Polymer adsorption onto random planar surfaces: Interplay of polymer and surface correlation

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by  Alexey Polotsky

Polymer adsorption onto random planar surfaces:
Interplay of polymer and surface correlations
5 Alexey Polotsky,∗ Friederike Schmid, and Andreas Degenhard
0
Fakult¨at fu¨r Physik, Universit¨at Bielefeld, Universit¨atsstraße 25, D-33615 Bielefeld, Germany
0
2
We study the adsorption of homogeneous or heterogeneous polymers onto heterogeneous planar

n surfaces with exponentially decaying site-site correlations, using a variational reference system ap-
a proach. As a main result, we derive simple equations for the adsorption-desorption transition line.
J Weshowthattheadsorptionthresholdisthesameforsystemswithquenchedandannealeddisorder.

5 Theresultsarediscussedwithrespecttotheirimplicationsforthephysicsofmolecularrecognition.

]
t
f I. INTRODUCTION significantly beyond the adsorptiontransition. In the bi-
o
ologicalcontext,atwo-stepadsorptionprocesswheread-
s
. Polymeradsorptionhasbeenstudiedintensivelyinthe sorption is followed by “freezing” possibly describes the
t
a lastdecadesbecauseofitswidepracticalapplicationsand physics of protein/DNA recognition, where the protein
m rich physics1,2. The early studies mostly focused on the slides on the DNA for a while before finding it’s specific
- adsorption onto physically and chemically homogeneous docking site. For other molecular recognition processes,
d such as antibody-antigen interactions, it is not only im-
surfaces. In nature, however,real objects are very rarely
n perfectly homogeneous with ideal geometry. Real sub- portant that a protein adsorbs to a given particle, but
o
equally crucial that it does not adsorb to other objects.
c strates usually contain some heterogeneity. Therefore, a
[ number of more recent studies have considered the influ- Inordertoassesstheimportanceofclustersizematching
inthosecases,onemustcalculatetheshiftofthe adsorp-
ence of random heterogeneities on the adsorption transi-
1 tion3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. tion transitionas afunction ofcorrelationlengths onthe
v
polymer and on the surface.
6 Fromamorefundamentalpointofview,theadsorption
6 of random heteropolymers (RHPs) onto heterogeneous In a previous publication27, we have calculated the
0 surfaces has attracted much attention, because RHPs adsorption/desorption transition of an ideal (non-self-
1 serveasbiomimeticpolymerstostudythephysicsofpro- interacting) RHP with sequence correlations on an im-
0 teinsandnucleicacids. StudyingtheinteractionofRHPs penetrable planar homogeneous surface. We have com-
5 with random heterogeneous surfaces can provide insight paredtwodifferentapproaches. First,wehavecalculated
0
into the physics of molecular recognition,a process play- numericallythe partitionfunctionofarandomsampleof
/
t ingakeyroleinlivingorganisms(e.g.,enzyme-substrate lattice RHPs2. Provided the sample is sufficiently large,
a
m or antigen-antibody interactions). this approachis basicallyexact. Second,wehaveapplied
The present work deals with the adsorption of ho- a reference system approach which was originally devel-
d- mopolymers and heteropolymers on surfaces with chem- oped by Chen28 and Denesyuk and Erukhimovich29 in
n ical heterogeneities. As a general rule, the introduction a similar context (RHP localization at liquid-liquid in-
o of chemical heterogeneities at constant averageload pro- terfaces). The second approach allowed to derive an ex-
c motes adsorption, since the polymers can adjust their plicit and surprisingly simple expression for the adsorp-
:
v conformations such that they touch predominantly the tion transition point (repeated here in Eq. (42)), which
Xi mostattractivesurfacesites. Theadsorptiontransitionis fitted the numerical results excellently.
thusshiftedtoalowervalueofthemeanpolymer/surface Having thus established the validity of the reference
r
a affinity. Theextentoftheshiftisdeterminedbythechar- system approach, we use it here to study the adsorption
acteristics of the randomness. In the present work, we of a single Gaussian chain onto a heterogeneous planar
focus on the influence of correlations on the adsorption surfacewithexponentiallydecayingsite-sitecorrelations.
transition. Weconsiderbothhomopolymersandrandomheteropoly-
Our study was motivated by the idea that clustering mers. As in the case of the RHP on an homogeneous
and the interplay of cluster sizes should contribute to surface, we shall derive simple analytic formulae for the
molecular recognition. This idea is supported, among adsorption/desorption transition point. This will allow
other, by a series of recent papers by Chakraborty to discuss how surface patchyness influences the adsorp-
and coworkers3,4,5,6,7,8,9,10,11. Using different analyt- tion behavior, and whether the interplay of surface and
ical approaches and Monte Carlo (MC) simulations, polymer correlationscan bring about something relevant
these authorshaveestablishedthatbelowthe adsorption to molecularrecognitionin the vicinity ofthe adsorption
threshold,statisticallyblockypolymersfavorstatistically transition.
blocky surfaces over uncorrelated surfaces. They found The problem of adsorption on heterogeneous surfaces
that, upon increasing the strength of the interactions, has already been considered theoretically by a number
the adsorptiontransition of heteropolymers on heteroge- of authors, both for homopolymers and heteropolymers
neous surfaces is followed by a second sharp transition, (block-copolymers, periodic or random copolymers).
wherethepolymersfreezeintoconformationsthatmatch Baumg¨artner and Muthukumar12 studied the effect of
the surfacepattern. This happens atadsorptionenergies chemicalandphysicalsurfaceroughnessonadsorbedho-
2
mopolymers. Using MC techniques, they calculated the Interestingapplicationsofinteractionsbetweencopoly-
adsorbed homopolymer amount on chemically heteroge- mers and heterogeneous surfaces have been examined
neous surfaces, as a function of temperature and deter- by Genzer23,24,25,26. Using a three dimensional self-
mined the shift of the adsorption-desorption transition consistent field model, he studied the adsorption of A-
temperature with decreasing concentration of adsorbing B copolymers from a mixture of A-B copolymers and
sites for different chain lengths. In addition, they pre- A homopolymers onto a heterogeneous planar substrate
sented a scaling analysis showing that lateral size of the composed of two chemically distinct regularly or ran-
adsorbed chain R depends on the distribution of re- domly distributed sites. He demonstrated that via sub-
||
pelling ”impurities” on the adsorbing surface and the straterecognition,the copolymercantranscribethe two-
strength of their repulsive interaction with the polymer. dimensional surface motif into three dimensions. The
R is unaffected by randomness and simply equal to the specific way the surface motif is translated was shown
||
radius of gyrationof a Gaussianchain in two dimensions to be strongly dictated by the copolymer sequence.
if impurities are very rare or weak. However, when the The present work differs from all these approaches in
impurities are not rare and repulsive barriers are high, that we focus on the adsorption transition, i. e., on the
R is of order of the mean distance between neighboring region in parameter space where adsorption just sets in,
||
impurities. and that we derive simple analytical expressions for the
Similar conclusions were reached by Sebastian and location of that transition. We shall see that the char-
Sumithra13,14, whodevelopedananalyticaltheoryofthe acteristic features observed in the regime of stronger ad-
adsorptionof Gaussian chains on random surfaces,using sorption do not necessarily carry over to this regime of
path integral methods combined with the replica trick weak adsorption.
and a Gaussian variational approach. They took surface The rest of the paper is organized as follows: Sec. II
heterogeneity into account by modifying de Gennes’ ad- introduces the model. The free energy calculation based
sorptionboundarycondition30,31,andanalyzedtheinflu- onthereferencesystemapproachisexplainedandcarried
ence of randomness on the conformation of the adsorbed out in Sec. III. Some aspects of the calculations in this
chains. section are verysimilar to those alreadydescribed in our
MCsimulationsofsinglepolymeradsorptionhavealso earlier publication, Ref. 27. Therefore, we will skip the
been performed by Moghaddam and Wittington15. The corresponding intermediate technical steps and refer the
sitesontheheterogeneoussurfaceweretakentobesticky reader to Ref. 27 for more details. Explicit analytic ex-
or neutral for the polymer. The temperature dependen- pressionsfortheadsorptiontransitionpointareobtained
cies of the energy and the heat capacity were calculated in Sec. IIIE. We summarize and discuss our final ex-
for different sticker fractions and compared for the cases pressions in Sec. IV. Some intermediate calculations are
of quenched and annealed surface disorder. Huber and described in three Appendices.
Vilgis16 have studied the case of physically rough sur-
faces.
Theabovementionedworksdealtwiththesingle-chain II. THE MODEL
problem. A number of other authors investigated the
adsorptionfromsolutiononto heterogeneoussurfacesus- We considersingle Gaussianpolymerschainconsisting
ing MC simulations17,18, numerical lattice self-consistent ofN monomerunitsnearanimpenetrableplanarsurface.
field (SCF) calculations19, and an analytical SCF the- The surface carries randomly distributed sites of differ-
ory20. ent chemical nature. The surface pattern is described
As mentioned earlier, the problem of heteropolymer by a locally varying interaction parameter σ(x). Here x
adsorption on heterogeneous surfaces has been studied denotes Cartesian coordinates (x,y). A particular (het-
extensively by A. Chakraborty and coworkers. Most of ero)polymer realization is described by a sequence ξ(n),
theresultsweresummarizedinarecentreviewarticle11. where n indicates the monomer number (n [0 : N]).
∈
Srebniketal. usedreplicameanfieldcalculationsandthe The strength and the sign of the monomer-surface inter-
replica symmetry breaking scheme to study the adsorp- actionisdeterminedbytheproductξ(n)σ(x)(a”+”sign
tion of Gaussian RHP chains for theta solvents3 and for refers to attraction, ”–” to repulsion).
good solvents10. MC simulations combined with simple The Hamiltonian of the system can, therefore, be rep-
non-replicacalculationsallowedtoanalyzetheroleofthe resented as follows:
statisticscharacterizingthesequenceandsurfacesitedis-
tribution, and that of intersegment interactions, for the H 3 N ∂r 2 N
= dn β dnV(z(n))ξ(n)σ(x(n)),
patternrecognitionbetweenadisorderedRHPandadis- kT 2a2 ∂n −
ordered surface4,5,6,7,8,9. Z0 (cid:18) (cid:19) Z0 (1)
TheadsorptionofglobularRHPs(e. g.,RHPsinabad wherer(n)isthechaintrajectoryinthespace,r x,z ,
≡{ }
solvent) on patterned surfaces has recently been consid- β =1/kT istheBoltzmannfactor,aistheKuhnsegment
ered by Lee and Vilgis with a variational approach21. A length of the chain, and V(z) describes the shape of the
MCsimulationstudyofsingleRHPchainadsorptiononto monomer-surface interaction potential. This potential is
random surfaces has also been performed by Moghad- taken to be a delta-function pseudopotential shifted at a
dam22. smallbutfinitedistancez fromtheimpenetrablesurface
0
3
and has the form where σ = σ is the mean load of the surface, and the
0
h i
site-site correlation function g(x ,x ) decays exponen-
1 2
δ(z z ) if z >0
V(z)= − 0 . (2) tially:
+ if z <0
(cid:26) ∞
The model Hamiltonian (1) can be used to describe g(x1,x2) (σ(x1) σ0)(σ(x2) σ0) (6)
≡ h − − i
interactions of both heteropolymers and homopolymers = g(x x )=∆2e−Γs|x1−x2|.
withbothheterogeneousandhomogeneoussurfaces. Ho- | 1− 2| s
mopolymers are described by a constant sequence func-
Here ∆ is again the variance of the single-site distri-
tion, i.e. ξ(n) ξ . The sequences ξ(n) of heteropoly- s
0
≡ bution on the surface and Γ is the inverse correlation
mers are taken to be randomly distributed according to s
length of the surface pattern. As before, g−1(x ,x )
a Gaussian distribution function 1 2
is the inverse function of g(x ,x ) with the property
1 2
P ξ(n) (3) dxg−1(x1,x)g(x,x2)=δ(x1 x2),andP σ(x) isnor-
{ }∝ malized. Homogeneoussurface−sare obtaine{din t}he limit
exp 1 Ndn dn (ξ(n ) ξ )c−1(n ,n )(ξ(n ) ξ ) , ∆R s 0, and are characterizedby σ(x) σ0.
1 2 1 0 1 2 2 0 → ≡
"−2ZZ0 − − #
where ξ = ξ characterizes the “mean charge” of the
0
chain, c(n ,nh )idescribes sequence correlations, and c−1
1 2 III. THE REFERENCE SYSTEM APPROACH
is the inverse of c, defined by dnc−1(n ,n)c(n,n ) =
1 2
δ(n n ). Here we considerexponentially decayingcor-
1 2
relati−ons, i. e., R A. Quenched Average via Replica Trick
c(n ,n ) (ξ(n ) ξ )(ξ(n ) ξ ) (4)
1 2 ≡ h 1 − 0 2 − 0 i Thefreeenergyofthesystemisgivenbythelogarithm
= c(|n2−n1|)=∆2pe−Γp|n2−n1|. of the conformational statistical sum averaged over all
realizations of the disorder βF = lnZ . Here
{σ(x),ξ(n)}
TheparameterΓpcorrespondstoaninverse“contourcor- and below angular brackets ... sthand ifor disorder aver-
relation length“ on the polymer, and ∆p gives the vari- age. To perform the quenchhediaverage we use the well-
ance of the single-monomer distribution. The propor- known replica trick
tionality factor in Eq. (3) is chosen such that P ξ(n)
{ }
is normalized. This heteropolymer model also covers ho-
Zm 1
moSpimoliylamrelyr,stihnetdhiestlrimibiutt∆iopn→of s0u.rface sites on heteroge- hlnZi{σ(x),ξ(n)} =mli→m0(cid:28) m− (cid:29){σ(x),ξ(n)} . (7)
neous surfaces is chosen to be Gaussian, with the proba-
bility
First we averageover the surface disorder σ(x). Intro-
P σ(x) (5) ducing m replicas of polymer chains and making use of
{ }∝ the properties of the Gaussian distribution, Eq. (5), we
1
exp dx dx (σ(x ) σ )g−1(x ,x )(σ(x ) σ ) , obtain
1 2 1 0 1 2 2 0
−2 − −
(cid:20) ZZ (cid:21)
m 3 m N ∂r 2
Zm = r (n) exp dn α (8)
h iσ(x) Zα=1D{ α } "−2a2 α=1Z0 (cid:18) ∂n (cid:19) #
Y X
N m 1 N
exp βσ dnξ(n) V(z (n))+ dn dn ξ(n )ξ(n )g(n ,n ) ,
0 α 1 2 1 2 1 2
× " Z0 α=1 2ZZ0 #
X
b
where α is the replica index and r(n) denotes inte- for clarity and future reference.
D{ }
grationoverallpossibletrajectoriesr(n) ofthe chain. In
R
Eq. (8) we have introduced the notation
Next we average over the sequence distribution ξ(n).
g(n ,n )= (9) This affects only the second exponential in (8). Again,
1 2
m exploitingthepropertiesofGaussianintegrals,weobtain,
β2 V(z (n ))V(z (n ))g(x (n ) x (n )) after some algebra (see appendix A)
b α 1 γ 2 α 1 − γ 2
X
4
m 3 m N ∂r 2 1
Zm = r (n) exp dn α (10)
h i{σ(x),ξ(n)} Z α=1D{ α } "−2a2 α=1Z0 (cid:18) ∂n (cid:19) # √detK
Y X
N m
exp σ ξ dn dn K−1(n ,n )β V(z (n ))
0 0 1 2 1 2 α 2
×
(cid:20) ZZ0 α=1 (cid:21)
X
ξ2 N
exp 0 dn dn dn K−1(n ,n )g(n ,n )
1 2 3 1 2 2 3
× 2
(cid:20) ZZZ0 (cid:21)
σ2 N m
exp 0 dn dn dn K−1(n ,n )bc(n n )β2 V(z (n ))V(z (n )) ,
1 2 3 1 2 3 1 α 2 γ 3
× 2 −
(cid:20) ZZZ0 α,γ=1 (cid:21)
X
where K(n ,n ) is defined as eropolymer chain adsorption on a homogeneous planar
1 2
surface, and shall be only sketched very briefly here. We
N
begin with recasting the quantity Zm from Eq. (10) in
K(n ,n )=δ(n n ) dn g(n ,n )c(n n ). (11)
1 2 1 2 3 1 3 3 2 h i
− − − the form
Z0
Eq. (10) covers the cases of hbomogeneous and hetero- Zm =Zm exp(A ) (m), (13)
geneous polymers adsorbing on homogeneous and het- h i 0 m 0
erogeneous surfaces. If at least one of the partners is whereZ isthepartitionfun(cid:2)ctionoft(cid:3)hereferencesystem,
0
homogeneous, the matrix K becomes unity, K(n1,n2)= and [ ](m) denotes the chain averagewith respect to m
δ(n1 n2),anddetK =1. Ifthepolymerishomogeneous indep·e·n·d0ent reference chains. The generalexpressionfor
−
(a homopolymer), the argument of the last exponential A is complicated, but can be obtained from Eq. (10)
m
of(10)vanishesandwerecoverEq.(8)inthespecialcase in a straightforward way. In the case of homopolymers
ξ(n) ≡ ξ0. If the surface is homogeneous, the argument adsorbed on a heterogeneous surface,the quantity Am is
of the second last exponential in (10) vanishes and we given by
recover Eq. (6) of Ref. 27.
So far the discussion has been fairly general. In the ξ2 N N m
A = 0 dn dn g(n ,n ) βv dn V(z (n)),
following, we shall mostly restrict ourselves to two spe- m 1 2 1 2 0 α
2 −
cific situations: Homopolymer adsorption on heteroge- ZZ0 Z0 αX=1
(14)
neous surfaces (c(n) 0,K(n ,n ) = δ(n n )), and b
≡ 1 2 1 − 2 and in the case of a neutral heteropolymer on a neutral
theadsorptionofoverallneutralheteropolymersonover-
surface, we get
all neutral heterogeneous surfaces (σ = ξ = 0). In the
0 0
second case, the last three exponentials in Eq. (10) can 1 N m
bedropped,andthe effectofthedisordercomesinsolely A = ln(detK) βv dn V(z (n)). (15)
m 0 α
−2 −
through the determinant detK =1. Z0 α=1
6 X
The free energy difference between the original system
and the reference system is given by:
B. Definition of the Reference System
β∆F = lim( Zm Zm)/m (16)
Toproceedwiththe calculations,weintroduce arefer- m→0 h i− 0
ence system28 with conformational and thermodynamic = lim([exp(A )](m) 1)/m.
properties reasonably close to those of the original sys- m→0 m 0 −
tem. A natural choice for the reference system is a sin- Unfortunately, the expression ([exp(A )](m) 1) cannot
gle homopolymer near an adsorbing homogeneous sur- m 0 −
becalculatedexactly. Therefore,weexpandituptolead-
face, interacting via an attractive potential of the same
ing order in powers of v , g, and c (i. e., ∆ and ∆ ). In
form V(z) as in (2), with the interaction strength w = 0 s p
0 the homopolymer case,Eq.(14), this is equivalentto ap-
σ ξ +v . The parameter v is a variational parameter
0 0 0 0 proximating exp(A) 1 A. In the heteropolymer case,
which can be adjusted such that the reference system is − ≈b
Eq. (15)) we also have to expand ln(detK):
as close as possible to the originalone. The Hamiltonian
of the reference system has the following form: ln(detK) = Tr(lnK) Tr(K 1) (17)
≈ −
H 3 N ∂r 2 N N
kT0 = 2a2 Z0 dn(cid:18)∂n(cid:19) −βw0Z0 dnV(z(n)). (12) = −ZZ0 dn1dn2g(n1,n2)c(n2−n1)
The next steps are very similar to those already de- The resulting expression for hobmopolymers on a het-
scribed in our earlier publication27 for the case of het- erogeneous surface reads
5
β∆F
βv [V(z(n))] (18)
0 0
N ≈ −
N
+β2ξ2 dk dx′dx′′g(x′ x′′)[V(z(n))δ(x(n) x′)V(z(n+k))δ(x(n+k) x′′)]
0 − − − 0
Z0 Z
β2ξ2 dx′dx′′g(x′ x′′)[V(z(n))δ(x(n) x′)] [V(z(n))δ(x(n) x′′)] N.
− 0 − − 0 − 0
Z
Thelasttermcorrespondstocontributionsfromdifferent
replicas, whereas the other two belong to single repli- r(n) exp[ βH0(r(n))]X(r(n))
[X] = D{ } − . (19)
cas. We have used the fact that averages for different 0 r(n) exp[ βH (r(n))]
R D{ } − 0
replicas factorize, since different replicated chains are
independent in the reference system. Thus [ ] denotes Likewise, we oRbtain for neutral heteropolymers on
0
the average with respect to a single refe·re·n·ce chain, neutral heterogeneous surfaces
β∆F
βv [V(z(n))] (20)
0 0
N ≈ −
N
+β2 dk c(k) dx′dx′′g(x′ x′′)[V(z(n))δ(x(n) x′)V(z(n+k))δ(x(n+k) x′′)]
0
− − −
Z0 Z
N
β2 dx′dx′′g(x′ x′′)[V(z(n))δ(x(n) x′)] [V(z(n))δ(x(n) x′′)] dk c(k)
0 0
− − − −
Z Z0
As in Eq. (18), the first two terms contain contributions G (x,x′;n)isthefreechainGaussiandistributionintwo
x
from the same replica, and the last term accounts for dimensions:
contributions from different replicas.
Atthispoint,theparameterv isstillafreevariational 3 3(x x′)2
parameterof the theory. In the0final step, we will choose Gx(x,x′;n)= 2πnaexp 2n−a2 . (26)
it such that the free energies of the reference system and (cid:20) (cid:21)
the original system are the same, i.e. ∆F =0.
For the z-dependent part of the Green’s function (25) in
the long-chain limit n 1, one obtains in the ground-
state dominance approx≫imation27,33
C. Green’s Function and Transition Point of the
Reference System
k /2
G (z,z′;n)= 0 e−ε0n. (27)
z
1+(2k k )z
To calculate the averages in (18) and (20) we need to b− 0 0
find the Green’s function of the reference system. It sat- e−kb|z−z0| e−kb(z+z0) e−kb|z′−z0| e−kb(z′+z0)
isfies the following differential equation31,32: − −
h i h i
∂G a2 where kb, determining the ground-state energy (ε0 =
−∂n =− 6 ∇2G−βw0δ(z−z0)G(r,r′;n), (21) kba2/6), is the root of the transcendental equation
with the appropriate initial and boundary conditions: 2k 6βw
0
=k with k = . (28)
1 e−2kz0 0 0 a2
G(r,r′;0) = δ(r r′), (22) −
−
G(r,r′;N)|r|→∞ = 0, (23) The complete Green’s function is calculated and dis-
|
G(r,r′;N) = 0. (24) cussed in the Appendix B.
z=0
|
Theadsorptiontransitioninthereferencesystemtakes
First, one can separate the variables x, y and z writing: place when k and ε vanish. This corresponds to the
b 0
condition27 z k = 1 or, according to the definition (28)
0 0
G(r,r′;n)=G (x,x′;n)G (z,z′;n), (25) for k , to 6βw /a2 =1/z .
x z 0 0 0
6
D. Free energy difference between the reference Eqs. (18) and (20) do no longer contain terms with con-
system and the original system tributions from different replicas. These terms, however,
aretheonlyoneswhichdistinguishbetweenquenchedand
With the results from the previous subsection, we annealed disorder. One can see that by directly carrying
can now evaluate the averages in (18) and (20). Using out an analogous calculation for annealed disorder. In
the ground-state dominance approximation (27) for the the annealed case, one has to average directly over the
partition function and calculate
Green’sfunction,weobtainthesameresultfor[V(z(n))]
0
as in Ref. 27:
βδF =ln Z lnZ =ln[exp(A )] . (31)
2k2/k annealed h i− 0 1 0
[V(z(n))] = b 0 . (29)
0
1+(2k k )z
b− 0 0 After expanding up to leading order in powers of v0, g
andc,oneobtainsthe sameexpressionsas(18)and(20),
Correspondingly,theaverages[V(z(n))δ(x(n) x′)] ap-
pearing in the last term of (18) and (20) are g−iven b0y except that the last term is missing. b
The assumption that quenched surface disorder is
1 2k2/k equivalent to annealed surface disorder has been used in
V(z(n))δ(x(n) x′) = b 0 . (30) replica meanfield calculations3 andMonte Carlosimula-
− 0 S 1+(2k k )z
(cid:20) b− 0 0(cid:21) tions5, based on the argument that the chain “samples
(cid:2) (cid:3)
where S is the surface area. The additional factor 1/S all significant surface patterns with a probability that
in (30), compared to (29), results from the translational asymptotically approachesthat ofan annealedsystem in
freedom of the polymer in the surface plane. Substitut- the adiabaticlimit” (cited fromRef. 5). Here we recover
ing this result into the last term of (18) and (20) and this equivalence in the limit of infinite surface area S.
performingtheintegrationover dx′dx′′ givesthefactor We turn to the calculation of
1/S. Since S , this contribution is infinitesimally [δ(x(n+k) x′)V(z(n+k))δ(x(n) x′′)V(z(n))] .
0
→ ∞ R − −
small and can be omitted in the following calculations. We assume the validity of the ground-state dominance
This seemingly “technical” point has an important approximation for the overall chain and for the side
physical interpretation. After dropping the last term, subchains. This gives
[δ(x(n+k) x′)V(z(n+k))δ(x(n) x′′)V(z(n))] (32)
0
− −
dr dr dr dr G(r ,r ;n)δ(r x′ )G(r ,r ;k)δ(r x′′ )G(r ,r ;N n k)
= 1 2 3 4 1 2 2− z0 2 3 3− z0 3 4 − −
dr dr dr dr G(r ,r ;n)G(r ,r ;k)G(r ,r ;N n k)
R 1 2 3 4 1 (cid:0)2 (cid:1) 2 3 3 (cid:0)4 (cid:1)
− −
1 2k2/k
= G (x′,x′′;Rk)G (z ,z ;k)e−|ε0|k b 0
x z 0 0
S 1+(2k k )z
b 0 0
−
Weconsiderfirstthehomopolymercase. Inserting(29) at z = z′ = z (i. e., at the level where the poten-
0
– (32) into (18), we obtain tial acts on the chain). The procedure of calculating
(x,z ,x′,z ; ε ) is briefly described in the Appendix
0 0 0
β∆F = 2kb2/k0 (33) GB. The result| ca|n be written in the following integral
N 1+(2k k )z form
b 0 0
−
β2ξ02 dxdx′ g(x x′) (x,z0,x′,z0, ε0 ) βv0 . (x,z ,x′,z ; ε )= 6 1 dqe−iq(x−x′) (35)
(cid:20) S Z − G | | |− (cid:21) G 0 0 | 0| a2(2π)2
Z
Here sinh z q2+k2
0 b
(r,r′,p) = Ndke−pkG(r,r′;k) (34) × q2+kb2ez0√q2+(cid:16)kb2 −pk0sinh (cid:17)z0 q2+kb2
G
Z0 p (cid:16) p (cid:17)
∞ where q is the two-dimensional vector and q q. In-
dke−pkG(r,r′;k) ≡ | |
≃ serting (35) into (33) and changing the order of integra-
Z0 tion over q and x yields the following expression for the
is the Laplace transformofthe ”complete” Green’s func- free energy difference between the reference system and
tionG(r,r′;k). Weareparticularlyinterestedinitsvalue a homopolymer adsorbed on a heterogeneous surface:
7
β∆F = 2kb2/k0 β2∆2sξ02 ∞dq Γsq 6sinh z0 q2+kb2 βv (36)
N 1+(2kb−k0)z0  a2 Z0 (q2+Γ2s)3/2 q2+kb2ez0√q2+(cid:16)kb2 −pk0sinh(z(cid:17)0 q2+kb2) − 0
 (cid:16)p p (cid:17) 
The second case of interest is the neutral heteropoly- monomer correlation function (4). Inserting expressions
meronthe neutralheterogeneoussurface. Thedifference (29 - 32) into (20), we obtain
between the Eqs. (18) and (20) is due to the monomer-
β∆F 2k2/k β2∆2
= b 0 p dxdx′ g(x x′) (x,z ,x′,z ,Γ + ε ) βv . (37)
0 0 p 0 0
N 1+(2kb−k0)z0 " S Z − G | | − #
This finally leads to
β∆F 2k2/k β2∆2∆2 ∞ Γ q 6sinh(z ω(q))
= b 0 s p dq s 0 βv , (38)
N 1+(2kb−k0)z0 " a2 Z0 (q2+Γ2s)3/2 ω(q)ez0ω(q)−k0sinh(z0ω(q)) − 0#
(cid:0) (cid:1)
with ω(q) q2+6Γ /a2+k2. Note that (36) can isdesorbedinboththeoriginalandthereferencesystem.
≡ p b
formally be obtained from (38) by setting ∆ = 1 and Fortheother,nontrivial,solutionweagainconsiderboth
p
p
Γ 0. This is, however, not a general rule. The cor- cases separately.
p
→
responding expression for a heteropolymer adsorbed on Inthehomopolymercase,thenontrivialsolutionforv
0
a homogeneous surface, Eq. (41) in Ref. 27 is not fully isgivenbysettingtheterminsquarebracketsinEq.(36)
contained in Eq. (38)34 tozero. Thetransitionpointforthereferencesystemcor-
responds to the condition 27 z k = 1 and k 0. Sub-
0 0 b
→
stituting these conditions and introducing, for the sake
E. Transition point of convenience, the dimensionless variables
Havingobtainedanapproximateexpressionforthedif- βv βσ β∆
0 0 s
v = , σ = , ∆ = ,
ference ∆F between the free energies of the original and 0 a 0 a s a
the reference systems, we adjust the auxiliary parameter
v suchthatthefreeenergyofthereferencesystemisthe e f f
0
best approximationfor that ofthe originalone. Thus we z
0
require ∆F(v∗) = 0. For the ∆F given by (36) or (38), Γs =aΓs, and z0 = (39)
0 a
this equation has one trivial solution k = 0 or w = 0, we obtain an equation for the transition point σ ξ )tr in
b 0 0 0
corresponding to the uninteresting case of a chain which the random systfem: e
f
(σ ξ )tr (σ ξ )tr = βv = 6∆ 2ξ2 ∞dq Γs sinh(z0q) , (40)
0 0 − 0 0 homo − 0 − s 0Z0 q2+Γ 2 3/2 (ez0q−sinh(z0q)/(z0q))
fs e
f f e f (cid:16) (cid:17) e
e e
f
where(σ ξ )tr 1/(6z ) is the transitionpointfor the We turn to the case of a neutral heteropolymer on a
0 0 homo ≡ 0
homopolymer near an attractive surface. Eq. (40) de- neutral heterogeneous surface. Here, the nontrivial so-
termines the relation between σ, Γ , and ∆ that corre- lution for v , is obtained by setting the term in square
f e s s 0
spondtotheadsorption-desorptiontransitionintheorig- brackets in Eq. (38) to zero. We insert again the con-
inal heterogeneous system. ditions for the transition point in the reference system,
8
k = 0 and k z = 1, and switch to the dimensionless dimensionless. Since σ = ξ = 0, the equation for the
b 0 0 0 0
variables according to Eq. (39). The parameters Γ transition point then reads:
p
and ∆ associated with the RHP statistics are already
p f
2
∞ 6 sinh z q2+6Γ Γ q∆ ∆2
βv =(σ ξ )tr = dq 0 p s s p . (41)
0 0 0 homo Z0 q2+6Γp ez0√q2+6(cid:0)Γep −psinh z0 (cid:1)q2+6Γp /z0 fq2+fΓs2 3/2
e f (cid:16)p e (cid:0) p (cid:1) (cid:17) (cid:16) (cid:17)
e e f
IV. SUMMARY AND DISCUSSION In order to complete this compilation of results, we
also give the general expression for the shift of the ad-
We have used a reference system approach to calcu- sorption transition in systems of heterogenous on het-
late the adsorption/desorption transition of polymers at erogeneous surfaces (not necessarily neutral). It con-
surfacesinsystemswithspatiallycorrelatedrandomness. tains the sum of the three contributions (42), (43), (44),
Theresultingexpressionsbecomeveryelegantinthelimit but has one extra term from the expansion of K−1 =
z0 ≪1,i. e., if the rangeof the surfacepotential is short ∞k=0( 0Ndn1 g(n1,n2) c(n2 − n1))k in (10). The cal-
compared to the statistical segment length of the poly- culation is absolutely analogous to the one presented in
P R
mer. We shall now compile these results, contrast them Sec. (III) and yields
e b
withthecorrespondingresultinourearlierpaper27,and
discuss them. Q(tr) Q(tr) = ∆2σ 2 6 ∆ 2ξ2 6 (45)
We briefly recall the meaning of the different param- − Homo − p 0 sΓp − s 0Γs
eters. ξ denotes the mean polymer charge per segment
and σ 0the mean surface charge per site; thus the mean ∆ 2f∆2C+ (f∆ 4,∆4)
0 − s p O s fp
affinity between polymer segments and surface sites is
2
givenfby the parameter Q = (σ0ξ0). Randomness shifts where the coefficient C of fthe higher ordefr term ∆s ∆2p
the critical affinity parameter at the adsorption transi- is given by
tionQ(tr) tolowervalues,compfaredtothecorresponding f
valueQ(tr) ofahomogeneoussystem. Theshiftdepends 6 1 1 1
Homo C = 72σ ξ +
0 0
on the variances ∆p and ∆s of the polymer and surface Γs+ 6Γp − ( 6Γp Γs Γs+ 6Γp
(cid:16) (cid:17)
disorder, and on the cluster parameters or inverse corre-
2
p 1 fΓ 1p1 p
lation lengths Γp and Γs. f f + G22 s ,f f (46)
The case of random heteropolymers on homogeneous πΓ 2 22 6Γp 21 32 )
surfaces was already cfonsidered in our earlier paper27. s (cid:16)f (cid:12)(cid:12) (cid:17)
with the Meijer G function(cid:12)G. The calculation of C is
The transition in the random system is shifted with re-
f
shortlysketchedintheAppendixC. NotethatEq.(45)is
spect to the homogeneous system by
an implicit equation for Q(tr), because the affinity Q =
6 (σ ξ ) depends both on the mean charge per segment ξ
Q(tr) Q(tr) = ∆2σ 2 . (42) 0 0 0
− Homo − p 0 sΓp of the polymer and the mean charge σ0 per site on the
surface.
f
Thecaseofhomopolymersonrandfomsurfacesiscovered The validity of the approximation z 1 is
f 0 ≪
by Eq. (40). In the limit z 1, the integral in (40) can demonstrated in Fig. 1, which shows some adsorption-
0
≪
be calculated explicitly and the solution takes the form desorption transition curves for homopolymers adsorbed
e
on heterogeneous surfaces at different values of z , as
Q(tr)−QH(torm)o =−∆s2ξ02Γ6 . (43) calculated from Eq. (40). As a function of the cor0rela-
s tion length 1/Γ , the curves become straight lines with
s e
Heterogeneous polymers on heterfogeneous surfaces were increasing 1/Γ with the same slope as the asymptotic
discussed in the special case of zero mfean affinity. The curve, Eq. (43). The slope of the curves deviates from
result, given by Eq. (41), simplifies considerably in the the asymptotic value only at small correlation lengths.
limit z 1. The heteropolymer is adsorbed, if Thesimplifiedexpression(43)approximatesthefullsolu-
0
≪
tion (40) reasonably for z <1. Thus the approximation
2 0
6 ∆s ∆2p Qtr =const. (44) is justified if the range of the potential is shorter than
Γ + 6Γ ≥ Homo the statistical segment leength. At first sight, this con-
s p
f dition seems somewhat arbitrary,given that the value of
This defines a phasepboundary in the parameterspace of thestatisticalsegmentlengthadependsontheparticular
f
variances ∆ and ∆ and cluster parameters Γ and Γ . choice of the “segment” unit. One can always increase a
p s p s
f f
9
pelling, and neutral sites and is kept neutral on average,
it is preferable to have strongly repelling and strongly
adsorbing sites rather than weakly repelling and weakly
adsorbing - ”almost neutral” - sites.
Wenoteanotherinterestingpropertyoftheconsidered
model. InSec.IIID,wehavealreadydiscussedtheequiv-
alence of quenched and annealed surface disorder. The
tr tr argument was based on the fact that all terms related to
(Q - Q )
Homo different replicas in Eqs. (18) and (20) contain an extra
factor 1/S, which vanishes in the limit of infinite surface
area S. Technically, the factor 1/S reflects the loss of
translational freedom of a replica, if it is restricted to
stay close to another replica by a coupling through the
correlationfunctiong(x x′). Theargumentisthusonly
−
valid for surface disorder. In general, quenched and an-
nealed disorder on the polymer are not equivalent.
At the transition point, however, terms related to dif-
ferent replicas always vanish, because every indepen-
dent replica also contributes a factor 2k2/k /(1+(2k
b 0 b−
FIG. 1: The shift (Qtr − QtHromo)/∆s2 of the adsorption- kp0re)zd0ic)ts∼th2aktb.theInadotshoreprtiwoonrdths,reosuhroldmoisdetlhecaslcaumlaetifoonr
desorption transition point for a random surface with expo-
quenchedandannealedsurfacedisorder. Webelieve that
nentiallydecayingsequencecorrelationfscomparedtothatofa
this is a general feature of the adsorption transition in
homogeneous attracting surface, vs. correlation length 1/Γ ,
s
disordered systems. Joanny finds the same equivalence
according to Eq. (40). The different curves show the results
for different values of z0 as indicated. The bold line corfre- when studying the adsorption of polyampholytes on sur-
spondsto z =0, Eq. (43). faces35withintheHartreeapproximation. Heclaimsthat
0
ithasbeenconjecturedforsimilarproblemsbyNieuwen-
e
e heuizen36. Indeed, polymers and surfaces have only very
few contacts at the adsorption transition. It is conceiv-
by choosing a more coarse grained polymer description,
ablethatthecontactscanbearrangedinanoptimalway
i.e.,combiningseveralsegmentstooneneweffectiveseg-
regardless of the particular type of disorder.
ment. Thereallengthscalesetbythenon-adsorbedpoly-
If this is true, it has obvious implications for the phe-
merisitsradiusofgyration. Applyingtheapproximation
nomenon ofmolecular recognition: It means that studies
z 1 in the calculation of the adsorption transition is
0
≪ oftheadsorptiontransitioninsystemswithquencheddis-
thusjustifiedwhenevertherangeofthepotentialismuch
order, averaged with general, unspecific site distribution
shorter than the radius of gyration of the polymer.
e functions such as Eqs. (3) or (5), can only provide very
Our results can be summarized as follows: The pres-
limited information about specific recognition processes.
enceofcorrelationsalwaysenhancesadsorption. Bothin-
The adsorptionthreshold of an annealed polymer, which
creasingthe correlationlengths,i. e., decreasingin Γ or
p hashadtheopportunitytoadaptitssequencetothesur-
Γ , and increasingthe strengthof the correlations(given
s face, is absolutely equivalent to that of a completely un-
by ∆ , ∆ ) results in a shift of the adsorption transition
p s adapted (quenched) polymer.
towards lower affinities.
We can also use our results to discuss the issue of the
Theresultsobtainedforhomopolymersonrandomsur- interplaybetweenclustersizesonthe surfaceandtheho-
faces (43) and those for heteropolymers on homogeneous mopolymer. Fig. 2 shows an example of an adsorption
surfaces (42) are very similar. The only difference are phase diagram, obtained from Eq. (44), for neutral het-
the exponents of Γs and Γp, which reflect the different eropolymersonneutralsurfacesasafunctionoftheclus-
”dimensions” of a planar surface and a linear polymer
ter parametersΓ and Γ . The phase boundary does not
p s
chain.
reflectparticularaffinitiesbetweencertainclusterparam-
The final result (44) for the adsorption of neutral het-
etersΓ onthepolymerfandΓ onthesurface. Regardless
p s
eropolymersonneutralheterogeneoussurfacesgivessome
ofthevalueofΓ ,theadsorptionisalwayshighestatlow
p
particularly interesting information. First, it is worth
Γ andviceversa. Wedonoftobserveanycomplementar-
noting that in Eq. (44), both Γ and Γ enter in the de- s
s p ity effects.
nominator with their ”characteristic” exponents (1 and
fTo summarize,we havebeen able to derive simple and
1/2, respectively). The other notable feature is that ∆
s elegantequationsfortheadsorptionthresholdinsystems
and ∆ enter in (44) as the product of their squares
p ofpolymersonsurfaceswithcorrelatedquenchedandan-
2
∆s ∆2p – a similar dependence on the varianceshas been nealed disorder. These may be useful for future studies.
observed by Srebnik et al. 3 for the case of short-range Wehavediscussedourresultsinthecontextofmolecular
cforrelated RHPs at a random surface. This means that recognition and found that the complex process of pat-
if the surface (heteropolymer) contains attractive, re- tern recognition between molecules and surfaces close to
10
0.1 and (3) and calculate
ξ(n) exp(ξσ¯+ 1ξgξ)exp( 1(ξ ξ¯)c−1(ξ ξ¯))
I = D{ } 2 −2 − −
ξ(n) exp( 1ξc−1ξ)
R D{ } −2
Γs−1 Adsorbed = exp 1ξ¯c−1ξ¯R+ 1(σ¯+bξ¯c−1)[c−1 g]−1(σ¯+c−1ξ¯)
−2 2 −
(cid:20) (cid:21)
ξ(n) e−21ξ[c−1−g]ξ b
D{ } (A1)
×R ξ(n) e−12ξc−1ξb
D{ }
Desorbed
The denoRminator corresponds to the normalization fac-
0
0 Γ −1 1
tor in Eq. (3), and the expression ξ(n) denotes
p D{ }
functional integration in sequence space. To sim-
R
plify the first exponential on the right hand side of
Eq. (A1), we define K = 1 gc (Eq. (11)) and
FIG. 2: Adsorption transition of neutral heteropolymers on replace [c−1 g]−1 = cK−1 th−us obtain the expres-
neutral surfaces as a function of Γp and Γs. The parameters sion exp ξ¯K−−1gξ¯/2+σ¯cK−1σ¯/2+ξ¯K−1σ¯ . This cor-
are ∆ =∆ =1 and Q(tr) =1. b
s p Homo responds to the last three exponentials in Eq. (10).
f (cid:2) b (cid:3)
To calculate the integrals over sequences ξ(n) in
f b { }
Eq. (A1), we set n back to be a discrete variable,
the adsorption threshold cannot be understood in terms ∞ ∞
ξ(n) = dξ dξ . This gives
of a simple model with quenched or annealed disorder. D{ } 0 1··· 0 N
Weconcludethatrecognitioneithertakesplaceathigher R ξR(n) e−21ξ[Rc−1−g]ξ det(c−1)
affinities, beyond the adsorption threshold, as suggested D{ } = (A2)
by Chakraborty et al. 11, or has to be studied by more R D{ξ(n)}e−21ξc−1ξb sdet(c−1−g)
sophisticatedmodels and methods. For example,the sit- 1 1
R = = .
uation might change if the probability distribution itself det[(c−1 g)c] det(K)b
reflects the pattern on the surface. Such considerations −
shall be the subject of future study. The final result, wphich enters Eq. (10)p, thus reads
b
1 1 1
I = exp ξ¯K−1gξ¯+ σ¯cK−1σ¯+ξ¯K−1σ¯ .
Acknowlegdement √detK 2 2
(cid:20) (cid:21)
(A3)
In matrix notation, it is ebasy to see that
The financial support of the Deutsche Forschungsge-
meinschaft (SFB 613) is gratefully acknowledged. A.P.
ln(det(K)) = ln(det(TKT−1)) (A4)
thanks RFBR (grant 02-03-33127).
= ln( λ )= ln(λ )=Tr(lnK),
n n
n n
Y X
APPENDIX A: PERFORMING THE SEQUENCE
whereT diagonalizesK,andλ aretheeigenvaluesofK.
AVERAGE n
We want to calculate surface disorder average in Eq APPENDIX B: COMPLETE GREEN’S
(8). Let us define, for convenience, FUNCTION
m
Our starting point is the equation(21) with the initial
σ¯(n)=βσ V(z (n)), ξ¯(n) ξ
0 α ≡ 0 condition (22) and the boundary conditions ((23) and
α=1
X (24)): Laplacetransformingwithrespecttonandtaking
into account the initial condition (22), we obtain
and adopt the matrix notation
p (x,z,x′,z′;p)+δ(x x′)δ(z z′) (B1)
N
− G − −
ξGξ :=ZZ0 dn1dn2ξ(n1) =−a62∇2G−βw0δ(z−z0)G(x,z,x′,z′;p),
where (x,z,x′,z′;p) is the Laplace transform of
N G(x,z,xG′,z′;n)withrespecttonwiththeboundarycon-
G(n ,n )ξ(n ) ξv := dnξ(n)v(n).
1 2 2
ditions
Z0
Thematricesc,c−1,andgaresymmetric. Inordertocal- (x,z,x′,z′;p)|x|,z→∞ =0,
G | (B2)
culatethe sequenceaverage,weneedtocombineEqs.(8) (x,z,x′,z′;p) =0
z→0
G |
b