On the Adsorption of Two-State Polymers
aa r X i v : . [ c ond - m a t . s o f t ] J u l On the Adsorption of Two-State Polymers
N. Yoshinaga
Department of Physics, Graduate School of Science,the University of Tokyo, Tokyo 113-0033, Japan.
E. Kats
Laue-Langevin Institute, F-38042, Grenoble, France,and L. D. Landau Institute for Theoretical Physics,RAS, 117940 GSP-1, Moscow, Russia
A. Halperin ∗ Structures et Propri´et´es dArchitectures Mol´eculaires,UMR 5819 (CEA, CNRS, UJF), INAC,CEA-Grenoble, 38054 Grenoble cedex 9, France (Dated: 25/5/08; Received textdate; Revised textdate; Accepted textdate; Published textdate)
Abstract
Monte Carlo(MC) simulations produce evidence that annealed copolymers incorporating twointerconverting monomers, P and H, adsorb as homopolymers with an effective adsorption energyper monomer, ǫ eff , that depends on the PH equilibrium constants in the bulk and at the surface.The cross-over exponent, Φ , is unmodified. The MC results on the overall PH ratio, the PH ratioat the surface and in the bulk as well as the number of adsorbed monomers are in quantitativeagreement with this hypothesis and the theoretically derived ǫ eff . The evidence suggests that theform of surface potential does not affect Φ but does influence the PH equilibrium. . INTRODUCTION The interest in polymer adsorption reflects both its practical importance[1] and the as-sociated theoretical challenges[2]. Within this domain the adsorption behavior of neutralwater soluble polymers (NWSP), exemplified by polyethylene oxide (PEO) and polyvinylpyrrolidone (PVP)[3], occupies a special position. On the applied side this is because of theirimportance in the formulation of water-based colloidal solutions[4, 5, 6] of practical interest.From a theoretical point of view the adsorption of NWSP poses a distinctive problem: WhileNWSP such as PEO are homopolymers comprising of chemically identical monomers, theyare modeled as ”two-state polymers” whose monomers interconvert between hydrophilic (P)and hydrophobic (H) states[7, 8, 9, 10, 11, 12, 13, 14, 15]. These two-state models are in-voked in order to rationalize a phase diagram exhibiting a closed insolubility loop, with bothupper and lower critical solution temperatures, that is thought to characterize NWSP. Themodels differ with respect to the precise identification of the two interconverting states. Nev-ertheless, within all of these models NWSP exhibit an annealed sequence of HP states with atemperature (T) dependent H/P ratio. The adsorption behavior of such annealed two-statepolymers is expected to differ from that of homopolymers comprising of single monomerspecies and of quenched random copolymers with a fixed sequence and composition. Thusfar, it was investigated using a self consistent field theory [16] of a laterally uniform adsorbedlayer, comprising many adsorbed chains, in contact with a polymer solution. In the followingwe consider, in contrast, the adsorption transition of a single, terminally anchored, two-statepolymer and compare it to the corresponding results for homopolymers [2, 17, 18, 19, 20]and for quenched copolymers [21, 22, 23, 24, 25]. In particular, we investigate the adsorp-tion of a non-cooperative two-state polymer using Monte Carlo simulation supplemented bysimple theory. Our discussion is mostly concerned with swollen chains under good solventconditions. Simulation evidence suggests that the adsorption of both homopolymers andof quenched copolymers exhibits identical scaling behavior upon introduction of the appro-priate effective adsorption energy per copolymer’s monomer, ǫ eff . The cross-over exponentΦ characterizing the second-order adsorption transition of these two systems is identical[21, 24, 25]. With this in mind we expect similar behavior for the annealed two-state poly-mers. Accordingly we first identify the ǫ eff for the annealed copolymers case and thenanalyze the simulation data assuming that such chains adsorb as homopolymers with ǫ eff N S onto universal curves and to reproduce the simula-tion results concerning the total H fraction as well as the H fraction in the bulk and atthe surface. Importantly, in distinction to quenched copolymers, the total H fraction aswell as the H fraction in the bulk and at the surface are not fixed. These results are ofinterest from two points of view. First, they complement earlier results on the adsorptiontransition of homopolymers[2, 17, 18, 19, 20] and quenched copolymers[21, 22, 23, 24, 25].Second, they provide a starting point for the formulation of a phenomenological free energydescription[20] of the adsorption of NWSP modeled as two-state polymers.In formulating the problem we aimed to capture the generic features common to thevarious two-state models. These differ with respect to the precise identification of the inter-converting states: polar vs apolar[8], hydrated vs nonhydrated[9, 10, 12, 14, 15], or clusteredvs nonclustered[11]. With this in mind we focused on the simplest case where the monomersundergo a unimolecular HP interconversion. In the following we confront simple theorywith off-lattice Monte Carlo simulations of the adsorption behavior of a single two-statepolymer within this minimal model. There is no explicit solvent in the simulation and themonomers states are modeled as Lennard-Jones particles having identical collision diame-ters. The interactions between of the various monomer-monomer pairs are identical but theinteraction parameters with the surface differ with the monomeric state. As a result thePH interconvesion at the surface and at the bulk are associated with different equilibriumconstants (Figure 1). Our model is thus closest to the one proposed by Karlstr¨om[8]. ǫ eff of the two-state model is obtained in section II using a partition function method similar tothat of Moghaddam and Whittington[22] as well as its free energy counterpart. The conse-quences for the scaling analysis and blob description are discussed in section III. In sectionIV we present the details of the simulation model and the simulation results are discussedin section V. 3 I. THE EFFECTIVE ADSORPTION ENERGY ǫ eff OF ANNEALED COPOLY-MERS
Two methods can be used to identify ǫ eff , the adsorption energy of an ”effective”monomer at the surface. In one, first used by Moghaddam and Whittington to obtain boundsfor quenched copolymers [22, 23], ǫ eff is obtained upon recasting the annealed copolymerpartition function into a homopolymer-like form containing factors of the form exp( N S ǫ eff kT )where k is the Boltzmann constant. Similarly, it is possible to consider the free energyof an annealed copolymer and recognize ǫ eff in a term of the form − N S ǫ eff . These twoequivalent methods yield as expected an identical ǫ eff . We briefly discuss the two versionsin order to establish the relationship both to the Moghaddam-Whittington method and tothe phenomenological free energy approach.We begin with the free energy of an adsorbed two-state chain. It comprises two terms, F conf and F seq . The first, F conf , allows for the adsorption induced confinement of the chain.It reflects the loss of configurational entropy of the flexible backbone upon confinement toa slab of thickness D R F where R F ∼ N / is the Flory radius of the swollen chain . F conf depends only on D, or equivalently N S , and its precise functional form is not important forthe first part of our discussion . The second term, F seq , accounts for the standard chemicalpotentials of the P and H monomers in the bulk and at the surface as well as the mixingentropies at the surface and in the bulk. In the good solvent regime the chains are swollen,the monomer concentration is dilute and monomer-monomer interactions have negligibleeffect on the P ⇄ H equilibrium. The P H sequences in the bulk and at the surface arethus modeled as ideal one-dimensional mixtures and F seq ( N S , x H , x SH ) = E − T S mix = N B f B + N S f S = N f B + N S ( f S − f B ) . (1)Here f B and f S are respectively the free energies per monomer in the bulk and at the surfaceas specified by f S = x SH µ SH + (1 − x SH ) µ SP + kT [ x SH ln x SH + (1 − x SH ) ln(1 − x SH )] , (2) f B = x H µ H + (1 − x H ) µ P + kT [ x H ln x H + (1 − x H ) ln(1 − x H )] , (3)where the N monomers in the chain comprise N S adsorbed monomers at the surface and N B = N − N S nonadsorbed monomers in the bulk. The H fraction among free monomers is4 H = N BH N B while for the adsorbed monomers it is x SH = N SH N S where N BH and N SH denoterespectively the number of H monomers in the bulk and at the surface. The standardchemical potentials of the different species are denoted by µ i . Note that f B and f S asspecified by (2) and (3) are similar to the Zimm-Bragg free energy for the helix-coil transitionin the case of zero cooperativity [26] . At this point f B and f S are completely decoupled fromeach other and from F conf . The equilibrium conditions ∂f B /∂x H = 0 and ∂f S /∂x SH = 0lead to the mass action laws in the bulk N BH N BP = x H − x H = K B = exp (cid:18) − µ H − µ P kT (cid:19) , (4)and at the surface N SH N SP = x SH − x SH = K S = exp (cid:18) − µ SH − µ SP kT (cid:19) . (5)Accordingly, a mass action law of the form K = x − x or x = K/ (1 + K ) leads, uponsubstitution in (2) and (3), to the equilibrium free energies per monomer at the surface andin the bulk f eqS = µ SP − kT ln(1 + K S ) = − kT ln (cid:20) exp (cid:18) − µ SP kT (cid:19) + exp (cid:18) − µ SH kT (cid:19)(cid:21) , (6) f eqB ≡ ǫ B = µ P − kT ln(1 + K B ) = − kT ln (cid:20) exp (cid:18) − µ P kT (cid:19) + exp (cid:18) − µ H kT (cid:19)(cid:21) . (7)In turn, these yield F seq ( N S , x H , x SH ) = N f eqB + N S ( f eqS − f eqB ) ≡ N ǫ B − N S ǫ eff at equilib-rium and ǫ eff = f eqB − f eqS = µ P − µ SP + kT ln (1 + K S )(1 + K B ) = kT ln exp (cid:16) − µ SP kT (cid:17) + exp (cid:16) − µ SH kT (cid:17) exp (cid:16) − µ P kT (cid:17) + exp (cid:16) − µ H kT (cid:17) . (8)Our sign convention for ǫ eff follows the custom in the phenomenological theories ofadsorption[20, 27, 28].Having obtained ǫ eff from the free energy of an adsorbed chain we turn to the partitionfunction argument. The partition function of an annealed two-state polymer at a surface is Z = X N S N SH = N S X N SH =0 N BH = N B X N BH =0 C + N ( N S )5 ( N − N S )! N BH !( N − N S − N BH )! (cid:20) exp (cid:18) − µ H kT (cid:19)(cid:21) N BH (cid:20) exp (cid:18) − µ P kT (cid:19)(cid:21) N − N S − N BH × N S ! N SH !( N S − N SH )! (cid:20) exp (cid:18) − µ SH kT (cid:19)(cid:21) N SH (cid:20) exp (cid:18) − µ SP kT (cid:19)(cid:21) N S − N SH . (9)Here C + N ( N S ) is the number of chain trajectories with N S surface contacts which is assumedto be identical to that of a homopolymer. N SP = N S − N SH is the number of surfacemonomers in P state and N BP = N − N S − N BH is the corresponding number among bulkmonomers. Z as given by (9) counts all possible free and adsorbed PH sequences and assignsto each sequence the appropriate Boltzmann factor. The summations over N BH and N SH yields Z = X N S C + N ( N S ) (cid:20) exp (cid:18) − µ H kT (cid:19) + exp (cid:18) − µ P kT (cid:19)(cid:21) N − N S × (cid:20) exp (cid:18) − µ SH kT (cid:19) + exp (cid:18) − µ SP kT (cid:19)(cid:21) N S . (10) Z HB = h exp (cid:16) − µ H kT (cid:17) + exp (cid:16) − µ P kT (cid:17)i N = exp (cid:0) − Nǫ B kT (cid:1) is the HP contribution to the bulkpartition function of a two-state chain comprising N monomers. It counts the HP sequencesin the absence of a surface or in the presence of a perfectly non-adsorbing surface. We nowintroduce e C + N ( N S ) = C + N ( N S ) exp (cid:0) − Nǫ B kT (cid:1) which allows for both the backbone configurationsand the HP sequence at a non-adsorbing surface. In turn, this yields a ”homopolymer-like” Z Z = X N S e C + N ( N S ) exp (cid:16) − µ SH kT (cid:17) + exp (cid:16) − µ SP kT (cid:17) exp (cid:16) − µ H kT (cid:17) + exp (cid:16) − µ P kT (cid:17) N S = X N S e C + N ( N S ) exp( N S ǫ eff kT ) , (11)allowing to identify ǫ eff , as given by equation (8).The free energy formulation translates into the partition function description via F = − kT ln Z . In particular, for each set N S , x H , x SH the sequence partitionfunction is approximately Z seq ( N S , x H , x SH ) = exp h − F seq ( N S ,x H ,x SH ) kT i and Z seq = P N S P x H P x SH C + N ( N S ) exp h − F seq ( N S ,x H ,x SH ) kT i where C + N ( N S ) containing the informationon N S corresponds roughly to exp (cid:16) − F conf kT (cid:17) . This correspondence is incomplete because itis based on the Stirling approximation and the implicit assumption that 1 ≪ N S < N ,1 ≪ N B < N , 1 ≪ N SH < N S and 1 ≪ N BH < N B . Alternatively, one may be-gin with the partition function and obtain the free energy formulation upon replacing6he factorials x ! in Z seq by exp( x ln x − x ) . As we discussed, ǫ eff can be obtained fromthe minimized free energy which corresponds to the maximal term of Z . Minimizationof F yields simultaneously ǫ eff and the equilibrium constants K B and K S . K B and K S are obtainable from Z upon minimizing of the individual terms subject to the caveatsnoted earlier. The Moghaddam-Whittington(MW) annealed approximation of the adsorp-tion of quenched copolymers [22] provided the starting point of our discussion of the an-nealed copolymers adsorption. Within this approximation a quenched copolymer with afixed average H fraction, N H /N = p, and a quenched sequence is modeled as a copoly-mer with a fixed p but with an annealed sequence [29]. The two treatments differ intwo respects: (i) within the MW treatment f B = 0 . (ii) The standard chemical po-tentials of monomers at the surface, µ SH and µ SP are replaced in the MW approach by˜ µ SH = µ SH − ln p and ˜ µ SP = µ SP − ln(1 − p ). Accordingly F seq ( N S , x H , x SH ) = N S f S where f S = x SH ˜ µ SH + (1 − x SH )˜ µ SP + kT [ x SH ln x SH + (1 − x SH ) ln(1 − x SH )] and the equilib-rium condition ∂f S /∂x SH = 0 leads to the mass action law x SH − x SH = e K S = p − p K S and toequilibrium x SH = pK S − p + pK S instead of the x SH = K S K S as obtained for the fully annealedpolymer considered by us. For the particular case considered by MW, where µ SP = 0 and µ SH /kT = − ǫ MW , this leads to ǫ eff kT = − f eqS kT = ln (cid:20) exp (cid:18) − e µ SP kT (cid:19) + exp (cid:18) − e µ SH kT (cid:19)(cid:21) = ln [ p exp ǫ MW + 1 − p ] , (12)while the partition function (9) is replaced by its MW counterpart Z = X N S N SH = N S X N SH =0 C + N ( N S ) N S ! N SH !( N S − N SH )! p N SH (1 − p ) N S − N SH exp ( N SH ǫ MW ) . (13)One may thus consider the MW partition function as a special case of equation (10) inwhich µ SH /kT = − ǫ MW − ln p , µ SP /kT = µ P /kT = − ln(1 − p ) and µ H /kT = − ln p . Thetwo procedures also differ because the MW approach is an approximation when applied toquenched copolymers while its counterpart, as described above, is rigorously applicable tothe annealed the two-state polymers considered here [30]. III. SCALING ANALYSIS AND ADSORPTION BLOBS
Having identified ǫ eff of two-state polymers we are in a position to discuss the scalinganalysis of such polymers and the corresponding blob picture. For two-state polymers it7ssumes, as is the case for quenched copolymers, that the polymers adsorb as homopolymersbut with an excess adsorption energy per monomer of ǫkT = ǫ eff − ǫ ceff . (14)Here ǫ ceff is a constant, model dependent, critical adsorption energy at the limit of τ = (cid:0) ǫ eff − ǫ ceff (cid:1) /ǫ ceff → , N → ∞ while ǫN Φ = const ′ [17]. In contrast to simple ho-mopolymers where ǫ ≈ const ′ [20] the T dependence of two-state polymers ǫ is ǫ ( T ) = ln exp (cid:16) − µ SH kT (cid:17) + exp (cid:16) − µ SP kT (cid:17) exp (cid:16) − µ H kT (cid:17) + exp (cid:16) − µ P kT (cid:17) − ǫ ceff kT . (15)Upon identifying ǫ, the ”homopolymer-like” scaling hypothesis for N S is standard. In par-ticular N S ≈ N Φ g s ( x ) where x = τ N Φ , (16)where g s ( x ) is a universal scaling function[17, 18, 20]. At the transition, where N S ≈ N Φ , we thus require g s ( x = 0) ≈
1. In the adsorption region, x >> , g s ( x ) follows a power lawbehavior, x q s . Since N S ∼ N when x >> q s = (1 − Φ) / Φ or N S ≈ N ǫ (1 − Φ) / Φ . Overall, a plot of N S /N Φ vs τ N Φ should collapse the data onto a single curve correspondingto g s ( τ N Φ ). To this end it is first necessary to determine the model dependent ǫ ceff . ForΦ = 1 / , the currently accepted value, N S N ∼ N − ∼ N , (17)at the critical point, ǫ eff = ǫ ceff . This indicates that curves of N S /N vs. ǫ eff for different N values intersect at ǫ eff = ǫ ceff thus allowing to determine ǫ ceff from the intersection point[31].We now return to the free energy per adsorbed chain, F . It comprises of two terms F = F conf + F seq . The adsorption free energy F seq ≈ − N S ǫkT allows for the attractivemonomer-surface contacts. F conf reflects the confinement of the polymer into a slab ofthickness D < R F . Within the blob picture[20] F conf ≈ kT N blob ≈ kT Ng where N blob is thenumber of confinement ” D blobs” comprising each of g monomers such that g ν a ≈ D . Since N S ≈ Ng g Φ we obtain g ≈ (cid:0) N S N (cid:1) − and thus F ( N S ) kT ≈ N (cid:18) N S N (cid:19) − Φ − N S ǫ, (18)8inimization with respect to N S than yields the equilibrium N S N S ≈ N ǫ − ΦΦ . (19)and the equilibrium free energy of the adsorbed chain F eq kT ≈ − N ǫ . (20)In turn, the equilibrium N S together with x H = K B / (1 + K B ) and x SH = K S / (1 + K S )determine the total number of H monomers N H = N S K S K S + ( N − N S ) K B K B or N H N ≈ ǫ − ΦΦ K S K S + (cid:16) − ǫ − ΦΦ (cid:17) K B K B . (21)As noted earlier, the recent consensus regarding the cross-over exponent suggests Φ = 1 / g ≈ /ǫ ; D ≈ a/ǫ / ; N S ≈ N ǫ. (22)However, simulation results suggest that finite size effects are important and for finite chainsΦ ≈ / g ≈ /ǫ / ; D ≈ a/ǫ ; N S ≈ N ǫ / . (23)For the adsorption of free chains, the form of ǫ eff leads to distinctive adsorption constantin dilute surface regime, when the adsorbed chains do not overlap. In this regime thechemical potential of the adsorbed chains is µ ads ≈ F eq + kT ln Γ where Γ is the activity ofthe adsorbed chains and F eq ≈ − kT N ǫ is the standard chemical potential of an adsorbedchain. The chemical potential of the free chains in the bulk is µ bulk ≈ kT ln c bulk where c bulk is the activity of the free chains. The adsorption isotherm, as obtained from µ ads = µ bulk , is Γ ≈ c bulk K ads where K ads ≈ exp( N ǫ ) is the adsorption constant[20]. K ads for two statepolymers and Φ = 1 / K ads = exp N ln exp (cid:16) − µ SH kT (cid:17) + exp (cid:16) − µ SP kT (cid:17)h exp (cid:16) − µ H kT (cid:17) + exp (cid:16) − µ P kT (cid:17)i exp (cid:16) ǫ ceff kT (cid:17) . (24)9he T dependence of K ads as given by (24) differs from that of K ads = exp (cid:0) N δ / Φ (cid:1) asobtained for homopolymer adsorption with ǫ = δ ≈ const ′ [20].As we shall see in section IV the approach discussed above accounts well for the simulationresults. Confrontation with experimental results is more difficult. The four parametersinvolved, µ H , µ P , µ SP and µ SH are specific both to the particular NWSP considered andto the model used in order to analyze the data. While µ H , µ P were determined for anumber of two-state models, the full set of parameters was only determined for PEO withinthe Karlstr¨om model by fitting phase boundaries and adsorption data[8, 16]. However, whilethis model is closest to the one we analyze, the two differ in a number of points. For example,in contrast to our model the PP, HH and PH interactions within the Karlstr¨om model arenot identical. With this caveat in mind, this set of µ H , µ P , µ SP and µ SH can be used toillustrate the behavior of ǫ eff for a ”PEO-like” case . In the Karlstr¨om model the standardchemical potential of the two-states in the bulk is given by µ i = U i − RT ln g i where U i isthe internal energy and g i is a degeneracy factor. The values per mole in KJ are µ P = 0and µ H = 5 . − RT ln 8[8] . For the adsorbed species µ = µ i − ∆ ǫ adi where the adsorptionenergy per mole ∆ ǫ adi at methylated silica are ∆ ǫ adP = 0 . KJ and ∆ ǫ adH = 1 . KJ [16].A rough idea concerning the physical consequences of the Karlstr¨om model may be gainedupon comparing the ǫ eff of a hypothetical P homopolymer, corresponding to PEO modeledas hydrophilic one-state polymer, to the annealed PH homopolymer model of PEO. Suchcomparison shows that ǫ eff of the two-state chain is shifted upwards by roughly a factor oftwo. IV. SIMULATION MODEL
We simulate a two-state polymer terminally anchored to a planar surface. FollowingBaumgartner [32] the polymer is modeled as freely jointed chain comprising N Lennard-Jones (LJ) particles. The monomers within this bead-spring model interact via a LJ poten-tial V LJ = 4 e ǫ X i,j "(cid:18) ar ij (cid:19) − (cid:18) ar ij (cid:19) , (25)where r ij = | r i − r j | is the distance between monomers i and j such that | i − j | ≥ r i is the position vector of the i th monomer. e ǫ specifies the depth of the potential10inimum at r = 2 / a and a , the collision diameter, is the separation for which V LJ = 0 . This potential exhibits a soft-core steric repulsion at r ≤ a , and steeply decaying attractionfor r ≥ a . The monomers exist in two interconverting states, P and H . It is thus necessaryto specify three LJ potentials, corresponding to the interactions between P P , HH , and P H monomer pairs, involving altogether six parameters. In the following all three potentials arecharacterized by the same a and we will thus express all distances in these units. The threeremaining parameters, e ǫ = e ǫ PP kT ; e ǫ PH kT ; e ǫ HH kT determine the strength of interactionsbetween the P P , P H , and HH monomers. For the simulation of self avoiding chains weset e ǫ PH = e ǫ HH = e ǫ PP = 0 .
20 so that the monomer-monomer interactions are dominated bythe excluded volume and contain effectively no attractive contribution. For simulations ofideal polymers e ǫ PH = e ǫ HH = e ǫ PP = 0 . In addition, the bulk HP states are characterizedby standard chemical potentials µ H = kT ∆ ǫ and µ P = 0 with ∆ ǫ >
0. Accordingly∆ µ = ∆ ǫkT specifies the difference in standard chemical potential ∆ µ = µ H − µ P betweennoninteracting H and P monomers. All monomer pairs, except nearest-neighbors, interactvia LJ potentials. Nearest-neighbor monomers along the chain are constrained to 0 . ≤ ( r i +1 − r i ) ≤ .
0. Separations outside this range incur an infinite energy penalty thusensuring connectivity.We have used two monomer-surface interaction potentials, U wall , both specified in termsof z , the distance between the monomer and the surface. One is the contact potential U wall = ∞ z ≤ − kT ∆ ǫ adi < z ≤ z > ǫ adi > i = P, H . We will mostly focus on the contact potential because it allows forunambiguous definition of adsorption i.e., a monomer having its center within the slab0 < z ≤ . U wall = kT ∆ ǫ adi "(cid:18) z (cid:19) − . (cid:18) z (cid:19) , (27)as obtained by integrating the LJ potentials between the top monolayer atoms of the sub-strate and a monomer at z . Here again ∆ ǫ adi > z < z dependent U wall . In both cases thevalues of ∆ ǫ adi for the P and H states, ∆ ǫ adH or ∆ ǫ adP , differ favoring the surface H state i.e., ∆ ǫ adP < ∆ ǫ adH . For the contact surface potentials we thus have µ SH /kT = µ H − ∆ ǫ adH = ∆ ǫ − ∆ ǫ adH and µ SP /kT = µ P − ∆ ǫ adP = − ∆ ǫ adP . (28)For the 10-4 potentials µ SH and µ SP are z dependent. However, due to the fast decay of thepotential µ SH and µ SP as given by (28) provide, as we shall see, a reasonable approximation.The simulations involves chains comprising N = 64 , ,
256 monomers. At each MonteCarlo step (MCs) we shift the position of every monomer in the chain and update its HPstate using the Metropolis algorithm. This procedure is thus repeated N times per MCs.For each set of parameters the simulation involves 2 × MCs. The system is equilibratedduring million MCs. The remaining 1 . × MCs are grouped into sets of 10 MCs whoseconfigurational and PH characteristics are averaged for analysis. In a conformational up-date of the monomer position, a monomer is chosen randomly, its position is shifted bya sufficiently small distance and the resulting energy difference kT ∆ E , accounting for LJinteractions, is calculated. When ∆ E ≤
0, the operation is accepted and we proceed to thenext monomer movement. On the other hand when ∆
E >
0, the operation is accepted withthe probability exp( − ∆ E ). The PH interconversion updates are implemented in two stagescorresponding to the bulk equilibration and its modification by the surface potential. In the”bulk stage” a monomer is randomly chosen and its state is updated with the probability p (P → H) = exp( − ∆ ǫ ) p (H → P) = 1 (29)where p (P → H) and p (H → P) denote respectively the transition probabilities from P to H,and from H to P. The detailed balance condition for the bulk equilibrium is p (P → H) N BP = p (H → P) N BH yields N BH N BP = x H − x H = exp( − ∆ ǫ ) ≡ K B . (30)This procedure is the counterpart of the trial motion in the conformational steps. ThePH states are subsequently updated, in z dependent fashion, allowing for the effect of thesurface potential on the PH equilibrium. For the contact potential, monomers with z = 1are converted following the transition probabilities12 P ( S P → SH) = exp(∆ ǫ adP − ∆ ǫ adH ) P ( S H → SP) = 1 (31)where P ( S H → SP) = 1 because ∆ ǫ adP < ∆ ǫ adH . The corresponding detailed balance conditionfor equilibrium at the surface, p (P → H) P ( SP → SH ) N SP = p (H → P) P ( SH → SP ) N SH yields N SH N SP = x SH − x SH = exp(∆ ǫ adH − ∆ ǫ − ∆ ǫ adP ) = K S . (32)In contrast, monomers at z > K B is accordingly unmod-ified. Similar procedure is used for the 10-4 wall potential with the distinction that the z dependent surface transition probabilities occur, in principle at all z values with subsequenteffect on K B as well as on K S . However, since the 10-4 potential decays fast the effect onthe bulk equilibrium is negligible. V. SIMULATION RESULTS
As opposed to the case of quenched copolymers, whose composition is fixed, the H fraction of our model two-state polymers depends on three tuning parameters ∆ ǫ, ∆ ǫ adH and ∆ ǫ adP . In the following ∆ ǫ adP is fixed while ∆ ǫ and ∆ ǫ adH are varied. Our results aremostly concerned with chains exhibiting self avoiding random walk (SAW) statistics andinteracting with the surface via contact potentials. In addition we will comment brieflyon the behavior of ideal chains exhibiting random walk (RW) statistics and the effect of10-4 monomer-surface potentials. Importantly, variation of ∆ ǫ adH affects N SH (Figure 2), N H (Figure 3) and N S (Figure 4). As seen from Fig 2, N SH /N S = K S / (1 + K S ) and N BH /N B = K B / (1 + K B ) in agreement with equations (4) and (5). To proceed further itis first necessary to determine ǫ ceff . As noted earlier, curves of N S /N vs. ǫ eff for different N values intersect at ǫ eff = ǫ ceff provided that Φ = 1 / . Alternatively, curves of N S /N vs. ∆ ǫ adH will intersect at ∆ ǫ cH for families of curves of identical ∆ ǫ, ∆ ǫ adP , but different N values. Within the accuracy of our data the curves do intersect at a single point (Figure 5)thus lending support to the assumption that the cross-over exponent Φ = 1 / ǫ ceff as obtained by substituting∆ ǫ cH and the corresponding ∆ ǫ, ∆ ǫ adP values into (8) is a constant (Figure 6) as expected13ithin the picture of ”homopolymer-like” adsorption. Having obtained ǫ ceff we find that the N S vs ∆ ǫ adH data collapses onto a universal curve (Figure 7) upon plotting N S /N / vs τ N / where τ = kTǫ ceff ln exp (cid:0) ∆ ǫ adH − ∆ ǫ (cid:1) + exp (cid:0) ∆ ǫ adP (cid:1) − ∆ ǫ ) − . (33)Furthermore, the N H /N data agrees with equation (21) as adapted to the simulation model(Figure 3) N H N ≈ ǫ exp(∆ ǫ + ∆ ǫ adP − ∆ ǫ adH )exp(∆ ǫ + ∆ ǫ adP − ∆ ǫ adH ) + 1 + (1 − ǫ ) exp(∆ ǫ )exp(∆ ǫ ) + 1 . (34)The above results concern SAW and contact potentials. The ǫ ceff values for RW are lowerthan those of SAW chains and depend on the wall potential (Figure 6). However, sinceΦ = 1 / N S /N / vs τ N / scaling with theproper choice of τ is obtained in both cases for adsorption due to contact potentials (Figure7-8).The adsorption behavior of homopolymers experiencing 10-4 potential and contact po-tential is indistinguishable[18]. Our discussion suggests that the adsorption of an annealedcopolymer is analogous to the adsorption of a homopolymer with ǫ eff given by (8). Ac-cordingly we expect the adsorption of annealed homopolymers to be insensitive to thechoice of the potential and of the z values used to define adsorbed monomers. Withinthe accuracy of our simulation, this is indeed the case for the adsorption of RW annealedcopolymers where the data is collapsed by plotting N S /N / vs τ N / (Figure 9). It is how-ever important to note that for this case the P H equilibrium is z dependent. Accordingly N SH /N S = K S / (1 + K S ) and N BH /N B = K B / (1 + K B ) as specified by (30) and (32) onlycapture the qualitative features of the PH equilibrium but do not yield quantitative agree-ment because µ SH and µ SP as approximated via (28) do not allow for the z dependenceof U wall . A quantitative agreement is achieved upon dividing ∆ ǫ adH or ∆ ǫ adP by 1.5 allowingpresumably for the average U wall experienced within the z < VI. DISCUSSION
The adsorption behavior of annealed copolymers may conceivably differ with the detailsof the model. Our discussion concerned the particular case of non-cooperative two-state14olymers involving unimolecular PH interconversion. For this case, our results suggest thatthe annealed copolymers adsorb as homopolymers once the appropriate effective adsorptionenergy per monomer at the surface, ǫ eff , is identified. This ǫ eff accounts for the averageadsorption energy as well as the mixing entropy of the surface PH monomers at equilib-rium. The adsorption of two-state polymers is described by the cross-over exponent Φ ofhomopolymer adsorption. In this respect, their behavior is identical to that of quenchedcopolymers. However, as opposed to quenched copolymers, where ǫ eff as obtained fromthe MW annealed approximation is determined by the fixed average H/P fraction[22, 25], ǫ eff in the annealed case depends on the P H equilibrium constants in the bulk and at thesurface, K B and K S . In other words ǫ eff varies with the H/P ratio which, in contrast to thequenched case, depends on the temperature, the adsorption energies and the bulk standardchemical potentials. This picture allows to calculate the number of adsorbed monomers N S ,the H/P ratio in the bulk and at the surface as well as the overall
H/P ratio. The calculatedvalues are in good agreement with the simulation results concerning SAW and RW experi-encing contact potentials. Simulation of homopolymer chains suggest that the form of thewall-monomer potential does not affect Φ and the scaling behavior of N S . Since annealedcopolymers adsorb as homopolymers characterized by ǫ eff one expects similar insensitivityto the details of the potential. In contrast, the H/P ratio is sensitive to the range of thewall potential. Both features are apparent from our results concerning the adsorption of RWannealed copolymers subject to a 10-4 wall potential. As noted earlier, our analysis utilizes asimplified version of the Karlstr¨om model. In contrast to this model all monomer-monomerinteractions are identical. New features may well occur upon introducing attraction betweenhydrophobic H monomers. In the case of free chain in the bulk such interaction can lead tophase separation[8] and to polymer collapse[33]. The preferential adsorption of H monomerstogether with HH attraction may lead to adsorption induced collapse.Our discussion thus far focused on the comparison between simulation results of annealedand quenched copolymers. The comparison of the experimental situation brings up a secondissue. The experimental realization of quenched copolymers is clear and studying the ad-sorption of the corresponding homopolymers allows to deduce the adsorption energies of thedifferent monomers involved. The situation is more difficult in the case of annealed copoly-mer as encountered in the modeling of NWSP. The corresponding one state homopolymersdo not exist. Furthermore, direct experimental evidence concerning the molecular identity15f the interconverting states is yet to emerge. As a result the interaction parameters areobtained indirectly by model dependent fitting of experimental data. With this in mindit is important to recall the experimental evidence for two-state models. Two observationsare of special interest. One is their ability to predict the qualitative features of the phasediagram and fit experimentally observed phase boundaries[7, 8, 9, 10, 12, 13, 14, 15]. Thesecond concerns the stretching of PEO chains in atomic force microscopy experiments. Thisreveals that different force laws characterize the strong stretching regime in water and inhexadecane. In particular, the chain extension in water exhibits a plateau characteristic oftwo-state polymers[34].
Acknowledgement 1
N.Y. benefited from fellowship No. 7662 from the Japan Society forthe Promotion of Science (JSPS) as well as from the hospitality of the theory group of theILL. Part of the numerical calculations in this work was carried out on Altix3700 BX2 atYITP in Kyoto University. [1] Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London,1983.[2] Eisenriegler, E. Polymers Near Surfaces: Conformation Properties and Relation to CriticalPhenomena; World Scientific Publishing Company 1993[3] Molyneux, P. Ed. Water-Soluble Synthetic Polymers: Properties and Behavior; Volume I &IICRC Press 1984[4] Glass J.E. Ed, Water-Soluble Polymers: Beauty with Performance; Oxford University Press,USA 1986[5] Glass, J.E. Ed., Hydrophilic Polymers: Performance with Environmental Acceptance; OxfordUniversity Press 1996[6] Williams, P. Ed. Handbook of Industrial Water Soluble Polymers; Wiley-Blackwell NY, 2007[7] Goldstein, R.
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New J. Phys. , 1, 6. IG. 1: Two configuration of an adsorbed two-state polymer undergoing P(full circles) H(emptycircles) interconversion in the bulk and at the surface.FIG. 2: N BH /N B and N SH /N S as a function of a contact potential ∆ ǫ − ∆ ǫ adH for various µ H − µ P = kT ∆ ǫ and the corresponding curves given by (30) and (32). IG. 3: Plot of N H /N as a function of of a contact potential ∆ ǫ adH for various µ H − µ P = kT ∆ ǫ and the corresponding curves given by (34).FIG. 4: N S /N vs ∆ ǫ adH of a contact potential for self avoiding chains with various µ H − µ P = kT ∆ ǫ and N . IG. 5: N S /N vs ∆ ǫ adH for various N and ∆ ǫ = 0.FIG. 6: ∆ ǫ cH and the corresponding ǫ ceff (as obtained by substituting ∆ ǫ cH into (8)) vs ∆ ǫ . IG. 7: N S /N / vs τ N / for SAW experiencing a contact potential.FIG. 8: N S /N / vs τ N / for RW experiencing a contact potential. IG. 9: N S /N / vs τ N / for RW experiencing a 10-4 potential.FIG. 10: N BH /N B and N SH /N S as a function of a 10-4 potential ∆ ǫ − ∆ ǫ adH / . µ H − µ P = kT ∆ ǫ and the corresponding curves given by (30) and (32).and the corresponding curves given by (30) and (32).