by solving␦˜e(k) from Eq.共4.16兲. This
gives ␦˜ e共k兲⫽ 4bb2 d
冋
k2R G 2⫺1⫹exp共⫺k2R G 2兲 k4b 2 RG 2⫹4 k2R G 2 ⫺4⫹4 exp共⫺k2R G 2兲册
, 共5.1兲 where the bulk correlation length, b⬅a/(3 vb)1/2. TheFourier transform of␦˜e(k) is ␦e共z兲⫽ bb d
冕
⫺⬁ ⬁ dk e2ikz/b ⫻冋
4 k 2⫺⫹ exp共⫺4k2/兲 4 k4⫹4 k2⫺⫹ exp共⫺4k2/兲册
, 共5.2兲 where we have introduced the parameter as the 共square of兲 the bulk correlation length, b, divided by the polymer’sThe segment density profile ␦(z) thus becomes ␦共z兲⫽2bdb
冕
0 ⬁ dk cos冉
2kz b冊
⫻冋
4 k 2⫺⫹ exp共⫺4k2/兲 4 k4⫹4 k2⫺⫹ exp共⫺4k2/兲册
. 共5.4兲 This expression is our final result for the segment density profile valid for any chain length. We first investigate Eq. 共5.4兲 by comparing it to the segment density profile obtained in MC simulations by de Joannis et al.23 In Fig. 2, the re-duced segment density ␦(z)/b is shown as a function ofz/RG. For the MC simulations 共circles兲, the following
pa-rameter values were used: chain length N⫽200, bulk density a3
b⫽0.0216, radius of gyration RG/a⫽9.76, and surface
interaction strengths⫽1.0 kBT.23For the evaluation of the analytical expression in Eq.共5.4兲 共solid curve兲, we have set
v⫽a3共good solvent兲, which gives ⯝0.46. The relation
be-tween the value ofs and the extrapolation length d is less
transparent; we have now set d⫽1/3 in units of a/
冑
6. The value of d determines the scale of␦(z) and is chosen such that the depth of the minima for the segment density profiles are approximately equal.Even though the adsorption strength is quite high in the MC simulations, Eq.共5.4兲 gives a fair description of the full polymer segment density profile 共inset Fig. 2兲. Good agree-ment is obtained for the ‘‘overshoot’’ of the polymer segagree-ment density profile, which may be further improved by shifting the density profile by a lattice distance a 共dashed curve兲.
We may further investigate the expression for the seg-ment density profile in Eq.共5.4兲 by expanding in Ⰶ1. Since small implies large N, we thus should reproduce the results derived previously in the ground state dominance approximation1 and those by Semenov et al.16,19 The three leading terms in the expansion in of Eq. 共5.4兲 are
␦共z兲⫽2bdb
冕
0 ⬁ dk冋
cos共2kz/b兲 k2⫹1册
⫺ bb 2d ⫻冕
0 ⬁ dk冋
cos共2kz/b兲 共k2⫹1兲2册
⫺ 3/2bb 4d冕
0 ⬁ dt cos冉
tz RG冊
⫻冋
1⫺e⫺t 2 ⫺t2e⫺t2 e⫺t2⫺1⫹t2册
⫹O共 2兲, 共5.5兲where we have defined t⬅2kRG/b. The first two integrals
can be carried out
␦共z兲⫽bdb
冋
e⫺2z/b⫺ 8冉
1⫹ 2z b冊
e⫺2z/b ⫺3/2冉
z RG冊
⫹O共 2兲册
, 共5.6兲where we have defined
共x兲⬅41
冕
0 ⬁ dt cos共xt兲冋
1⫺e ⫺t2 ⫺t2e⫺t2 e⫺t2⫺1⫹t2册
. 共5.7兲 The leading term in Eq. 共5.6兲 corresponds to the segment density profile in the ground state dominance approximation. One may verify that this profile indeed satisfies the Euler– Lagrange equation and boundary condition in Eq.共4.3兲.Both the leading term and the first correction in are exponentially decaying functions of z/b; the next term,
however, varies as a function of z/RG. This means that even
as this term is subdominant in , at distances much larger than the bulk correlation length, it becomes the dominant contribution,16,19
␦共z兲⫽⫺bdb3/2
冉
zRG
冊
⫹O共2兲 共zⰇ
b兲. 共5.8兲
In Fig. 3, we have plotted(z/RG)共circles兲 as a function of
z/RG.关To show(z/RG) on a logarithmic scale the absolute
value has been taken.兴 One finds that at large distances,
(z/RG) exhibits an oscillatory behavior. These oscillations
FIG. 2. Polymer segment density profiles,␦(z)/b, as a function of z/RG
from MC simulations共Ref. 23兲 共circles兲 with the analytical expression in Eq. 共5.4兲 共solid curve兲. In the MC simulations: N⫽200, a3b⫽0.0216;
RG/a⫽9.76; surface interaction strength s⫽1.0 kBT 共Ref. 23兲. In Eq. 共5.4兲: v⫽a3; ⯝0.46; d⫽(1/3)a/冑6. The inset shows the same polymer
segment density profile on a full scale. The dashed curve is Eq.共5.4兲 shifted by a lattice distance a.
FIG. 3. Numerical solution for兩(z/RG)兩 as a function of z/RG共circles兲;
the solid curve is the approximate expression for the asymptotic behavior (zⰇRG) given by Eq.共5.9兲.
are also observed28in numerical solutions of the density pro-file in the Scheutjens and Fleer lattice model for polymers.
It is interesting to further quantify the behavior of the density profile at large distances, zⰇRG. One may show
that, for large x 关see also Eq. 共118兲 in Ref. 16兴,
共x兲⫽C0e⫺A0xsin共B0x⫹␣0兲 共xⰇ1兲, 共5.9兲 where A0⬅Im关W共1,⫺1/e兲⫹1兴1/2⬇2.217792... , B0⬅Re关W共1,⫺1/e兲⫹1兴1/2⬇1.682188... , C0⬅⫺ 1 4
冑
A0 2⫹B 0 2⬇⫺0.695897... , ␣0⬅arctan共A0/B0兲⬇0.921879... , 共5.10兲 with W(k,x) the Lambert W-function.29The polymer density profile therefore falls as an oscillating exponential on the scale of the polymer’s radius of gyration. The asymptotic behavior as given by Eq.共5.9兲 is shown as the solid curve in Fig. 3.The asymptotic behavior of the density profile was de-termined under the assumption that is small. It is interest-ing to investigate the asymptotic behavior of the density pro-file directly from the full expression in Eq. 共5.4兲. One finds that also then the density profile is a decaying sinusoid,
␦共z兲⬀e⫺ 共Az/RG兲sin
冉
Bz RG⫹␣
冊
共zⰇRG兲. 共5.11兲The coefficients A and B, that determine the exponential decay and the oscillation period of the polymer density pro-file, respectively, depend 共moderately兲 on 共see Fig. 4兲. In the limit→0, A and B are equal to the values A0 and B0in Eq. 共5.10兲. The value of the coefficients A and B were phe-nomenologically determined in Ref. 28 by fitting Eq. 共5.11兲 to their numerically obtained density profiles. It was found that Anum⯝ 1/0.19
冑
6⯝2.1 and Bnum⯝ 2/1.5冑
6⯝1.7 in excellent agreement with the analytical results in Eq.共5.10兲.B. Surface tension
The surface tension is obtained by inserting the density profile in Eq. 共5.2兲 into the expression for the free energy given by Eq. 共4.19兲 关or the density profile in Eq. 共5.4兲 into Eq.共4.20兲兴. However, the expression for the free energy may be simplified by first using the Euler–Lagrange equation in Eq.共4.17兲. Multiplying both sides in Eq. 共4.17兲 by 12␦e(z),
integrating over z, and adding the result to the free energy in Eq. 共4.19兲, one finds
Fe A kBT ⫽2 kBT ⫽⫺a 2 3 e共0兲 d ⫹ a2 6 1 d␦e共0兲, 共5.12兲 so that the surface tension is given by
kBT⫽⫺ a2 6 b d ⫺ a2 12 1 d␦共0兲. 共5.13兲
The first term in this expression is the constant contribution to the surface tension that remains even when the density profile is equal to the bulk density everywhere, (z)⫽b.
To show more directly the influence of polymer adsorption on the surface tension, it is therefore convenient to subtract this constant, ⌬ kBT ⬅ kBT ⫹a 2 6 b d ⫽⫺ a2 12 1 d␦共0兲. 共5.14兲
Inserting z⫽0 in Eq. 共5.4兲 thus gives for the surface tension, ⌬ kBT ⫽⫺a 2 6 bb d2 1
冕
0 ⬁ dk ⫻冋
4 k 2⫺⫹ exp共⫺4k2/兲 4 k4⫹4 k2⫺⫹ exp共⫺4k2/兲册
. 共5.15兲 This is our final result for the surface tension valid for all values of the parameter . In Fig. 5, the rescaled surface tension, ⌬/⌬⫽0, is shown as a function of 共solid curve兲. As expected, the contribution to the surface tension due to polymer adsorption decreases when the polymer chain length becomes shorter (→⬁).FIG. 4. The coefficients A共solid curve兲 and B 共dashed curve兲, that deter-mine the exponential decay and the oscillation period of the polymer density profile, respectively关see Eq. 共5.11兲兴, as a function of ⫽2/(vbN).
FIG. 5. Reduced surface tension ⌬/⌬⫽0 as a function of
⫽2/(vbN)共solid curve兲. The dashed curve is the asymptotic expression
It is interesting to compare this expression to previous results obtained in the limit of large polymer length. In an expansion in we find for ⌬,
⌬ kBT ⫽⫺a 2 12 bb d2
冋
1⫺ 8⫺共0兲 3/2⫹O共2兲册
, 共5.16兲 with 共0兲⫽41冕
0 ⬁ dt冋
1⫺e ⫺t2 ⫺t2e⫺t2 e⫺t2⫺1⫹t2册
⬇0.154042... . 共5.17兲 For ⫽0, the ground state dominance result in the weak adsorption limit is recovered.1The asymptotic formula in Eq. 共5.16兲 is shown as the dashed curve in Fig. 5. The expansion deviates from the full expression when⬇1.VI. INTERACTION BETWEEN TWO PLANAR SURFACES
In this section, we consider the density profile and inter-action energy of two planar surfaces separated at a distance h. This calculation is of particular interest for the description of the interaction between colloidal particles mediated by a polymer solution.
A. Polymer density profile
The calculation presented here is closely related to the case of polymer adsorption onto a single planar surface. Again, it is convenient to extend the polymer density profile 共between 0⭐z⭐h) to the whole space. Here, we extend the density profile periodically关see Fig. 1共c兲兴. Instead of a single delta function, the external potential that induces such a pe-riodic profile is an infinite sum of delta functions. The Euler–Lagrange equation for the extended density profile,
␦e(z;h), thus becomes关cf. Eq. 共4.17兲兴,
1 4bN
冕
⫺⬁ ⬁ dz⬘
␦e⬙
共z⬘
;h兲␣D冉
z⫺z⬘
2RG冊
⫽v␦e共z;h兲⫺ a2 3 1 dn⫽⫺⬁兺
⬁ ␦共z⫺nh兲. 共6.1兲The total free energy is then 关cf. Eq. 共4.19兲兴, Fe关␦e兴 A kBT ⫽ ⫺1 8bN
冕
⫺⬁ ⬁ dz冕
⫺⬁ ⬁ dz⬘
␦e共z;h兲␦e⬙
共z⬘
;h兲 ⫻␣D冉
z⫺z⬘
2RG冊
⫹ v 2冕
⫺⬁ ⬁ dz关␦e共z;h兲兴2 ⫺a 2 3 1 dn⫽⫺⬁兺
⬁冕
⫺⬁ ⬁ dze共z;h兲␦共z⫺nh兲. 共6.2兲The density profile ␦e(z;h) is obtained by solving the
Euler–Lagrange equation in Eq.共6.1兲. Again, it is convenient to Fourier transform the Euler–Lagrange equation in Eq. 共6.1兲, and then solve for␦˜e(k;h). One finds
␦˜ e共k;h兲⫽ 4bb2 d n⫽⫺⬁
兺
⬁ einhk ⫻冋
k 2R G 2 ⫺1⫹exp共⫺k2R G 2兲 k4b2RG2⫹4 k2RG2⫺4⫹4 exp共⫺k2RG2兲册
. 共6.3兲 The back Fourier transform gives the segment density pro-file, ␦共z;h兲⫽bb 2 h d n⫽⫺⬁兺
⬁ cos冉
2nz h冊
⫻冋
4 kn 2 ⫺⫹ exp共⫺4kn 2 /兲 4 kn4⫹4 kn2⫺⫹ exp共⫺4kn2/兲册
, 共6.4兲 where we have defined kn⬅nb/h. This is the finalexpres-sion for the segment density profile between two plates at distance h valid for all . One may verify that at infinite separation (h→⬁), the summation reduces to an integration and Eq.共6.4兲 reduces to the previous expression in Eq. 共5.4兲.
Again, one may expand the segment density in,
␦共z;h兲⫽bb d
再
cosh共共2z⫺h兲/b兲 sinh共h/b兲 ⫺ 8冋
共h/b兲cosh共共2z⫺h兲/b兲 sinh2共h/b兲 ⫹cosh共共2z⫺h兲/b兲⫺共2z/b兲sinh共共2z⫺h兲/b兲 sinh共h/b兲册
⫺3/2冉
z RG; h RG冊
⫹O共 2兲冎
, 共6.5兲where we have defined
共x;y兲⬅4y1
兺
n⫽⫺⬁ ⬁ cos共xtn兲冋
1⫺e ⫺tn 2 ⫺tn 2e⫺t n 2 e⫺tn 2 ⫺1⫹tn 2册
, 共6.6兲 and where tn⬅2n/y . The function(x; y ) is defined suchthat it reduces to(x) defined in Eq.共5.7兲 when y→⬁. The leading term in Eq.共6.5兲 is the ground state dominance result for the polymer density profile between two planar polymer-adsorbing surfaces in the weak adsorption limit.
B. Interaction potential
We now turn to the derivation of the interaction potential between two planar surfaces. Consider again the free energy for the whole system as given by Eq.共6.2兲. Like the surface tension for the single surface, this expression may be reduced by first using the Euler–Lagrange equation in Eq.共6.1兲. One then obtains, Fe共h兲 A kBT⫽⫺ a2 6 1 dn⫽⫺⬁
兺
⬁ 关2b⫹␦e共nh;h兲兴. 共6.7兲For the free energy of the two-surface system, one therefore has
F共h兲 A kBT⫽⫺ a2 3 1 db⫺ a2 6 1 d␦共0;h兲. 共6.8兲
The interaction potential U(h) is defined as the free energy 共per unit area兲 of the two-surface system with the free energy at infinite separation subtracted,
U共h兲 kBT ⬅ F共h兲 A kBT⫺ 2 kBT⫽⫺ a2 6 1 d关␦共0;h兲⫺␦共0;⬁兲兴. 共6.9兲 The interaction potential is obtained by inserting the expres-sion for the density profile given in Eq. 共6.4兲,
U共h兲 kBT ⫽⫺ a2 6 bb d2
再
b h n⫽⫺⬁兺
⬁冋
4 kn2⫺⫹ exp共⫺4kn2/兲 4 kn4⫹4 kn2⫺⫹ exp共⫺4kn2/兲册
⫺冕
⫺⬁ ⬁ dk冋
4 k 2⫺⫹ exp共⫺4k2/兲 4 k4⫹4 k2⫺⫹ exp共⫺4k2/兲册
冎
. 共6.10兲An alternative but equivalent expression may be derived in terms of the density profile of the single surface. This expression is derived by using the fact that the density profile of the two-surface system is given by the sum of all the single-surface density profiles, ␦(z;h)⫽兺n␦(z⫹nh).
The interaction potential may therefore also be written as U共h兲 kBT ⫽⫺a 2 3 1 dn
兺
⫽1 ⬁ ␦共nh兲 ⫽⫺2 a 2 bb 3d2 n兺
⫽1 ⬁冕
0 ⬁ dk cos冉
2nhk b冊
⫻冋
4 k 2⫺⫹ exp共⫺4k2/兲 4 k4⫹4 k2⫺⫹ exp共⫺4k2/兲册
. 共6.11兲 This expression, or the equivalent expression in Eq.共6.10兲, is our final result for the interaction potential valid for all . One may expand in to compare with previous resultsU共h兲 kBT ⫽⫺a 2 bb 3 d2
冋
1 e2h/b⫺1 ⫺ 32 共2h/b兲⫹1⫺e⫺2h/b sinh2共h/b兲 ⫺3/2兺
n⫽1 ⬁ 冉
nhR G冊
⫹O共 2兲册
. 共6.12兲The leading term in Eq.共6.12兲 corresponds to the interaction potential in the ground state dominance approximation. The negative sign indicates that the interaction between two plates is attractive.18,30
The leading term and the first correction in, are expo-nentially decaying functions of h/b; the next term,
how-ever, varies as a function of h/RG. Therefore, at distances
much larger than the bulk correlation length, it becomes the dominant contribution, U共h兲 kBT ⫽ a2bb 3 d2 3/2
兺
n⫽1 ⬁ 冉
Rnh G冊
⫹O共 2兲 共hⰇ b兲. 共6.13兲 In Fig. 6,兺n(nh/RG) is plotted as a function of h/RG.关Toshow兺n(nh/RG) on a logarithmic scale the absolute value
has been taken.兴 One may identify different regimes; at in-termediate distances, bⰆhⰆRG, an approximate
expres-sion for (x) is derived by replacing the term in square brackets in Eq. 共5.7兲 with a Gaussian for small t. The sum-mation over n is subsequently approximated by an integral,
兺
n⫽1 ⬁ 冉
Rnh G冊
⬇ 1 4 n兺
⫽1 ⬁冕
0 ⬁ dt cos冉
nht RG冊
e⫺ 共t2/3兲 ⫽ ) 8冑
n兺
⫽1 ⬁ e⫺ 共3h2n2/RG2兲⬇RG 8 h. 共6.14兲 This approximation for intermediate distances is shown as the solid curve in Fig. 6. For the full interaction potential in this region we thus haveU共h兲 kBT ⫽ bb 4 4 d2N 1 h 共bⰆhⰆRG兲. 共6.15兲
The sign of this contribution is positive indicating that be-yond a certain distance the interaction becomes repulsive. This distance is of the order of the bulk correlation length.
At large separations, hⰇRG, the summation over n may
be limited to the first term only, 兺n(nh/RG)⬇(h/RG).
The asymptotic behavior of (x) was previously given in Eq. 共5.9兲 and is shown as the solid curve in Fig. 6. We thus find for the interaction potential,
FIG. 6. Numerical solution for 兩兺n⬁⫽1(nh/RG)兩 as a function of h/RG 共circles兲; the solid curves are the approximate expressions for the asymptotic
behavior in the regionsbⰆhⰆRGand hⰇRG, given by Eqs.共6.14兲 and
U共h兲 kBT ⫽⫺ 2bb 4 d2N RG C0e⫺ 共A0h/RG兲sin
冉
B0h RG ⫹␣0冊
共hⰇRGⰇb兲. 共6.16兲At these large separations, the interaction between two pla-nar surfaces is oscillatory19 and falls of exponentially with h/RG. This feature is preserved also when is not small,
U共h兲⬀e⫺ 共Ah/RG兲sin
冉
BhRG⫹␣
冊
共hⰇRG⬇b兲, 共6.17兲 with the numerical value of the coefficients A and B, that determine the exponential decay and the oscillation period of the interaction potential, given previously in Fig. 4.VII. DISCUSSION
In this article, we have investigated the surface tension, segment density profile, and interaction potential for polymer adsorbing共planar兲 surfaces. Analytical expressions for these quantities, valid for any polymer length, are derived in the limit of weakly adsorbing polymers. Particular attention is paid to the behavior of the segment density profile at large distances and the interaction potential between two polymer-adsorbing surfaces at large separations.
The relevant parameter connected to the polymer chain length is the square of the ratio of the bulk correlation length and the polymer’s radius of gyration, ⬅b2/RG2 ⫽2/(vbN). WhenⰆ1, i.e., RGⰇb, the contributions to
the segment density profile and the interaction potential may be investigated in an expansion in. The three leading con-tributions dominate the polymer density profile in three dif-ferent spatial regions: the zeroth order term, corresponding to the ground state dominance, dominates the density profile due to loops near the wall; the leading order correction of
O(1/N) due to tails gives a description at an intermediate
region of the order of the bulk correlation length; terms of
O(1/N3/2), which dictate the density profile at the region of the order of the polymer coil size RG.16,19This last term is
responsible for the共oscillatory兲 decay of the density profile and the interaction potential on the scale of the polymer’s radius of gyration.
The picture that emerges from the expansion in is the following: the polymer density profile decays exponentially as z/b close to the solid surface, ‘‘overshoots’’ at a certain
distance共typically, a fewb), and then decays exponentially
as z/RG in an oscillatory way with a period determined by
RG. This overshoot in the segment density profile was
pre-viously obtained in analytical work and computer simulations23,31and agrees quantitatively with our expression for the segment density profile in Eq.共5.4兲 共see Fig. 2兲. The oscillatory decay on the scale of RG that follows the over-shoot was previously observed28 in numerical solutions of the Scheutjens and Fleer lattice model for polymers. The parameters that describe the exponential decay and the oscil-lation period of the polymer density profile determined from the numerical profiles28 are in excellent agreement with our analytical results 关Eq. 共5.10兲兴. An interpretation of these os-cillations in terms of a liquidlike ordering of the polymer
coils is tempting, but it should be kept in mind that the poly-mer coils are still highly entangled forming a transient net-work of mesh sizeb.
A similar picture emerges for the interaction potential between two polymer-adsorbing surfaces; at close separa-tion, the interaction is attractive and decays exponentially as h/b. At a separation of typically a fewb, the interaction
becomes repulsive and then oscillates between an exponen-tially decreasing attractive/repulsive interaction on a scale determined by RG. The repulsive interaction was previously
obtained in the context of the two-order parameter model;18 whereas the presence of oscillations on the scale of RGwas previously noted by Semenov.19
We have assumed that deviations from bulk behavior are small—an assumption which holds for weak adsorption, but which also holds for strong adsorption at large distances. The description given for the distal behavior of the density profile and interaction potential is therefore also valid for strong polymer adsorption, up to a possible shift in the distance to the solid surface 共see, e.g., Fig. 2兲. Furthermore, an impor-tant benefit from the expressions derived here, is that it is also possible to investigate the distal behavior of the segment density profile and the interaction potential when is not small. These results are especially relevant when the polymer chain length is not so large or, even when N is large, when the bulk polymer concentration is low.
ACKNOWLEDGMENTS
We have benefited greatly from discussions with A. Johner and J.-F. Joanny. We are grateful to D. J. Bukman and B. Widom for help with obtaining the asymptotic behavior of the density profile. J. de Joannis kindly provided us with the MC data in Fig. 2. The Dutch Science Foundation共N.W.O.兲 is acknowledged for supplying travel funds for one of us 共J.B.A.兲.
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