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Dependence of the Critical Adsorption Point on Surface and 
Sequence Disorders for Self-Avoiding Walks Interacting with a 
Planar Surface  
Jesse D. Ziebarth1,  Yongmei Wang1*, Alexey Polotsky2, Mengbo Luo3 
1Department of Chemistry, the University of Memphis, Memphis, Tennessee 
38152, USA 
2 Sérvice de Physique de l'Etat Condensé CEA Saclay, 91191 Gif-sur-Yvette 
Cedex, France. 
3Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China  
 Abstract. The critical adsorption point (CAP) of self-avoiding walks (SAW) interacting 
with a planar surface with surface disorder or sequence disorder has been studied. We present 
theoretical equations, based on ones previously developed by Soteros and Whittington (J. Phys. 
A.: Math. Gen. 2004, 37, R279-R325), that describe the dependence of CAP on the disorders 
along with Monte Carlo simulation data that are in agreement with the equations. We also show 
simulation results that deviate from the equations when the approximations used in the theory 
break down. Such knowledge is the first step toward understanding the correlation of surface 
disorder and sequence disorder during polymer adsorption. 
*Corresponding author: email: ywang@memphis.edu. Tel: 901-678-2629. Fax: 901-678-
3447.
1. Introduction 
Adsorption of polymers on surfaces plays a key role in many technological applications 
and is also relevant to many biological processes. As a result, it has been studied for more than 
three decades1 and continues to receive intense interest.2 The field is rich and contains a wide 
variety of topics, from equilibrium properties of adsorbed layers and conformations of adsorbed 
polymer chains to dynamic properties and non-equilibrium processes in adsorption.2 For polymer 
adsorption on planar surfaces, it is well-known that there exists a critical adsorption point (CAP) 
that marks the transition of a polymer chain, in contact with a surface, from a non-adsorbed state 
to an adsorbed state.3 Scaling laws for a variety of quantities below, above and at the CAP for a 
homopolymer in contact with a planar surface were developed by Eisenriegler, Kremer, and 
Binder (EKB).4 For example, when the chain goes from a non-adsorbed state to an adsorbed 
state, the energy of the chain E changes from an intensive variable independent of chain length N 
to an extensive variable dependent on N. At the CAP, E is expected to scale with Nφ where φ is 
the crossover exponent. Numerical studies, including exact enumeration,5 the scanning method6,7 
and the multiple Markov chain method8 have been performed to determine the location of the 
CAP and the crossover exponent φ. The values reported are however not completely in 
agreement with each other and are still under debate, especially the crossover exponent φ. The 
disagreement may be traced, as suggested by a recent article,9 to different methods used for 
determining the CAP and the crossover exponent φ.  
While many studies focused on adsorption of homopolymers on planar homogeneous 
surfaces, adsorption of polymers on chemically or physically heterogeneous surfaces has also 
received a fair amount of studies.10-20 Some were inspired by specific applications such as 
segregation of polymer chains on patterned surfaces,10 or pattern transfer via surface 
adsorption,21,22 others were motivated by a desire to understand how the presence of surface or 
sequence disorders may influence adsorption.13,14,16,17,23-25 For example, Sebastian and Sumithra 
developed an analytical theory of the adsorption of Gaussian chains on random surfaces using 
Gaussian variational approach.24,25 They took surface heterogeneity into account by modifying 
de Genne’s adsorption boundary condition and analyzed influence of randomness on the 
conformation of the adsorbed chains. Adsorption of heteropolymers on heterogeneous surfaces, 
in particular, has been studied because of its relevance to molecular recognition in biological 
process. The concept of “pattern matching” was proposed26 and has been investigated with 
different approaches.12,20,26,27 Muthukumar for example derived an equation for the critical 
condition of adsorption of a polyelectrolyte to an oppositely charged patterned surface.26 
Golumbfski et al.12 showed that a statistical blocky chain was selectively adsorbed on a patchy 
surface while a statistically alternating chain was selectively adsorbed on an alternating surface. 
Jayaraman et al.19 described a simulation method to design surfaces for recognizing specific 
monomer sequences in heteropolymers. Recently Polotsky et al18 considered adsorption of 
Gaussian heteropolymer chains onto heterogeneous surface. They found that the presence of 
correlations between sequence and surface heterogeneity always enhances adsorption. However, 
the dependence of the critical adsorption point on either surface disorder or sequence disorder is 
not well-understood. Lack of this knowledge hampers further understanding on the correlation 
between sequence disorder and surface disorder during adsorption.  
Here we present theoretical equations that describe the dependence of CAP on the surface 
disorder or sequence disorder, along with Monte Carlo simulation data in agreement with the 
derived equations. The current study does not address the correlation between sequence disorder 
and surface disorder. We only consider cases where the disorder is either present randomly on 
the surface (i.e. adsorption of homopolymers on random heterogeneous surface) or on the 
sequence (i.e., adsorption of random copolymer on homogeneous surface). The correlation 
between sequence disorder and surface disorder will be the subject of future publications. In the 
following, we first present the theory that predicts the dependence of CAP on surface disorder 
and sequence disorder. Then we present details of Monte Carlo simulation methods used to 
determine the CAP, followed by simulation data that agree with the derived equations. Finally, 
we discuss implications of these results on practical applications such as chromatographic 
separations of polymers.  
2. Theory 
2.1 Adsorption of a homopolymer on a homogeneous surface 
We first consider adsorption of a homopolymer chain on a homogeneous surface. This 
can be represented by a self-avoiding walk (SAW) in a three-dimensional lattice interacting with 
a plane and restricted to lie on one side of the plane. The vertices of the walks interact with the 
surface sites with an attractive energy εw. The partition function for a N-step SAW interacting 
with a homogeneous surface is given by 
( )∑=
wNw vvcNZ εε exp)(),(homo  (1) 
where cN(v) is the number of SAWs that lie above the surface with v visits to the surface. 
Hammersley et al.28 have shown that the model exhibits a phase transition at a critical adsorption 
energy, εc, with a desorbed state for εw  < εc, and an adsorbed state for εw  > εc. They have shown 
that the limiting monomer free energy f(εw) 
),(log
lim)( homo wNw NZN
=  (2) 
exists and is a convex non-decreasing continuous function of εw. Moreover, f(εw)=κ  for εw ≤ 0, 
where κ is the lattice connective constant, and f(εw) is a strictly increasing function of εw when 
εw  > εc. Therefore, f(εw) is non-analytic at εw = εc. εc has also been determined to be greater than 
zero and, based on the best-known connective constant for the simple cubic lattice29, to have an 
upper bound of 0.5738. The lattice connective constant κ is also the limiting monomer free 
energy of the SAWs in bulk solution. Hence the CAP can be understood as the condition where 
the limiting monomer free energy of a chain attached to the surface becomes equal to the limiting 
monomer free energy of the chain in the bulk solution.  
2.2 Adsorption of a homopolymer on a random heterogeneous surface 
Now we consider the adsorption of a homopolymer interact with a heterogeneous surface 
consisting of two types of surface sites, A and B. The interaction energy of the vertices with the 
two surface sites are εwA and εwB. Following Soteros and Whittington23, and express the partition 
function of a N-step SAW interacting with a heterogeneous surface that consists of A and B 
surface sites as: 
( ) ( )∑ ∑
ANBAhet BvfAvfAv
vcffZ
)()( )((exp)(exp
)(),( εε            (3) 
where cN(v) is the number of walks that have v surface contacts, v(A) is the number of monomers 
interacting with the A sites, and v(B) is the number of monomers interacting with the B sites, fA 
and fB = 1 - fA are the fractions of A and B sites on the surface, respectively.  Here the partition 
function is averaged over random distributions of the surface sites, i.e. the so called annealed 
approximation. Physically the annealed disorder means that the type of surface sites may change 
while the system attains equilibrium state. However, it has been previously suggested11,15 that the 
annealed approximation is valid if the chain can visit a large area of the surface and hence 
samples all distributions of surface patterns. Furthermore, the surface sites are randomly 
distributed. If there is a correlation between surface disorders, such as those present in patchy 
surface or alternating surface, then Eq. (3) will not be valid, as Eq. (3) gives equal weight to all 
possible surface labelings, while correlations restrict possible labelings. Summing over v(A), 
equation (3) can be simplified to 
( )∑ +=
wANBAhet ffvcffZ )exp()exp()(),( εε  (4) 
A comparison of equations (1) and (4) reveals that the partition functions for homogeneous and 
annealed random heterogeneous surface become equivalent if    
( ) ( ) )exp(expexp BwBAwAw ff εεε +=  (5) 
From Eq. (5), we derive the following equation that gives the dependence of CAP on the surface 
disorder: 
( ) ( ) ))(exp(exp)1()(exp ccffcc BwBAwBhw εεε +−=  (6) 
 where εwh(cc) is the CAP of a homopolymer above a homogeneous surface, εwB(cc) is the CAP 
of a homopolymer above a heterogeneous surface while the surface interaction energy εwA held 
constant. It can be easily seen from this equation, that the dependence of the CAP on the 
percentage of attractive sites on the surface is not expected to be linear, in contrast to the 
conclusion drawn by an earlier study.13 Equation (6) is expected to be valid as long as the two 
conditions are met: (i) the chain has enough mobility to visit a large area of surface so that the 
annealed approximation is valid, and (ii) the surface sites are randomly distributed (i.e. 
uncorrelated).  
2.3 Adsorption of a random heteropolymer on a homogeneous surface 
The same approach can be extended to consider the adsorption of a random 
heteropolymer interacting with a homogeneous surface.  We will use the same notation as in 
previous section except now fA and fB represent fractions of A and B monomers present on the 
heteropolymer. We will only consider random copolymers composed by A and B monomers. 
The sequence of a random copolymer can be represented by χ ={χ1, χ2, … χN} where χi are 
independently and identically distributed random variables with χi =A with a probability of fA 
and χi=B with a probability of 1-fA.  A sequence order parameter λ can be defined to characterize 
the sequence randomness.12,27  
BAAB pp −−= 1λ  (7) 
where pij is the nearest neighbor transition probabilities which is the probability that a monomer 
of type i is followed by a monomer of type j. When λ=0, the sequence is random. When 
λ>0, then the sequence is statistically blocky, and when λ<0, the sequence is statistically 
alternating. We note that a given random sequence designated by χ may have non-zero values of 
λ. More discussions will be given in the later section. 
The partition function of N-step SAWs with the given sequence above a homogenous 
surface is written as: 
)exp()|,(),,( BwB
BANBAhetpoly vvvvCffZ εεχχ += ∑  (8) 
There are two different ways to average over different distributions of random sequences, 
namely the annealed average and the quenched average. With the annealed average, the partition 
function in Eq. (8) is first averaged over different distributions of χ. This then leads to a partition 
function, Zhetpoly(fA, fB), which is exactly the same as in Eq. (3). With the annealed 
approximation, we derive the same equation as given by Eq. (6) for the CAP of a random 
heteropolymer interacting with a homogeneous surface, provided that fA and fB now represent the 
fractions of A and B monomers on the chain.  
In the following, we will present Monte Carlo simulation data that conform to the two 
equations and also results that do not conform to the equations because of the invalidation of the 
approximations used in deriving the equations.  
3. Monte Carlo Simulation Methods 
In our simulations, polymer chains are modeled as SAWs with N vertices on a simple 
cubic lattice of dimensions 250a × 250a × 100a, where a is the lattice spacing.  Each vertex 
represents a monomer on the polymer chain. Chain lengths studied are in the range of N = 25 to 
250.  There is an impenetrable wall in the z = a plane representing the surface.  One monomer, 
picked randomly from the chain, is first placed on a site adjacent to the wall (in the z = 2a plane). 
The rest of the chain is then grown using the biased chain insertion method.30 Monomers that are 
in the z = 2a plane are considered to be adsorbed on the surface.  For all adsorbed monomers, an 
attractive polymer-surface interaction, εw, is applied.  The standard chemical potential of the 
chain (since it does not contain translation entropy), µ0, is calculated from the Rosenbluth-
Rosenbluth weighting factor, W(N), which is given by30  
0 ln)(lnβµ  and 
)exp( β
 (9) 
where z is the lattice coordination number (z = 6 for simple cubic lattice), Ej is the energy of ith 
inserted monomer in the jth potential direction. We note that µ0 calculated is the free energy per 
chain, and µ0/N is free energy per monomer discussed in equation (2). Typically, the chemical 
potential is determined based on about twenty million copies of trial chain conformations. 
We obtained the standard chemical potentials of a chain with at least one monomer 
attached to the surface, µads0, and compared that against a chain grown in a bulk solution, µbulk0. 
The bulk solution is modeled by a 100a × 100a × 100a lattice with periodic boundary conditions 
applied in all three directions.  All chemical potentials calculated are reduced by the Boltzmann 
factor, β=1/kBT=1. A coefficient K, similar to partition coefficient if the chain was placed in a 
pore instead of near a surface, is calculated by K =exp(-∆µ0), where ∆µ0 = µads0 − µbulk0. The way 
we determined the CAP is based on the dependence of K on the chain length N and will be 
presented in the results section.  
Heterogeneous surfaces were modeled by making the z = a plane composed of two 
different types of sites, which have different values for polymer-surface interactions. The 
designations εwA and εwB will be used to distinguish between interaction energies of different site 
types.  Simulations were performed using surfaces with different fractions of A and B sites.  
Surfaces were created by randomly assigning each site as A or B based on the probabilities, pA 
and pB, where pA and pB are, respectively, the desired fractions of A and B sites on the surface.  
Because of size of the surface, this procedure resulted in the real surface composition 
percentages matching the desired percentages within 0.1%.  For a given surface composition, the 
surface was randomly created once and was subsequently used in all simulations that determine 
the chemical potential of a chain above that surface. The surfaces displayed quenched 
randomness, i.e. the surface pattern remained unchanged throughout the simulations.  However, 
the first bead of chain was placed randomly over the surface during the chain insertion, and 
hence the chemical potential determined has been averaged over different surface randomness. 
Therefore, the annealed approximation used in deriving Eq. (6) was met in the simulations. In a 
few cases, patchy and alternating surfaces were created by simulating a two-dimensional Ising 
model at appropriate conditions.  
Heteropolymers were modelled as SAWs consisting of two types of monomers, A and B 
with specified fractions fA and fB=1-fA.. Chains were created by randomly selecting N*fB different 
positions along the chain to be B beads, while the remaining beads were assigned as A beads, 
ensuring that the chain had the exact composition called for by fA and fB. The sequence order 
parameter, λ, in generated random sequences exhibits a Gaussian distribution with zero mean. 
Examples of distributions are presented in Figure 1.  The longer the chain, the narrower the 
distribution is. For a given chain length N, we typically generate 5000 copies of random 
sequences with specified fA. Each sequence is then used in biased insertion for 5000 or more 
copies to obtain the Rosenbluth-Rosenbluth weighting factor. Letting W(N, χ) stands for the 
sequence order parameter λ
-1.0 -0.5 0.0 0.5 1.0
N=100
N=200
Figure 1: Distribution of sequence order parameters obtained from 5000 copies of 
random sequences generated with fA = fB = 0.50 for three different chain lengths. Lines 
are smooth fit to the data. 
Rosenbluth-Rosenbluth weighting factor obtained for a given sequence χ, the chemical potential 
of a chain can be obtained using two different averages over sequences: 
),(ln)(0 χβµ NWNads −=  (10) 
),(ln),()( 00 χχβµβµ NWNN adsads −==  (11) 
The first approach is the annealed average, while the second approach is the quenched average. 
The two chemical potentials calculated differ slightly from each other. More discussion of the 
quenched versus annealed averages will be given later. For the determination of CAP, we have 
used annealed chemical potentials. 
4. Results and Discussion 
4.1. Method Used to Determine the Critical Adsorption Point 
The method we used to determine the CAP follows our earlier papers31,32 and is briefly 
sketched out. We obtain the difference in standard chemical potential ∆µ0 at different surface 
interaction εw for a set of chains with different lengths.  An example of data is presented in 
Figure 2(a) for a homopolymer above a homogeneous surface. The lines for different length N 
nearly intersect at a common point, which is estimated to be at εc=0.276 ± 0.005. A convenient 
way to identify this intersection point is to plot the standard deviation of all ∆µ0, σ(∆µ0), for a 
given range of chain length studied versus εw, which yields a minimum in a plot shown in Figure 
2(b). The minimum identified is directly related to the critical condition point employed in liquid 
chromatography at the critical condition (LCCC) 32-34. In LCCC, the critical condition was 
defined as the co-elution point of homopolymers with different molecular weights, which, 
corresponding to computer simulation, is the point where K has least dependence on chain 
length. If K is truly independent of chain length, then σ(∆µ0)  will be zero and will be the 
minimum in a plot in Figure 2(b). The critical condition point bracketed in this fashion depends 
slightly on the range of chain length included in the calculation of σ(∆µ0).  However, in the 
current study we fixed the range of chain lengths used.  
Since this common intersection point does not occur at ∆µ0 =0, one may wonder if it is 
the critical adsorption point discussed in the literature. We have applied the same method for 
random walks above a planar surface in simple cubic lattice31. The intersection point found was 
at εc = 0.183± 0.002, in excellent agreement with expected CAP for random-walks, εc = -ln(5/6)= 
0.1823.1 On the other hand, CAP could be understood as the point where the limiting monomer 
free energy for a chain attached to the surface f(ε) equals to the limiting monomer free energy of 
an unattached chain in the bulk solution. Therefore, we may define a CAP at a finite chain 
0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
N=100
N=200
0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
Figure 2: (a) Plot of ∆µ0 versus εw for SAW chains with N =25, 50, 100 and 200 above a 
homogeneous surface. The critical adsorption point is identified as the common intersection 
point, εw(cc)=0.276±0.005. (b) Plot of deviation in ∆µ0 for the given range of N versus εw. The 
minimum in the plot is the critical adsorption point. 
length, εc (N), at which ∆µ0(N)=0. From Figure 2(a), we extract such εc (N).  This εc (N) is 
expected to depend on N in a scaling law, εc (N) = εc(∞) –αN−φ, and εc(∞) is the CAP at infinite 
chain length limit. Assuming φ = 0.5, Figure 3 shows the linear fitting of εc (N) versus N−0.5 
which yields εc (∞) = 0.274 ± 0.005. The εc (∞) identified is within the error bars of the common 
intersection point. 
The CAP of SAWs in simple cubic lattice has been studied by others.6-8 The reported literature 
value for the CAP of SAWs on the simple cubic lattice ranged from ~0.37 by Ma et al.35 down to 
0.288 ± 0.02 by Janse van Rensburg and Rechnitzer8.  The value reported by Ma et al. was 
considered to be too high, probably due to chains analyzed being too short. Methods used to 
determine the CAP varied in the literature. Meirovitch and Livne6 obtained the CAP for SAW in 
simple cubic lattice with Monte Carlo simulations with the scanning method. They plotted 
E(T)/N against N and found the exponent α in E(T)/N~Nα over three different ranges of chain 
length (N = 20-60, 60-170, and 170-350).  Then, the critical point was located by finding the 
N-0.5
0.00 0.05 0.10 0.15 0.20 0.25
Figure 3: Plot of εc(N) versus N-0.5 where εc(N) is extracted from figure 1(a) as 
the point when ∆µ(N) = 0. The extrapolated εc(∞) =0.274 ± 0.005.  
value of the reciprocal temperature Θ that resulted in the exponent α being constant for the three 
different ranges of chain lengths. Their reported Θc, which is equivalent to our εc, was 0.291 + 
0.001. Their method for determining Θc was based on the scaling theory developed by EKB.4 As 
stated earlier, at CAP, E(T)/N is expected to scale with Nφ-1 where φ is the crossover exponent.  
The value of this crossover exponent was debated. EKB first showed that φ  ≈ ν ≈ 0.59, where ν 
is the Flory’s exponent. Several recent reports suggest that φ = 0.5 even for SAW chains, the 
same as φ for random-walks.8,36 In Meriovitch and Levin’s study, φ was left as an adjustable 
parameter. The reported φ value in their study was =0.530+ 0.007, slightly larger than recent 
reported values φ=0.5.  If we were to take φ=0.5, then their data would suggest a lower Θc. 
Recently Decase et al.9 explored four different ways to determine the CAP, mostly based on the 
scaling idea. They found that a slight change of εc lead to large deviations in the resulting φ. 
Therefore, simultaneous determination of εc and φ may not give the true location of CAP. Janse 
van Rensburg and Rechnitzer8 studied CAP for SAWs in two and three dimensions using a 
variety of methods, including studying the energy ratios of walks of different lengths and the 
specific heats of the chains. They found that analysis of the specific heat data in three dimensions 
were fraught with difficulty. The energy ratios of different lengths and the free energy method 
yielded εc within the error bars. They reported a value for the CAP, εc=0.288 + 0.020 and a 
crossover exponent φ = 0.5005 + 0.0036. Our CAP is within the error bars of their reported 
value. Interestingly, if they assume that the convergence of the energy ratios of different chain 
lengths is proportional to N1 , the yielded εc = 0.276 + 0.029, exactly the same as in our study.  
The above discussion suggests that the critical condition determined with our approach is 
the CAP. Our approach to determine the CAP does not depend on knowledge of φ and therefore 
does not suffer from the uncertainty in εc when both εc and φ need to be determined 
simultaneously. In the remainder of the paper, we will use this method to determine the CAP of 
SAWs above a planar heterogeneous surface and SAWs for heteropolymers above a planar 
homogeneous surface.  
4.2. Homopolymers above Heterogeneous Surfaces with Attractive and Non-Interacting Sites 
Here we consider adsorption of homopolymers above a heterogeneous surface. The first 
type of heterogeneous surface studied consists of a surface composed of two types of sites.  One 
type of the surface sites, which will be called A sites, did not interact with the polymer chains; 
that is, εwA = 0. The other type of surface site, the B sites, had an attractive interaction with the 
polymer chains, εwB.  The value of εwB was varied to locate the CAP.  Figure 4 shows a plot of 
the standard deviations in β∆µ0 over all chain lengths for each value of εwB scanned. The 
minimum in standard deviations occurs for εwB(cc) = 0.49± 0.01, where the error was based on 
the energy increment scanned. The same method was used to determine the CAP for surfaces 
with 10%, 15%, 20%, 25%, and 75% attractive sites. Table I summarizes the CAP of 
homopolymers over heterogeneous surfaces along with the data over a homogeneous surface. 
Figure 5 presents the plot of CAP, εwB(cc), as a function of fB along with the theoretical 
prediction according to Eq. (8) with εwA = 0 and εwh(cc) = 0.276.  
It is clear that a good agreement between Eq. (6) and simulation data is observed. Also 
we note that CAP is not linearly dependent on fB over the entire range but is well-described by 
Eq.(6). Earlier study by Sumithra and Baumgaertner13 focused on surfaces with fB above the 
percolation threshold. Within that limited range of fB, a linear dependence may be obtained. This 
study is the first to confirm the dependence of CAP on the surface disorder over a wide range of 
fB.  
0.44 0.46 0.48 0.50 0.52 0.54 0.56
Figure 4: Plot of deviation in ∆µ0 against εwB for a homogeneous chain 
adsorbing on a surface with 50% attractive sites and 50% non-interacting 
sites. The CAP occurs at εwB = 0.49 + 0.01. 
 As discussed in the theory section, one of the assumptions used in deriving Eq. (6) is that 
the interacting surface sites are randomly distributed. We have tested this assumption by 
studying adsorption of homopolymers over a 50% surface with alternating and patchy patterns. 
For a surface with 50% of A and B, an order parameter O.P. can be defined (readers are referred 
to literature for the definition).19 If O.P.=0, the surface is random; if O.P.=+1, then the surface is 
patchy; and if O.P.=-1, the surface is alternating. The data are also included in Table I and are 
indicated in Figure 4. The two points deviate from the line described by Eq. (6). The CAP 
obtained over a 50% alternating surface is larger than that over a 50% random surface.  On the 
other hand, the CAP obtained over a 50% patchy surface is smaller than over a 50% random 
surface. These results can be easily understood. When a chain is adsorbed on the surface, it 
forms trains, loops and tails.1 Formation of trains lowers the energy of a chain to overcome the 
entropy loss during the adsorption. When a chain is in contact with an alternating surface, it is 
however difficult to form trains as no adsorbing sites are adjacent, while this is possible for 
Percent B sites
0 20 40 60 80 100
Figure 5: Plot of the CAP, εwB(cc), against the percent of attractive B sites, fB.  The 
symbols are the CAP determined by the simulation, and the solid line is from equation 
(6) with εWA =0.0 and εwh(cc) =0.276. Circles are CAP over random surfaces, the cross 
(×) is the CAP over a strictly alternating surface, and the upper triangle (∆) is the CAP 
over a patchy surface with O.P. =+0.94.  
random and patchy surfaces. Therefore, chains attraction to the alternating surface is lessened, 
and adsorption over a 50% alternating surface has to occur at a larger value of εw. On the other 
hand, a chain over a patchy surface can selectively sample patches of the surface composed of 
adsorbing sites, so the adsorption over patchy surface can occur at a smaller value of εw. 
Another assumption used in deriving Eq. (6) is the annealed approximation. This 
approximation is strictly met if the surface pattern in contact with the chain changes during the 
chain adsorption,11 hence averaging over different distributions can be performed as done in Eq. 
(3). The surface in this case is said to contain annealed randomness. If the surface pattern can not 
change, then the surface is said to contain quenched randomness. In our simulations, the surface 
contains quenched randomness. In fact, we have used only one realization of a quenched random 
surface. However, the chain was placed randomly over different surface sites, making the 
annealed approximation applicable to our simulations. We note that Sumithra and 
Baumgaertner13, in their studies, averaged over 50 different realizations of quenched randomness 
and they compared the results with that of a single surface realization. They did not find major 
difference between these two approaches, especially if the temperature is high. Moghaddam and 
Whittington16 investigated the difference between the quenched average and the annealed 
average for homopolymer adsorption on heterogeneous surface and random copolymer 
adsorption on homogeneous surface. Their data show that there was no difference between the 
two averages in the case of adsorption on random surfaces but there were differences for 
adsorption of random copolymers especially at low temperature. It has been argued that 
quenched and annealed averages are equivalent in cases where the quenched surface is large in 
comparison with the polymer.11,15 Polotsky et. al18 have also found that the CAP for quenched 
and annealed surface disorders are the same. In our simulations, the surface is large in 
comparison with the size of the polymer, and the attachment of the polymer to the surface occurs 
at many random places on the surface. Therefore, the chain can effectively interact with many 
different random arrangements of surface sites, and the system approaches the annealed average.  
4.3. Homopolymers above Heterogeneous Surfaces with All Sites Interacting 
In order to assess whether the equation derived for the CAP of random surfaces was valid 
in more general cases, random surfaces that contained all attractive sites were prepared.  For 
these surfaces, the polymer-surface interaction for the A sites, εwA, was set at a relatively weak 
attractive strength, 0.10, and the interaction for the B surface sites was varied to find the CAP.  
Additionally, surfaces with repulsive A sites (εwA = -.10) were also investigated. 
Percent B Sites
0 20 40 60 80 100
Figure 6: Plot of the critical adsorption point, εwB(cc), against the percent of attractive 
B sites for surfaces with attractive or repulsive A sites.  The dashed line and open 
symbols are for surfaces with slightly repulsive A sites, εwA=-0.10.  The solid line and 
closed symbols are for surfaces with slightly attractive A sites, εwA=+0.10.  The 
symbols are simulation results, while the lines are from equation (6) with the 
corresponding εwA values. 
Figure 6 shows the values of εwB(cc) determined for these two cases, as well as the prediction of 
the value of εwB(cc) given the values of fB and εwA used in the simulation.  As can be seen in the 
figure, there is a good agreement between the data and the equation, indicating that the equation 
is valid for surfaces with many different types of surfaces, not just surfaces with attractive and 
non-interacting sites. 
4.4. Random Copolymers above Homogeneous Surfaces  
Critical adsorption point for random copolymers adsorbing on homogeneous surfaces 
were also determined.  In these systems, polymer chains are considered to be composed of two 
different types of monomers, A’s and B’s, interacting with a surface composed of only one type 
of site. B monomers were attracted to the surface, while A monomers do not interact with the 
surface, i.e. εwA = 0. Table 2 shows the values of the CAP, εwB(cc), for various values of  fB along 
with results obtained for homopolymers, alternating copolymers and block copolymers. Here we 
have used annealed chemical potentials to determine the CAP. Figure 7 presents the plot of 
B(cc) as a function of fB along with the theoretical prediction according to Eq. (6) with εwA = 0 
and εwh(cc) = 0.276.  The data fit the equation well for situations in which sequences are 
randomly specified. However, similar to homopolymer adsorption on heterogeneous surfaces, 
the equation does not apply when the chain sequence is not random. For a diblock copolymer, 
where the first half of the chain is all A monomers while the second half of the chain is all B 
monomers, a weaker attraction is required to reach the CAP than for a random 50% copolymer 
chain.  An alternating copolymer requires a slightly stronger attraction to reach the CAP. Again, 
these results can be explained by considering the tendency of forming trains during adsorption.  
The diblock copolymer is a homogeneous string of adsorbing B monomers attached to a string of 
A monomers.  The B section of the chain is able to interact with the surface like a homogeneous 
chain, while the A section does not adsorb and slightly repels the chain from the surface, 
indicating that the value of εwB(cc) for a diblock chain should be similar to a homogeneous chain 
on a homogeneous surface. In fact, εwB(cc) = 0.30 for diblock copolymers, a value only slightly 
higher than for homopolymer adsorption, and much lower than εwB(cc) for a 50% random 
copolymer chain.  For an alternating chain, consecutive attractive interactions are not possible, 
resulting in the necessity of a stronger εwB(cc) than for a random chain. 
Finally we compare the chemical potential determined with annealed approximation 
versus quenched average. We found that the chemical potential of a random copolymer above 
the surface, µ0ads, obtained via the annealed average in Eq. (10) was smaller than the quenched 
Percent B monomers
0 20 40 60 80 100
Figure 7: Plot of the CAP, εwB(cc), of copolymers over a homogenous surface against the 
percent of attractive B monomers, fB.  The symbols are the CAP determined by the 
simulation, and the solid line is the plot according to equation (6) with εWA =0.0 and 
h(cc) =0.276. Circles are CAP of random copolymers, the cross (×) is the CAP of block 
copolymers, and (∆) is the CAP of alternating copolymers. 
average in Eq. (11). This has been suggested in the literature.23 Annealed approximation implies 
that the chain sequence can change when it interacts with the surface. As a result, the chemical 
potential is lowered when compared with a chain with a fixed sequence. Figure 8 below shows 
the distribution of µ0ads, obtained based on trial insertions of a given random sequence, against 
the sequence order parameter λ. As discussed in section 3, a generated random sequence may not 
correspond to exactly λ=0, therefore resulting a distribution of µ0ads against λ. Figure 8 shows 
that within the range of λ spanned by random sequences, the chemical potential is seen to depend 
on λ. The µ0ads is higher for negative λ and is lower for positive λ. This is consistent with the 
results in Table II. A negative λ implies the random copolymer chain exhibits statistically 
alternating behaviour. A higher µ0ads implies that the chain is more difficult to be adsorbed on the 
surface; therefore, it needs a stronger attraction to reach CAP.  
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
Figure 8: The distribution of µ0ads versus the sequence order parameter λ of a random 
copolymer. Each data point represent one µ0ads based on insertion of one given random 
sequence for 5000 times and the figure contains data for 5000 random sequences. Chain 
length N =100, fA = fB = 0.5, and εwA =0.0 and εWB =-0.5.   
5. Summary Remarks 
Polymer adsorption at surfaces is relevant to many practical applications and has thus 
received extensive experimental investigation. However, interest in the CAP, to a large degree, 
has, until recently, remained a theoretical exercise. There were neither experimental methods that 
directly measure the CAP, nor were there applications that depended on the exact location of the 
CAP.  This has now changed as interesting applications in liquid chromatography separations 
have been developed.37,38 In particular, liquid chromatography at the critical condition (LCCC), 
first reported in the 1980’s, has now widely used for characterization of polymer systems that 
contain structural and chemical heterogeneities. The critical condition in LCCC experiments was 
defined as the point where homopolymers of a specific type co-elute regardless of their 
molecular weights. By erasing the dependence of elution on the molecular weights of one 
species, other species, differing either chemically or structurally, can then be analyzed. 
Experimentalists39 have mostly regarded this critical condition as the CAP. Our earlier Monte 
Carlo simulations largely support this view.31-34 The current study provides knowledge on the 
dependence of CAP on sequence disorder or surface disorder and such knowledge will be useful 
to develop chromatographic methods for analyzing random copolymers.  
We note that several earlier studies 13,14,16,17 have examined the adsorption of polymers on 
surfaces with either surface disorder or sequence disorder. These studies examined influence of 
disorder on a variety of properties related to polymer adsorption, such as the change of heat 
capacity, energy of the chain, and radius of gyration of the chain. Very few, however, have tried 
to determine the dependence of CAP on the disorder. One of possible reasons that hamper these 
earlier studies to study the dependence of CAP on the disorder may be due to the lack of a 
convenient way to determine the CAP. As we have discussed in the theory section, CAP was 
typically understood as the phase transition of an infinitely long chain near a surface. Earlier 
studies trying to determine the CAP need to wrestle with the difficulty in extrapolation of results 
to the limit of infinitely long chain. On the other hand, validity of our studies hinges on the way 
we determine the CAP. In the case of adsorption of homopolymers over homogeneous surface, 
we discussed the relationship between the CAP determined by our method with reported 
literature values. Abundant evidence that supports the validity of our approach was presented in 
section 4.1.  However, for the adsorption over heterogeneous surface, the nature of this CAP is 
not well-understood. Can a long chain in contact with a surface with few adsorbing sites still 
exhibit a phase transition similar as that of homopolymers over homogeneous surface? If it does, 
is the transition first-order or second-order? These questions therefore may cast some doubt on 
the CAP determined by our approach in the presence of disorder. However, the CAP we 
determined is directly related to the critical condition point in LCCC. Hence, even though the 
physical meaning of the CAP determined in this study in the presence of disorder could be 
subjected to further scrutiny, the importance of our results is not undermined.  
Table 1: Critical Adsorption Point for Homopolymers above Heterogeneous Surfaces with 
Attractive B Sites and Non-interacting A Sites. 
Percentage of Attractive Sites εwB(cc) 
100% 0.276 + 0.005 
75% 0.35 + 0.01 
50% 0.49 + 0.01 
25% 0.82 + 0.01 
20% 0.96 + 0.01 
15% 1.15 + 0.01 
10% 1.45 + 0.01 
50% alternating surface 0.55+ 0.01 
50% patchy surface (O.P=0.94) 0.31+ 0.01 
Table 2: Critical Adsorption Point for Heteropolymers with Attractive B Monomers and 
Non-interacting A Monomers over Homogeneous Surface 
Percentage of B monomers εwB(cc) 
100% 0.276 + 0.005 
75% 0.36 + 0.01 
50% 0.49 + 0.01 
25% 0.84 + 0.01 
15% 1.16 + 0.01 
50% alternating copolymers 0.55 + 0.01 
50% block copolymers 0.30 + 0.01 
References: 
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Polymers at Interfaces; Chapman & Hall: London, UK, 1993. 
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Graphics to be used for the Table of Contents 
Percent B sites
0 20 40 60 80 100

  The critical adsorption point (CAP) of self-avoiding walks (SAW) interacting
with a planar surface with surface disorder or sequence disorder has been
studied. We present theoretical equations, based on ones previously developed
by Soteros and Whittington (J. Phys. A.: Math. Gen. 2004, 37, R279-R325), that
describe the dependence of CAP on the disorders along with Monte Carlo
simulation data that are in agreement with the equations. We also show
simulation results that deviate from the equations when the approximations used
in the theory break down. Such knowledge is the first step toward understanding
the correlation of surface disorder and sequence disorder during polymer
adsorption.

Introduction 
Adsorption of polymers on surfaces plays a key role in many technological applications 
and is also relevant to many biological processes. As a result, it has been studied for more than 
three decades1 and continues to receive intense interest.2 The field is rich and contains a wide 
variety of topics, from equilibrium properties of adsorbed layers and conformations of adsorbed 
polymer chains to dynamic properties and non-equilibrium processes in adsorption.2 For polymer 
adsorption on planar surfaces, it is well-known that there exists a critical adsorption point (CAP) 
that marks the transition of a polymer chain, in contact with a surface, from a non-adsorbed state 
to an adsorbed state.3 Scaling laws for a variety of quantities below, above and at the CAP for a 
homopolymer in contact with a planar surface were developed by Eisenriegler, Kremer, and 
Binder (EKB).4 For example, when the chain goes from a non-adsorbed state to an adsorbed 
state, the energy of the chain E changes from an intensive variable independent of chain length N 
to an extensive variable dependent on N. At the CAP, E is expected to scale with Nφ where φ is 
the crossover exponent. Numerical studies, including exact enumeration,5 the scanning method6,7 
and the multiple Markov chain method8 have been performed to determine the location of the 
CAP and the crossover exponent φ. The values reported are however not completely in 
agreement with each other and are still under debate, especially the crossover exponent φ. The 
disagreement may be traced, as suggested by a recent article,9 to different methods used for 
determining the CAP and the crossover exponent φ.  
While many studies focused on adsorption of homopolymers on planar homogeneous 
surfaces, adsorption of polymers on chemically or physically heterogeneous surfaces has also 
received a fair amount of studies.10-20 Some were inspired by specific applications such as 
segregation of polymer chains on patterned surfaces,10 or pattern transfer via surface 
adsorption,21,22 others were motivated by a desire to understand how the presence of surface or 
sequence disorders may influence adsorption.13,14,16,17,23-25 For example, Sebastian and Sumithra 
developed an analytical theory of the adsorption of Gaussian chains on random surfaces using 
Gaussian variational approach.24,25 They took surface heterogeneity into account by modifying 
de Genne’s adsorption boundary condition and analyzed influence of randomness on the 
conformation of the adsorbed chains. Adsorption of heteropolymers on heterogeneous surfaces, 
in particular, has been studied because of its relevance to molecular recognition in biological 
process. The concept of “pattern matching” was proposed26 and has been investigated with 
different approaches.12,20,26,27 Muthukumar for example derived an equation for the critical 
condition of adsorption of a polyelectrolyte to an oppositely charged patterned surface.26 
Golumbfski et al.12 showed that a statistical blocky chain was selectively adsorbed on a patchy 
surface while a statistically alternating chain was selectively adsorbed on an alternating surface. 
Jayaraman et al.19 described a simulation method to design surfaces for recognizing specific 
monomer sequences in heteropolymers. Recently Polotsky et al18 considered adsorption of 
Gaussian heteropolymer chains onto heterogeneous surface. They found that the presence of 
correlations between sequence and surface heterogeneity always enhances adsorption. However, 
the dependence of the critical adsorption point on either surface disorder or sequence disorder is 
not well-understood. Lack of this knowledge hampers further understanding on the correlation 
between sequence disorder and surface disorder during adsorption.  
Here we present theoretical equations that describe the dependence of CAP on the surface 
disorder or sequence disorder, along with Monte Carlo simulation data in agreement with the 
derived equations. The current study does not address the correlation between sequence disorder 
and surface disorder. We only consider cases where the disorder is either present randomly on 
the surface (i.e. adsorption of homopolymers on random heterogeneous surface) or on the 
sequence (i.e., adsorption of random copolymer on homogeneous surface). The correlation 
between sequence disorder and surface disorder will be the subject of future publications. In the 
following, we first present the theory that predicts the dependence of CAP on surface disorder 
and sequence disorder. Then we present details of Monte Carlo simulation methods used to 
determine the CAP, followed by simulation data that agree with the derived equations. Finally, 
we discuss implications of these results on practical applications such as chromatographic 
separations of polymers.  
2. Theory 
2.1 Adsorption of a homopolymer on a homogeneous surface 
We first consider adsorption of a homopolymer chain on a homogeneous surface. This 
can be represented by a self-avoiding walk (SAW) in a three-dimensional lattice interacting with 
a plane and restricted to lie on one side of the plane. The vertices of the walks interact with the 
surface sites with an attractive energy εw. The partition function for a N-step SAW interacting 
with a homogeneous surface is given by 
( )∑=
wNw vvcNZ εε exp)(),(homo  (1) 
where cN(v) is the number of SAWs that lie above the surface with v visits to the surface. 
Hammersley et al.28 have shown that the model exhibits a phase transition at a critical adsorption 
energy, εc, with a desorbed state for εw  < εc, and an adsorbed state for εw  > εc. They have shown 
that the limiting monomer free energy f(εw) 
),(log
lim)( homo wNw NZN
=  (2) 
exists and is a convex non-decreasing continuous function of εw. Moreover, f(εw)=κ  for εw ≤ 0, 
where κ is the lattice connective constant, and f(εw) is a strictly increasing function of εw when 
εw  > εc. Therefore, f(εw) is non-analytic at εw = εc. εc has also been determined to be greater than 
zero and, based on the best-known connective constant for the simple cubic lattice29, to have an 
upper bound of 0.5738. The lattice connective constant κ is also the limiting monomer free 
energy of the SAWs in bulk solution. Hence the CAP can be understood as the condition where 
the limiting monomer free energy of a chain attached to the surface becomes equal to the limiting 
monomer free energy of the chain in the bulk solution.  
2.2 Adsorption of a homopolymer on a random heterogeneous surface 
Now we consider the adsorption of a homopolymer interact with a heterogeneous surface 
consisting of two types of surface sites, A and B. The interaction energy of the vertices with the 
two surface sites are εwA and εwB. Following Soteros and Whittington23, and express the partition 
function of a N-step SAW interacting with a heterogeneous surface that consists of A and B 
surface sites as: 
( ) ( )∑ ∑
ANBAhet BvfAvfAv
vcffZ
)()( )((exp)(exp
)(),( εε            (3) 
where cN(v) is the number of walks that have v surface contacts, v(A) is the number of monomers 
interacting with the A sites, and v(B) is the number of monomers interacting with the B sites, fA 
and fB = 1 - fA are the fractions of A and B sites on the surface, respectively.  Here the partition 
function is averaged over random distributions of the surface sites, i.e. the so called annealed 
approximation. Physically the annealed disorder means that the type of surface sites may change 
while the system attains equilibrium state. However, it has been previously suggested11,15 that the 
annealed approximation is valid if the chain can visit a large area of the surface and hence 
samples all distributions of surface patterns. Furthermore, the surface sites are randomly 
distributed. If there is a correlation between surface disorders, such as those present in patchy 
surface or alternating surface, then Eq. (3) will not be valid, as Eq. (3) gives equal weight to all 
possible surface labelings, while correlations restrict possible labelings. Summing over v(A), 
equation (3) can be simplified to 
( )∑ +=
wANBAhet ffvcffZ )exp()exp()(),( εε  (4) 
A comparison of equations (1) and (4) reveals that the partition functions for homogeneous and 
annealed random heterogeneous surface become equivalent if    
( ) ( ) )exp(expexp BwBAwAw ff εεε +=  (5) 
From Eq. (5), we derive the following equation that gives the dependence of CAP on the surface 
disorder: 
( ) ( ) ))(exp(exp)1()(exp ccffcc BwBAwBhw εεε +−=  (6) 
 where εwh(cc) is the CAP of a homopolymer above a homogeneous surface, εwB(cc) is the CAP 
of a homopolymer above a heterogeneous surface while the surface interaction energy εwA held 
constant. It can be easily seen from this equation, that the dependence of the CAP on the 
percentage of attractive sites on the surface is not expected to be linear, in contrast to the 
conclusion drawn by an earlier study.13 Equation (6) is expected to be valid as long as the two 
conditions are met: (i) the chain has enough mobility to visit a large area of surface so that the 
annealed approximation is valid, and (ii) the surface sites are randomly distributed (i.e. 
uncorrelated).  
2.3 Adsorption of a random heteropolymer on a homogeneous surface 
The same approach can be extended to consider the adsorption of a random 
heteropolymer interacting with a homogeneous surface.  We will use the same notation as in 
previous section except now fA and fB represent fractions of A and B monomers present on the 
heteropolymer. We will only consider random copolymers composed by A and B monomers. 
The sequence of a random copolymer can be represented by χ ={χ1, χ2, … χN} where χi are 
independently and identically distributed random variables with χi =A with a probability of fA 
and χi=B with a probability of 1-fA.  A sequence order parameter λ can be defined to characterize 
the sequence randomness.12,27  
BAAB pp −−= 1λ  (7) 
where pij is the nearest neighbor transition probabilities which is the probability that a monomer 
of type i is followed by a monomer of type j. When λ=0, the sequence is random. When 
λ>0, then the sequence is statistically blocky, and when λ<0, the sequence is statistically 
alternating. We note that a given random sequence designated by χ may have non-zero values of 
λ. More discussions will be given in the later section. 
The partition function of N-step SAWs with the given sequence above a homogenous 
surface is written as: 
)exp()|,(),,( BwB
BANBAhetpoly vvvvCffZ εεχχ += ∑  (8) 
There are two different ways to average over different distributions of random sequences, 
namely the annealed average and the quenched average. With the annealed average, the partition 
function in Eq. (8) is first averaged over different distributions of χ. This then leads to a partition 
function, Zhetpoly(fA, fB), which is exactly the same as in Eq. (3). With the annealed 
approximation, we derive the same equation as given by Eq. (6) for the CAP of a random 
heteropolymer interacting with a homogeneous surface, provided that fA and fB now represent the 
fractions of A and B monomers on the chain.  
In the following, we will present Monte Carlo simulation data that conform to the two 
equations and also results that do not conform to the equations because of the invalidation of the 
approximations used in deriving the equations.  
3. Monte Carlo Simulation Methods 
In our simulations, polymer chains are modeled as SAWs with N vertices on a simple 
cubic lattice of dimensions 250a × 250a × 100a, where a is the lattice spacing.  Each vertex 
represents a monomer on the polymer chain. Chain lengths studied are in the range of N = 25 to 
250.  There is an impenetrable wall in the z = a plane representing the surface.  One monomer, 
picked randomly from the chain, is first placed on a site adjacent to the wall (in the z = 2a plane). 
The rest of the chain is then grown using the biased chain insertion method.30 Monomers that are 
in the z = 2a plane are considered to be adsorbed on the surface.  For all adsorbed monomers, an 
attractive polymer-surface interaction, εw, is applied.  The standard chemical potential of the 
chain (since it does not contain translation entropy), µ0, is calculated from the Rosenbluth-
Rosenbluth weighting factor, W(N), which is given by30  
0 ln)(lnβµ  and 
)exp( β
 (9) 
where z is the lattice coordination number (z = 6 for simple cubic lattice), Ej is the energy of ith 
inserted monomer in the jth potential direction. We note that µ0 calculated is the free energy per 
chain, and µ0/N is free energy per monomer discussed in equation (2). Typically, the chemical 
potential is determined based on about twenty million copies of trial chain conformations. 
We obtained the standard chemical potentials of a chain with at least one monomer 
attached to the surface, µads0, and compared that against a chain grown in a bulk solution, µbulk0. 
The bulk solution is modeled by a 100a × 100a × 100a lattice with periodic boundary conditions 
applied in all three directions.  All chemical potentials calculated are reduced by the Boltzmann 
factor, β=1/kBT=1. A coefficient K, similar to partition coefficient if the chain was placed in a 
pore instead of near a surface, is calculated by K =exp(-∆µ0), where ∆µ0 = µads0 − µbulk0. The way 
we determined the CAP is based on the dependence of K on the chain length N and will be 
presented in the results section.  
Heterogeneous surfaces were modeled by making the z = a plane composed of two 
different types of sites, which have different values for polymer-surface interactions. The 
designations εwA and εwB will be used to distinguish between interaction energies of different site 
types.  Simulations were performed using surfaces with different fractions of A and B sites.  
Surfaces were created by randomly assigning each site as A or B based on the probabilities, pA 
and pB, where pA and pB are, respectively, the desired fractions of A and B sites on the surface.  
Because of size of the surface, this procedure resulted in the real surface composition 
percentages matching the desired percentages within 0.1%.  For a given surface composition, the 
surface was randomly created once and was subsequently used in all simulations that determine 
the chemical potential of a chain above that surface. The surfaces displayed quenched 
randomness, i.e. the surface pattern remained unchanged throughout the simulations.  However, 
the first bead of chain was placed randomly over the surface during the chain insertion, and 
hence the chemical potential determined has been averaged over different surface randomness. 
Therefore, the annealed approximation used in deriving Eq. (6) was met in the simulations. In a 
few cases, patchy and alternating surfaces were created by simulating a two-dimensional Ising 
model at appropriate conditions.  
Heteropolymers were modelled as SAWs consisting of two types of monomers, A and B 
with specified fractions fA and fB=1-fA.. Chains were created by randomly selecting N*fB different 
positions along the chain to be B beads, while the remaining beads were assigned as A beads, 
ensuring that the chain had the exact composition called for by fA and fB. The sequence order 
parameter, λ, in generated random sequences exhibits a Gaussian distribution with zero mean. 
Examples of distributions are presented in Figure 1.  The longer the chain, the narrower the 
distribution is. For a given chain length N, we typically generate 5000 copies of random 
sequences with specified fA. Each sequence is then used in biased insertion for 5000 or more 
copies to obtain the Rosenbluth-Rosenbluth weighting factor. Letting W(N, χ) stands for the 
sequence order parameter λ
-1.0 -0.5 0.0 0.5 1.0
N=100
N=200
Figure 1: Distribution of sequence order parameters obtained from 5000 copies of 
random sequences generated with fA = fB = 0.50 for three different chain lengths. Lines 
are smooth fit to the data. 
Rosenbluth-Rosenbluth weighting factor obtained for a given sequence χ, the chemical potential 
of a chain can be obtained using two different averages over sequences: 
),(ln)(0 χβµ NWNads −=  (10) 
),(ln),()( 00 χχβµβµ NWNN adsads −==  (11) 
The first approach is the annealed average, while the second approach is the quenched average. 
The two chemical potentials calculated differ slightly from each other. More discussion of the 
quenched versus annealed averages will be given later. For the determination of CAP, we have 
used annealed chemical potentials. 
4. Results and Discussion 
4.1. Method Used to Determine the Critical Adsorption Point 
The method we used to determine the CAP follows our earlier papers31,32 and is briefly 
sketched out. We obtain the difference in standard chemical potential ∆µ0 at different surface 
interaction εw for a set of chains with different lengths.  An example of data is presented in 
Figure 2(a) for a homopolymer above a homogeneous surface. The lines for different length N 
nearly intersect at a common point, which is estimated to be at εc=0.276 ± 0.005. A convenient 
way to identify this intersection point is to plot the standard deviation of all ∆µ0, σ(∆µ0), for a 
given range of chain length studied versus εw, which yields a minimum in a plot shown in Figure 
2(b). The minimum identified is directly related to the critical condition point employed in liquid 
chromatography at the critical condition (LCCC) 32-34. In LCCC, the critical condition was 
defined as the co-elution point of homopolymers with different molecular weights, which, 
corresponding to computer simulation, is the point where K has least dependence on chain 
length. If K is truly independent of chain length, then σ(∆µ0)  will be zero and will be the 
minimum in a plot in Figure 2(b). The critical condition point bracketed in this fashion depends 
slightly on the range of chain length included in the calculation of σ(∆µ0).  However, in the 
current study we fixed the range of chain lengths used.  
Since this common intersection point does not occur at ∆µ0 =0, one may wonder if it is 
the critical adsorption point discussed in the literature. We have applied the same method for 
random walks above a planar surface in simple cubic lattice31. The intersection point found was 
at εc = 0.183± 0.002, in excellent agreement with expected CAP for random-walks, εc = -ln(5/6)= 
0.1823.1 On the other hand, CAP could be understood as the point where the limiting monomer 
free energy for a chain attached to the surface f(ε) equals to the limiting monomer free energy of 
an unattached chain in the bulk solution. Therefore, we may define a CAP at a finite chain 
0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
N=100
N=200
0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34
Figure 2: (a) Plot of ∆µ0 versus εw for SAW chains with N =25, 50, 100 and 200 above a 
homogeneous surface. The critical adsorption point is identified as the common intersection 
point, εw(cc)=0.276±0.005. (b) Plot of deviation in ∆µ0 for the given range of N versus εw. The 
minimum in the plot is the critical adsorption point. 
length, εc (N), at which ∆µ0(N)=0. From Figure 2(a), we extract such εc (N).  This εc (N) is 
expected to depend on N in a scaling law, εc (N) = εc(∞) –αN−φ, and εc(∞) is the CAP at infinite 
chain length limit. Assuming φ = 0.5, Figure 3 shows the linear fitting of εc (N) versus N−0.5 
which yields εc (∞) = 0.274 ± 0.005. The εc (∞) identified is within the error bars of the common 
intersection point. 
The CAP of SAWs in simple cubic lattice has been studied by others.6-8 The reported literature 
value for the CAP of SAWs on the simple cubic lattice ranged from ~0.37 by Ma et al.35 down to 
0.288 ± 0.02 by Janse van Rensburg and Rechnitzer8.  The value reported by Ma et al. was 
considered to be too high, probably due to chains analyzed being too short. Methods used to 
determine the CAP varied in the literature. Meirovitch and Livne6 obtained the CAP for SAW in 
simple cubic lattice with Monte Carlo simulations with the scanning method. They plotted 
E(T)/N against N and found the exponent α in E(T)/N~Nα over three different ranges of chain 
length (N = 20-60, 60-170, and 170-350).  Then, the critical point was located by finding the 
N-0.5
0.00 0.05 0.10 0.15 0.20 0.25
Figure 3: Plot of εc(N) versus N-0.5 where εc(N) is extracted from figure 1(a) as 
the point when ∆µ(N) = 0. The extrapolated εc(∞) =0.274 ± 0.005.  
value of the reciprocal temperature Θ that resulted in the exponent α being constant for the three 
different ranges of chain lengths. Their reported Θc, which is equivalent to our εc, was 0.291 + 
0.001. Their method for determining Θc was based on the scaling theory developed by EKB.4 As 
stated earlier, at CAP, E(T)/N is expected to scale with Nφ-1 where φ is the crossover exponent.  
The value of this crossover exponent was debated. EKB first showed that φ  ≈ ν ≈ 0.59, where ν 
is the Flory’s exponent. Several recent reports suggest that φ = 0.5 even for SAW chains, the 
same as φ for random-walks.8,36 In Meriovitch and Levin’s study, φ was left as an adjustable 
parameter. The reported φ value in their study was =0.530+ 0.007, slightly larger than recent 
reported values φ=0.5.  If we were to take φ=0.5, then their data would suggest a lower Θc. 
Recently Decase et al.9 explored four different ways to determine the CAP, mostly based on the 
scaling idea. They found that a slight change of εc lead to large deviations in the resulting φ. 
Therefore, simultaneous determination of εc and φ may not give the true location of CAP. Janse 
van Rensburg and Rechnitzer8 studied CAP for SAWs in two and three dimensions using a 
variety of methods, including studying the energy ratios of walks of different lengths and the 
specific heats of the chains. They found that analysis of the specific heat data in three dimensions 
were fraught with difficulty. The energy ratios of different lengths and the free energy method 
yielded εc within the error bars. They reported a value for the CAP, εc=0.288 + 0.020 and a 
crossover exponent φ = 0.5005 + 0.0036. Our CAP is within the error bars of their reported 
value. Interestingly, if they assume that the convergence of the energy ratios of different chain 
lengths is proportional to N1 , the yielded εc = 0.276 + 0.029, exactly the same as in our study.  
The above discussion suggests that the critical condition determined with our approach is 
the CAP. Our approach to determine the CAP does not depend on knowledge of φ and therefore 
does not suffer from the uncertainty in εc when both εc and φ need to be determined 
simultaneously. In the remainder of the paper, we will use this method to determine the CAP of 
SAWs above a planar heterogeneous surface and SAWs for heteropolymers above a planar 
homogeneous surface.  
4.2. Homopolymers above Heterogeneous Surfaces with Attractive and Non-Interacting Sites 
Here we consider adsorption of homopolymers above a heterogeneous surface. The first 
type of heterogeneous surface studied consists of a surface composed of two types of sites.  One 
type of the surface sites, which will be called A sites, did not interact with the polymer chains; 
that is, εwA = 0. The other type of surface site, the B sites, had an attractive interaction with the 
polymer chains, εwB.  The value of εwB was varied to locate the CAP.  Figure 4 shows a plot of 
the standard deviations in β∆µ0 over all chain lengths for each value of εwB scanned. The 
minimum in standard deviations occurs for εwB(cc) = 0.49± 0.01, where the error was based on 
the energy increment scanned. The same method was used to determine the CAP for surfaces 
with 10%, 15%, 20%, 25%, and 75% attractive sites. Table I summarizes the CAP of 
homopolymers over heterogeneous surfaces along with the data over a homogeneous surface. 
Figure 5 presents the plot of CAP, εwB(cc), as a function of fB along with the theoretical 
prediction according to Eq. (8) with εwA = 0 and εwh(cc) = 0.276.  
It is clear that a good agreement between Eq. (6) and simulation data is observed. Also 
we note that CAP is not linearly dependent on fB over the entire range but is well-described by 
Eq.(6). Earlier study by Sumithra and Baumgaertner13 focused on surfaces with fB above the 
percolation threshold. Within that limited range of fB, a linear dependence may be obtained. This 
study is the first to confirm the dependence of CAP on the surface disorder over a wide range of 
fB.  
0.44 0.46 0.48 0.50 0.52 0.54 0.56
Figure 4: Plot of deviation in ∆µ0 against εwB for a homogeneous chain 
adsorbing on a surface with 50% attractive sites and 50% non-interacting 
sites. The CAP occurs at εwB = 0.49 + 0.01. 
 As discussed in the theory section, one of the assumptions used in deriving Eq. (6) is that 
the interacting surface sites are randomly distributed. We have tested this assumption by 
studying adsorption of homopolymers over a 50% surface with alternating and patchy patterns. 
For a surface with 50% of A and B, an order parameter O.P. can be defined (readers are referred 
to literature for the definition).19 If O.P.=0, the surface is random; if O.P.=+1, then the surface is 
patchy; and if O.P.=-1, the surface is alternating. The data are also included in Table I and are 
indicated in Figure 4. The two points deviate from the line described by Eq. (6). The CAP 
obtained over a 50% alternating surface is larger than that over a 50% random surface.  On the 
other hand, the CAP obtained over a 50% patchy surface is smaller than over a 50% random 
surface. These results can be easily understood. When a chain is adsorbed on the surface, it 
forms trains, loops and tails.1 Formation of trains lowers the energy of a chain to overcome the 
entropy loss during the adsorption. When a chain is in contact with an alternating surface, it is 
however difficult to form trains as no adsorbing sites are adjacent, while this is possible for 
Percent B sites
0 20 40 60 80 100
Figure 5: Plot of the CAP, εwB(cc), against the percent of attractive B sites, fB.  The 
symbols are the CAP determined by the simulation, and the solid line is from equation 
(6) with εWA =0.0 and εwh(cc) =0.276. Circles are CAP over random surfaces, the cross 
(×) is the CAP over a strictly alternating surface, and the upper triangle (∆) is the CAP 
over a patchy surface with O.P. =+0.94.  
random and patchy surfaces. Therefore, chains attraction to the alternating surface is lessened, 
and adsorption over a 50% alternating surface has to occur at a larger value of εw. On the other 
hand, a chain over a patchy surface can selectively sample patches of the surface composed of 
adsorbing sites, so the adsorption over patchy surface can occur at a smaller value of εw. 
Another assumption used in deriving Eq. (6) is the annealed approximation. This 
approximation is strictly met if the surface pattern in contact with the chain changes during the 
chain adsorption,11 hence averaging over different distributions can be performed as done in Eq. 
(3). The surface in this case is said to contain annealed randomness. If the surface pattern can not 
change, then the surface is said to contain quenched randomness. In our simulations, the surface 
contains quenched randomness. In fact, we have used only one realization of a quenched random 
surface. However, the chain was placed randomly over different surface sites, making the 
annealed approximation applicable to our simulations. We note that Sumithra and 
Baumgaertner13, in their studies, averaged over 50 different realizations of quenched randomness 
and they compared the results with that of a single surface realization. They did not find major 
difference between these two approaches, especially if the temperature is high. Moghaddam and 
Whittington16 investigated the difference between the quenched average and the annealed 
average for homopolymer adsorption on heterogeneous surface and random copolymer 
adsorption on homogeneous surface. Their data show that there was no difference between the 
two averages in the case of adsorption on random surfaces but there were differences for 
adsorption of random copolymers especially at low temperature. It has been argued that 
quenched and annealed averages are equivalent in cases where the quenched surface is large in 
comparison with the polymer.11,15 Polotsky et. al18 have also found that the CAP for quenched 
and annealed surface disorders are the same. In our simulations, the surface is large in 
comparison with the size of the polymer, and the attachment of the polymer to the surface occurs 
at many random places on the surface. Therefore, the chain can effectively interact with many 
different random arrangements of surface sites, and the system approaches the annealed average.  
4.3. Homopolymers above Heterogeneous Surfaces with All Sites Interacting 
In order to assess whether the equation derived for the CAP of random surfaces was valid 
in more general cases, random surfaces that contained all attractive sites were prepared.  For 
these surfaces, the polymer-surface interaction for the A sites, εwA, was set at a relatively weak 
attractive strength, 0.10, and the interaction for the B surface sites was varied to find the CAP.  
Additionally, surfaces with repulsive A sites (εwA = -.10) were also investigated. 
Percent B Sites
0 20 40 60 80 100
Figure 6: Plot of the critical adsorption point, εwB(cc), against the percent of attractive 
B sites for surfaces with attractive or repulsive A sites.  The dashed line and open 
symbols are for surfaces with slightly repulsive A sites, εwA=-0.10.  The solid line and 
closed symbols are for surfaces with slightly attractive A sites, εwA=+0.10.  The 
symbols are simulation results, while the lines are from equation (6) with the 
corresponding εwA values. 
Figure 6 shows the values of εwB(cc) determined for these two cases, as well as the prediction of 
the value of εwB(cc) given the values of fB and εwA used in the simulation.  As can be seen in the 
figure, there is a good agreement between the data and the equation, indicating that the equation 
is valid for surfaces with many different types of surfaces, not just surfaces with attractive and 
non-interacting sites. 
4.4. Random Copolymers above Homogeneous Surfaces  
Critical adsorption point for random copolymers adsorbing on homogeneous surfaces 
were also determined.  In these systems, polymer chains are considered to be composed of two 
different types of monomers, A’s and B’s, interacting with a surface composed of only one type 
of site. B monomers were attracted to the surface, while A monomers do not interact with the 
surface, i.e. εwA = 0. Table 2 shows the values of the CAP, εwB(cc), for various values of  fB along 
with results obtained for homopolymers, alternating copolymers and block copolymers. Here we 
have used annealed chemical potentials to determine the CAP. Figure 7 presents the plot of 
B(cc) as a function of fB along with the theoretical prediction according to Eq. (6) with εwA = 0 
and εwh(cc) = 0.276.  The data fit the equation well for situations in which sequences are 
randomly specified. However, similar to homopolymer adsorption on heterogeneous surfaces, 
the equation does not apply when the chain sequence is not random. For a diblock copolymer, 
where the first half of the chain is all A monomers while the second half of the chain is all B 
monomers, a weaker attraction is required to reach the CAP than for a random 50% copolymer 
chain.  An alternating copolymer requires a slightly stronger attraction to reach the CAP. Again, 
these results can be explained by considering the tendency of forming trains during adsorption.  
The diblock copolymer is a homogeneous string of adsorbing B monomers attached to a string of 
A monomers.  The B section of the chain is able to interact with the surface like a homogeneous 
chain, while the A section does not adsorb and slightly repels the chain from the surface, 
indicating that the value of εwB(cc) for a diblock chain should be similar to a homogeneous chain 
on a homogeneous surface. In fact, εwB(cc) = 0.30 for diblock copolymers, a value only slightly 
higher than for homopolymer adsorption, and much lower than εwB(cc) for a 50% random 
copolymer chain.  For an alternating chain, consecutive attractive interactions are not possible, 
resulting in the necessity of a stronger εwB(cc) than for a random chain. 
Finally we compare the chemical potential determined with annealed approximation 
versus quenched average. We found that the chemical potential of a random copolymer above 
the surface, µ0ads, obtained via the annealed average in Eq. (10) was smaller than the quenched 
Percent B monomers
0 20 40 60 80 100
Figure 7: Plot of the CAP, εwB(cc), of copolymers over a homogenous surface against the 
percent of attractive B monomers, fB.  The symbols are the CAP determined by the 
simulation, and the solid line is the plot according to equation (6) with εWA =0.0 and 
h(cc) =0.276. Circles are CAP of random copolymers, the cross (×) is the CAP of block 
copolymers, and (∆) is the CAP of alternating copolymers. 
average in Eq. (11). This has been suggested in the literature.23 Annealed approximation implies 
that the chain sequence can change when it interacts with the surface. As a result, the chemical 
potential is lowered when compared with a chain with a fixed sequence. Figure 8 below shows 
the distribution of µ0ads, obtained based on trial insertions of a given random sequence, against 
the sequence order parameter λ. As discussed in section 3, a generated random sequence may not 
correspond to exactly λ=0, therefore resulting a distribution of µ0ads against λ. Figure 8 shows 
that within the range of λ spanned by random sequences, the chemical potential is seen to depend 
on λ. The µ0ads is higher for negative λ and is lower for positive λ. This is consistent with the 
results in Table II. A negative λ implies the random copolymer chain exhibits statistically 
alternating behaviour. A higher µ0ads implies that the chain is more difficult to be adsorbed on the 
surface; therefore, it needs a stronger attraction to reach CAP.  
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
Figure 8: The distribution of µ0ads versus the sequence order parameter λ of a random 
copolymer. Each data point represent one µ0ads based on insertion of one given random 
sequence for 5000 times and the figure contains data for 5000 random sequences. Chain 
length N =100, fA = fB = 0.5, and εwA =0.0 and εWB =-0.5.   
5. Summary Remarks 
Polymer adsorption at surfaces is relevant to many practical applications and has thus 
received extensive experimental investigation. However, interest in the CAP, to a large degree, 
has, until recently, remained a theoretical exercise. There were neither experimental methods that 
directly measure the CAP, nor were there applications that depended on the exact location of the 
CAP.  This has now changed as interesting applications in liquid chromatography separations 
have been developed.37,38 In particular, liquid chromatography at the critical condition (LCCC), 
first reported in the 1980’s, has now widely used for characterization of polymer systems that 
contain structural and chemical heterogeneities. The critical condition in LCCC experiments was 
defined as the point where homopolymers of a specific type co-elute regardless of their 
molecular weights. By erasing the dependence of elution on the molecular weights of one 
species, other species, differing either chemically or structurally, can then be analyzed. 
Experimentalists39 have mostly regarded this critical condition as the CAP. Our earlier Monte 
Carlo simulations largely support this view.31-34 The current study provides knowledge on the 
dependence of CAP on sequence disorder or surface disorder and such knowledge will be useful 
to develop chromatographic methods for analyzing random copolymers.  
We note that several earlier studies 13,14,16,17 have examined the adsorption of polymers on 
surfaces with either surface disorder or sequence disorder. These studies examined influence of 
disorder on a variety of properties related to polymer adsorption, such as the change of heat 
capacity, energy of the chain, and radius of gyration of the chain. Very few, however, have tried 
to determine the dependence of CAP on the disorder. One of possible reasons that hamper these 
earlier studies to study the dependence of CAP on the disorder may be due to the lack of a 
convenient way to determine the CAP. As we have discussed in the theory section, CAP was 
typically understood as the phase transition of an infinitely long chain near a surface. Earlier 
studies trying to determine the CAP need to wrestle with the difficulty in extrapolation of results 
to the limit of infinitely long chain. On the other hand, validity of our studies hinges on the way 
we determine the CAP. In the case of adsorption of homopolymers over homogeneous surface, 
we discussed the relationship between the CAP determined by our method with reported 
literature values. Abundant evidence that supports the validity of our approach was presented in 
section 4.1.  However, for the adsorption over heterogeneous surface, the nature of this CAP is 
not well-understood. Can a long chain in contact with a surface with few adsorbing sites still 
exhibit a phase transition similar as that of homopolymers over homogeneous surface? If it does, 
is the transition first-order or second-order? These questions therefore may cast some doubt on 
the CAP determined by our approach in the presence of disorder. However, the CAP we 
determined is directly related to the critical condition point in LCCC. Hence, even though the 
physical meaning of the CAP determined in this study in the presence of disorder could be 
subjected to further scrutiny, the importance of our results is not undermined.  
Table 1: Critical Adsorption Point for Homopolymers above Heterogeneous Surfaces with 
Attractive B Sites and Non-interacting A Sites. 
Percentage of Attractive Sites εwB(cc) 
100% 0.276 + 0.005 
75% 0.35 + 0.01 
50% 0.49 + 0.01 
25% 0.82 + 0.01 
20% 0.96 + 0.01 
15% 1.15 + 0.01 
10% 1.45 + 0.01 
50% alternating surface 0.55+ 0.01 
50% patchy surface (O.P=0.94) 0.31+ 0.01 
Table 2: Critical Adsorption Point for Heteropolymers with Attractive B Monomers and 
Non-interacting A Monomers over Homogeneous Surface 
Percentage of B monomers εwB(cc) 
100% 0.276 + 0.005 
75% 0.36 + 0.01 
50% 0.49 + 0.01 
25% 0.84 + 0.01 
15% 1.16 + 0.01 
50% alternating copolymers 0.55 + 0.01 
50% block copolymers 0.30 + 0.01 
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Graphics to be used for the Table of Contents 
Percent B sites
0 20 40 60 80 100
