Properties of branched polymer chains adsorbed on a patterned surface

Polymer 53 (2012) 1741e1746

Contents lists available at SciVerse ScienceDirect

Polymer journal homepage: www.elsevier.com/locate/polymer

Properties of branched polymer chains adsorbed on a patterned surface Stanis1aw Jaworski y, Andrzej Sikorski* Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 November 2011 Received in revised form 1 February 2012 Accepted 22 February 2012 Available online 3 March 2012

The aim of this study was to investigate the properties of polymer chains strongly adsorbed on a planar surface. Model macromolecules were constructed of identical segments, the positions of which were restricted to nodes of a simple cubic lattice. The chains were in good solvent conditions, thus, the excluded volume was the only interaction between the polymer segments. The polymer model chain interacted via a simple contact potential with an impenetrable flat surface with two kinds of points: attractive and repulsive (the latter being arranged into narrow strips). The properties of the macromolecular system were determined by means of Monte Carlo simulations with a sampling algorithm based on the local conformational changes of the chain. The structure of adsorbed chains was found to be strongly dependent on the distance between the repulsive strips, whenever this distance was very short. The mobility of the chains was also studied and it was found that diffusion across repulsive strips was suppressed for large distances between the strips. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Lattice models Monte Carlo method Polymer adsorption

1. Introduction Systems containing adsorbed polymer chains have recently become the subject of many experimental and theoretical works, owing to their practical importance, e.g. lubrication, colloidal stabilization, chromatography etc. [1]. Adsorption of macromolecules is also interesting from a theoretical point of view, as the presence of the attractive surface changes the properties of the adsorbed chain, compared with a non-adsorbed chain in solution. Most of the theoretical works were devoted to adsorbed linear polymer chains, while mean field theories are most commonly applied to describe adsorbed macromolecules [2e5]. Several dynamic properties of adsorbed linear polymer chains were determined by Binder et al. from off-lattice Monte Carlo simulations [6e8]. Studies of DNA molecules adsorbed on a lipid bilayer, performed by means of fluorescence microscopy showed that the diffusion coefficient scales with the length of the molecule (number of base pairs) as N1, which corresponds to the scaling for a twodimensional chain model [9]. On the other hand, Granick et al. found much stronger scaling, namely N3/2 for adsorbed polyethylene glycol [10,11]. The properties of branched polymer chains differ from the properties of their linear counterparts and this effect is particularly visible at interfaces [12]. The simplest model of a non-linear * Corresponding author. Tel.: þ48 22 822 0211; fax: þ48 22 822 5996. E-mail address: [email protected] (A. Sikorski). y Deceased author. 0032-3861/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2012.02.039

macromolecule is a regular star-branched polymer consisting of three equal length branches originating from a common point e the branching point. These star-branched polymer chains can be synthesized, their properties can be studied experimentally, and they are particularly useful for theoretical considerations [12,13]. Recently, Kosmas showed that the differences in adsorption of starbranched polymers with several arms, and of linear polymers, are almost negligible: for stars with a small number of arms, the adsorption is slightly (6%) higher than for linear chains [14]. As the number of arms increases, so does the degree of adsorption of the polymer chain due to compactness of star polymers, as compared to linear ones. Scaling analysis of adsorbed star-branched polymers was performed by Ohno and Binder [15,16]. They showed that the density profile exponents depend on the number of arms. It was also found that in the case of strong attraction, the chain becomes almost two-dimensional. The distribution of polymer segments near the adsorbing surface, as well as its dependence on the strength of adsorption were studied; the competition between this interaction and the intra-chain attraction was investigated; and certain dynamic properties of such polymers were also determined. Coarse-grained models of adsorbed star-branched polymer chains on homogenous planar surfaces have recently been developed and studied by means of Monte Carlo simulations [17e19]. The size and structure of adsorbed star-branched polymer chains, as compared to linear polymers, were studied and the simulations revealed that the degree of adsorption was similar for star-branched and linear chains, and was ca. 40% larger for ring polymers. The abovementioned model was also extended by introducing a second

1742

S. Jaworski, A. Sikorski / Polymer 53 (2012) 1741e1746

adsorbing surface, which served to confine the chains in the slit [20]. The structure of star-branched chains in confinement was determined and discussed there. At low temperatures and for sufficiently narrow slits, the chains were fully adsorbed on one surface, but it was found that they could jump from one surface to another [20]. Star copolymers were also studied employing a mean field theory [21,22]. The development of nanotechnology enables one to design systems consisted of a surface with adsorbed polymers. Macromolecules adsorbed on patterned surfaces found numerous applications as antireflectors, biosensors, or in microelectronic etc. The influence of nanopatterns on the conformation and other properties of polymer chains are also of theoretical interest. Recently, the adsorption of polymer and protein chains on patterned surfaces has also become the subject of studies. Such systems were studied experimentally [23e29], theoretically [30e33] and by means of computer simulation [34e44]. Homopolymers, copolymers, copolymer blends, polyelectrolytes and polymer brushes constituted the objects of these studies. The process of adsorption on the patterned surface can be considered as consisted of two-stages: binding and rearrangement. Both these processes are well known but the dynamics of the chains in such conditions are still not described and understood. Star-branched polymers on patterned surfaces are scarcely investigated and they seemed to be an interesting object of computational studies. Moreover, the structure and certain dynamic properties of these macromolecules can be compared with the findings for homogenous attractive surfaces reported in Refs. [17,18]. In this work we examine regular star-branched homopolymers adsorbed on a strip-patterned surface. The model macromolecules were adsorbed on a patterned surface containing attractive and repulsive points, arranged into strips. The chains consisted of f ¼ 3 arms (branches) and were embedded in a simple cubic lattice. The model chains were studied in good solvent conditions e the excluded volume was the only intra-chain potential and no attractive polymer segment nor polymer segment interactions were introduced. This choice was made in order to avoid competition between adsorption and chain collapse, which is the subject of forthcoming studies. The Monte Carlo algorithm used employed local micromodifications of chains of a Verdier-Stockmayer type [45,46]. The paper is organized as follows: The model and the calculation method outlines the assumptions of the model and the details of the simulation technique. Results and discussion presents the results concerning mostly the size and structure of chains, as well as their long-time dynamic properties. Conclusions section contains the most important concluding remarks. 1.1. The model and the calculation method The star-branched chains under consideration consisted of f ¼ 3 linear chains of equal length (‘arms’). These arms were built of sequences of identical segments originating from the branching point. Each polymer segment can be treated as a united atom representing several monomers of a real polymer. In order to make the calculations more efficient, the spatial positions of polymer segments were restricted to the vertices of a simple cubic lattice. The exclusion of the double occupancy of the same lattice site by polymer segments was introduced into the model, which was equivalent to the effect of the excluded volume in the system. This resembled good solvent conditions, where the excluded volume was not compensated by the attraction between non-bonded polymer segments. The simulations were carried out for single chains, i.e. for the case of an infinitely dilute polymer solution. A single chain of a given length was located near the planar surface, parallel to the xy plane and set at z ¼ 0. The surface was

impenetrable to polymer segments and a simple square-well contact potential V was assumed between it and the polymer segment:


Vðxi ; zi ¼ 1Þ ¼

εr εa

for for

modðxi ; dv Þ ¼ 0 modðxi ; dv Þs0

(1)

where xi and zi are the coordinates of the ith polymer bead along the x and z axes, respectively. The interaction takes place when a polymer bead is located in the layer adjacent to the surface (z ¼ 1). εa is the value of the attractive contact potential, while εr is the repulsive contact potential. The repulsive points on the surface are arranged in strips and dv is the distance between the strips parallel to the y-axis. The strips are infinitely long and the width of each strip is one lattice unit. This kind of surface patterning is rather common in real experiments, where the surface is covered periodically by polar (silicon oxide, ethylene oxide) and non-polar (alkylsilan, polystyrene) compounds [23e29]. This periodic patterning of the substrate can be a factor governing the selfassembly of polymers and proteins. The schematic of a polymer chain at the adsorbing surface with the repulsive strips is shown in Fig. 1a. Periodic boundary conditions were imposed along the x and y directions only. The length L of the Monte Carlo box was chosen large enough to void the influence of its size on the results, i.e. the chain did not interacted with itself. The Monte Carlo simulations of the model system were carried out using a sampling algorithm in which the conformation of the chain was locally changed by the following set of micromodifications [46]: (a) 2-segment motion, (b) 2-segment end reorientations, (c) 3-segment motion, (d) 3-segment crankshaft motion, and (e) the collective motion of the branching point. All these local moves involving one or two polymer beads are schematically presented on Fig. 1b. A fragment of the chain to be modified was selected at random during the simulation, while one average attempt of every micromodification per one polymer segment defined the time unit. It was shown that the choice of this set of local motions can reproduce the dynamics of real macromolecules, i.e. the Rouse-like behavior. The use of equal frequency of all motions is justified by the efficiency of the algorithm [45,47]. The conformation of a new chain was accepted according to chain connectivity and the excluded volume, with the probability proportional to its Boltzmann factor, as found from the Metropolis scheme with respect to changes of the adsorption energy of the system:

Pold/new ¼ min½1; expð  DEa =kTÞ

(2)

where Pold/new is the probability of the transition from the old to the new configuration, DEa is the energy difference between the old and the new conformation, k is the Boltzmann constant and T is the temperature. Each simulation run consisted of 109e1010 time units and was preceded by an equilibration run, which consisted of 107e108 time units. This procedure was repeated 10e15 times, starting from different conformations of the chain. The use of the dynamic Monte Carlo method and of the lattice model can be justified by the fact that simulations of atomistic models with Brownian/Molecular Dynamics on such a large time scale are still beyond current computing capabilities. Since we studied the structure of entire chains and the dynamic behavior for time scales considerably longer than those on which conformational changes take place, the method is quite reliable. 2. Results and discussion Most of the simulations were carried out for star-branched chains consisting of n ¼ 67 segments in any arm, which implies

S. Jaworski, A. Sikorski / Polymer 53 (2012) 1741e1746

Fig. 1. a. A scheme of a star-branched polymer chain adsorbed on a patterned surface. The repulsive strips on the surface are shown with thick lines. b. Local moves used in the algorithm: 2-segment motion (a), 2-segment end reorientations (b), 3-segment motion (c), 3-segment crankshaft motion (d) and the collective motion of the branching point (e).

that the total number of segments in the macromolecule was N ¼ f(n1) þ 1 ¼ 199. Such length was chosen because it was previously shown that it is long enough to avoid the influence of the artifacts of the lattice model, and many calculations for adsorbed polymers of this length were performed [17,20]. In order to check the influence of the chain length on the macromolecular structure we have also simulated chains with N ¼ 49, 100, 400 and 799. The distance between the repulsive strips on the attractive surface was varied between dv ¼ 2 and dv ¼ 50. The distance of 2 lattice units is the shortest possible on a square lattice (dv ¼ 1 corresponds to a purely repulsive surface), while the distance of 50 lattice units is larger than the diameter of the adsorbed polymer chain of the length under consideration [16]. The length L of the Monte Carlo box was chosen to be large enough to avoid the interaction of the chain with itself. Previous works concerning star-branched chains on an attractive surface [16] and initial studies of the model with repulsive strips indicated that the diameter of the macromolecule, i.e. 2 < S2>½ was considerably smaller than L ¼ 100 for the chain lengths under consideration. Thus, the calculations were

1743

performed for systems with L  100. In order to keep a constant distance between the repulsive strips in the entire space, the length of the Monte Carlo box was varied between L ¼ 100 and 120 e satisfying the condition mod (L, dv) ¼ 0. For longer chains we used larger box with L between 200 and 240. The attractive potential of the surface was chosen as εa ¼ 1, which corresponded to the strong adsorption regime [17], and the repulsive potential was chosen to be of the same magnitude εr ¼ 1 (both expressed in kT units). The properties of the system under consideration determined for the potential < 1 < εa < 0.6, i.e. in the strong adsorption regime, are similar. For εa > 0.6, the transition to the weak adsorption regime takes place when a chain has several contacts with the surface, while for εa < 1, the sampling algorithm becomes inefficient and, therefore, the study of this region requires a different simulation technique [17]. The mean-square radius of gyration of the entire star , and the mean-square distance from the branching point to the end of an arm described the size of the polymer chain, as usual. In Fig. 2 we present both the size parameters and the reduced value of * as functions of the distance between the repulsive strips dv (all distances discussed here are in lattice units). The reduced center-to-end distance * ¼  0.4 was used for the plot, in order to make its value comparable to the radius of gyration e for a free star-branched chain one can find / ¼ 0.4 [16]. The calculated parameters describing the mean size of the macromolecule were statistically stable: the relative standard deviation did not exceed 3%. One can observe that both curves are very similar, but not monotonic. The rapid increase in the size of the chain for lower values of dv is caused by an opportunity for the packing of the chain along the y-axis (impossible for dv ¼ 2, as only the rod structures could be packed without energetic losses). For both curves, the minimum appears near dv ¼ 35, the reason for this behavior is discussed below. The inset of Fig. 2 shows the changes in the z-contribution to the mean-square radius of gyration with the distance between the strips. The values of z are very small (between 0.1 and 0.2) for most of the distances dv, and only for very short distances between strips (or higher numbers of repulsive points) their values increase significantly, although their contribution to never exceeds 2%. The adsorbing surface contained the repulsive patterns along the y-axis and thus, it was anisotropic. Hence, we also studied

Fig. 2. The mean-square radius of gyration and the reduced mean-square centerto-end distance * as a function of the distance between the repulsive strips dv. The inset shows the z-contribution to the mean-square radius of gyration z as a function of the distance dv. All distances are in lattice units.

1744

S. Jaworski, A. Sikorski / Polymer 53 (2012) 1741e1746

chains adsorbed on a patterning surface, one can show that the balance between entropic losses and energetic gains governs the size of the chain. This is particularly visible when the distances between the repulsive strips are short. The behavior of the centerto-end distance is basically the same, which means that the arms are influenced by the pattern to the same extent as the entire chains. The influence of the chain length on its properties was also studied. The scaling behavior of the mean-square radius of gyration of chains on a patterned surface was different than in the case of two-dimensional chains, where w N3/2. For short distances between the repulsive strips (dv ¼ 5), we found that w N1.610.01 and an increase in this distance to dv ¼ 30 led to a decrease in the scaling exponent: w N1.410.02. The scaling exponent of the x contribution to the radius of gyration is considerably lower than the one along the y-axis, i.e. along the repulsive strips as the macromolecules tend to extend in this direction. The two short chains under consideration (N ¼ 49 and 100) have comparable S2x and S2y contributions, because their size is smaller than the distance between the strips (dv ¼ 30). The influence of the adsorbed chain length on its size is presented in Fig. 4. In order to make the results comparable for chains of different lengths, reduced quantities were used here. The reduced size of the chain is expressed as * ¼ /0, where the subscript ‘0’ refers to the chain adsorbed on a pure adsorbing surface, and the reduced distance between the repulsive strips was defined as d*v ¼ dv/ (2 < S2>1/2), where 2 1/2 is the approximate diameter of the chain. The behavior of all chains is very similar and the minima of the curves are located at almost the same d*v . The minima were found for values of d*v > 1, i.e. for distances longer that the mean chain diameter. In order to study the influence of the repulsive pattern on the motion of the macromolecule, we calculated the autocorrelation function which describes the mean-square displacement of the center of mass of the system gcm(t):

changes in the contributions to the mean-square radius of gyration and the mean-square center-to-end distance along both the x and y axes. Fig. 3 shows these contributions as functions of the distance between the repulsive strips for the chain length N ¼ 199. It was found that the x component of and increases with the increase in the distance dv, with the exception of the vicinity of dv ¼ 35, where a shallow minimum appears. This shape of S2x and S2y curves is responsible for the minimum on the S2 curve (Fig. 2), which is quite unexpected, but reproducible. This non-monotonic behavior is probably caused by the instantaneous asymmetry of the chain. The polymer is aspherical in a dilute solution and this shape is preserved in the case of a strongly attractive surface, where the chain is almost two-dimensional (a pancake e see Fig. 2 and the discussion above). The instantaneous shape of such chain resembles an ellipse and the decrease of the S2x curve starts when the longer axis of the ellipse becomes comparable to the distance between repulsive strips, i.e. above dv ¼ 25. Further increase of dv up to 30 causes the decrease of the S2x ; some configurations with longer axis along the x-axis are still discriminated. The changes in the y contributions are somewhat different. For the narrowest attractive strips (dv ¼ 2), S2y is slightly larger than S2x , which implies that despite the presence of the repulsive strips, almost no direction is preferred. This can be explained by the fact that for such narrow distances dv, the chain has to cross the strips many times, because the probability of forming conformations with no energy losses (with few segments perpendicular to the y-axis: rod-like or Greek key-like) is rather small. A moderate increase in the distance between the repulsive strips (from dv ¼ 3 to dv ¼ 5) leads to a dramatic increase (approximately threefold) in the size of the chain along the y-axis. This widening of the attractive areas allows more polymer segments to remain perpendicular to the y-axis. The above explanation is confirmed by an observed decrease in the number of strips, simultaneously adsorbed by a chain with an increase in the width of a strip (with the exception of the case dv ¼ 2). The same non-monotonic behavior of the size of the chain in the direction parallel to the strips was found for charged linear homopolymers [34] and for linear stiff chains [36]. A further increase in the parameter dv leads to a decrease in the radius of gyration: this decrease is more pronounced for dv  15 and less pronounced for dv > 15. For longer distances dv, both contributions to the radius of gyration converge to the same value, i.e. the chain becomes isotropic along the xy plane. Significantly, in the case of

where r(t) and r(0) are the vectors describing the position of the center of mass of the polymer chain at time t and 0, respectively. The function gcm(t) was calculated for the xy components, because it describes the diffusion of the chain along the attractive surface,

Fig. 3. The x and y contributions to the mean-square radius of gyration and the reduced mean-square center-to-end distance * as a function of the distance between the repulsive strips dv. All distances are in lattice units.

Fig. 4. The reduced mean-square radius of gyration * as a function of the reduced distance between the repulsive strips d*v (see text for details). The chain lengths are given in the inset. All distances are in lattice units.

gcm ðtÞ ¼

E D ½rðtÞ  rð0Þ2

(3)

S. Jaworski, A. Sikorski / Polymer 53 (2012) 1741e1746

Fig. 5. The reduced self-diffusion coefficients Dx/Do and Dy/Do as functions of the distance between the repulsive strips dv. All distances are in lattice units.

whereas the motion in the perpendicular direction is highly suppressed. From the autocorrelation function one can determine the self-diffusion coefficient according to the formula

Dx ¼

x gcm ; 2t

Dy ¼

y gcm 2t

(4)

where t stands for time. The determination of the diffusion coefficient along the y-axis was performed for the larger displacements, where the behavior of the chain is Rouse-like, i.e. gcm scaled as t1 and thus the motion of the center of mass was diffusive. In the x direction the motion is also Rouse-like, but slower, which is expected, since the motions are hindered by the presence of the

1745

repulsive strips. Fig. 5 presents the self-diffusion coefficient along the x and y directions as a function of the distance between the repulsive strips dv. The error in Dx and Dy did not exceed 8% for displacements smaller than 10 . In order to compare the motion on our patterned surface with that on homogeneous attractive surface we used the reduced diffusion coefficients Dx/Do and Dy/Do, where Do ¼ 1.25  103 was the coefficient for homogenous surface. The fastest motion along the x-axis was observed for very short distances between the repulsive strips, such as dv ¼ 5. There are two main reasons for such high mobility. First of all, the chain is extended over many repulsive strips and at all times it has many repulsive contacts or loops over the strips, and thus its motion does not lead to a significant lowering of the energy. Therefore, the Metropolis criterion does not hinder its motion. Next, the chain is not stretched in the y direction, and therefore the number of trans conformations is small, which leads to more effective conformational changes. In the remaining cases, the translational motion of the chain was considerably slower/smaller (by more than two orders of magnitude). The increase of the distance between the repulsive strips leads to the decrease of the mobility along the x-axis what implies that the motion in this direction is limited, i.e. the repulsive barriers become almost impenetrable. The mobility of the polymer chain along the y-axis is considerably higher: the ratio Dx/Dy increases with the distance dv from 2 (dv ¼ 5) to 50 (dv ¼ 50). The diffusion along the y-axis does not depend monotonically on the distance between the repulsive strips. For short distances dv  25 the mobility of the chain is near 0.4Do and slightly decreases. A sufficiently wide distance between adsorbing strip allows the faster motion of the chain along these strips and its mobility increases and goes towards the value of Do. In this case the chain is not stretched and does not have to lose its attractive contacts during the motion (dv ¼ 50). The influence of the chain length on its long-time dynamic behavior was also studied. The self-diffusion constant for the motion along the y direction

Fig. 6. The trajectory of the polymer center of mass for the distance between the repulsive strips dv ¼ 5 (a), 15 (b), 30 (c) and 50 (d). All distances are in lattice units.

1746

S. Jaworski, A. Sikorski / Polymer 53 (2012) 1741e1746

scales as Ng and the scaling exponents are g ¼ 0.99  0.02 (dv ¼ 5); and 0.96  0.04 (dv ¼ 50). This exponent is close to theoretically predicted g ¼ 1 [1]. The motion along the x direction is limited for longer distances between repulsive stripes but for short dv one can find g ¼ 0.87  0.03 (dv ¼ 5). The motion of the macromolecule on the surface can be visualized in order to confirm our conclusions. As we considered displacements of the chains considerably larger than their size (gcm >> 2 1/2), using/plotting the trajectory of the center of mass seems appropriate. In Fig. 6, the trajectories of the center of mass of the chain are presented for several values of dv. Since the motion perpendicular to the surface (along the z axis) is suppressed, we considered the xy contribution only. For short distances between the repulsive strips (dv ¼ 5), the polymer chain moves in all directions (for the reasons described above) and the reflection of the repulsive pattern cannot be observed in this trajectory (Fig. 6, panel (a)). For dv ¼ 15, the motion of the chain along the repulsive strips dominates, and the center of mass is often located in the adsorbing areas, but the crossings of the repulsive strips are frequent (Fig. 6, panel (b)). A further increase in the distance between the repulsive strips (Fig. 6, panel (c), dv ¼ 15 e the size of the chain is still greater than the distance dv) leads to a confinement of the macromolecule in the attractive areas for a longer period of time. For the longest distance between the repulsive strips under consideration (Fig. 6, panel (d), dv ¼ 50), the dynamic behavior of the macromolecule is similar, but the number of jumps over the repulsive strips was considerably reduced, and the fast motion at short distances along the x-axis was observed. 3. Conclusions In this paper we present selected initial results concerning computer simulations of star-branched polymer chains adsorbed on a flat surface. The macromolecules were embedded in a simple cubic lattice with the united atom representation of chains. The system was in good solvent conditions and the excluded volume was the only intra-chain potential. The adsorbing surface was patterned with narrow and parallel repulsive strips. The properties of the model system were calculated using Monte Carlo simulations with a Metropolis-like sampling algorithm. It was shown that the introduction of the attractive surface changed the size and structure of chains in a significant way e strong adsorption led to a two-dimensional polymer structure. The differentiation of surface points by the introduction of the repulsive strips led to further changes in the size of the chain. These changes were dramatic only for very short distances between the repulsive strips. For longer distances between the strips, the chain was able to accommodate in the attractive areas. Non-monotonic changes of the size of the chain with the distance between the repulsive strips were rather unexpected, but they appeared to be almost independent of the chain length. A similar universal behavior was found for macromolecules in confinement, where the ratio of the size of the slit to the chain diameter was found to be a good normalization factor [20]. The use of a coarse-grained model allowed us to study long-time dynamics of the polymeric system. The dynamic properties of chains were also affected by the presence of the repulsive strips. The motion in the direction perpendicular to the strips was fastest for short distances between the strips, while the motion along the strips was fastest for the longest distances under

consideration. The mobility of chains was also found to depend non-monotonically on the distance between the repulsive strips, and we found the fastest motion across the repulsive strips to occur for very short distances between them. Acknowledgments Helpful discussion with Prof. Andrey Milchev from Johannes Gutenberg University in Mainz is gratefully acknowledged. The computational part of this work was performed using the computer cluster at the Computing Center of the Department of Chemistry at the University of Warsaw. This work was supported by the Polish Ministry of Science and Higher Education under grant N N507 326536. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

Eisenriegler E. Polymers near surfaces. Singapore: World Scientific; 1993. Scheutjens JMHM, Fleer GJ. J Phys Chem 1979;83:1619e35. Scheutjens JMHM, Fleer GJ. J Phys Chem 1980;84:178e90. Semenov AN, Bonet-Avalos J, Johner A, Joanny JF. Macromolecules 1996;29: 2179e96. Kritikos G, Terzis AF. Polymer 2007;48:638e51. Milchev A, Binder K. Macromolecules 1996;29:343e54. Baschnagel J, Binder K. J Phys I (Fr) 1996;6:1271e94. Pandey RB, Milchev A, Binder K. Macromolecules 1997;30:1194e204. Maier B, Rädler JO. Phys Rev Lett 1999;82:1911e4. Sukhishvili SA, Chen Y, Müller JD, Gratton E, Schweizer KS, Granick S. Nature 2000;406:146. Sukhishvili SA, Chen Y, Müller JD, Gratton E, Schweizer KS, Granick S. Macromolecules 2002;35:1776e84. Likos CN. Soft Matter 2006;2:478e98. Freire JJ. Adv Pol Sci 1999;143:35e112. Kosmas MK. Macromolecules 1990;23:2061e5. Ohno K, Binder K. J Chem Phys 1991;95:5444e58. Ohno K, Binder K. J Chem Phys 1991;95:5459e73. Sikorski A. Macromol Theor Simul 2001;10:38e45. Sikorski A. Macromol Theor Simul 2002;11:359e64. Charmuszko K, Sikorski A. J Non-Crystal Solids 2009;355:1408e13. Romiszowski P, Sikorski A. J Chem Phys 2005;123:104905. Kritikos G, Terzis AF. Polymer 2008;49:3601e9. Kritikos G, Terzis AF. Polymer 2009;50:5314e25. Rockford L, Liu Y, Mansky P, Russel TP. Phys Rev Lett 1999;82:2602e5. Lei S, Wang C, Wan L, Bai C. J Phys Chem B 2004;108:1173e5. Shi F, Wang Z, Zhao N, Zhang X. Langmuir 2005;21:1599e602. Denis FA, Pallandre A, Nysten B, Jonas AM, Dupont-Gillain CC. Small 2005;1: 984e91. Pallandre A, De Meersman B, Blondeau F, Nysten B, Jonas AM. J Am Chem Soc 2005;127:4320e5. Wang Y, Liu Z, Huang Y. Langmuir 2006;22:1928e31. Glynos E, Chremos A, Petekidis G, Camp PJ, Koutsos V. Macromolecules 2007; 40:6947e58. Petera D, Muthukumar M. J Chem Phys 1998;109:5101e7. Balazs A, Singh C, Zhulina EB. Macromolecules 1998;31:6369e79. Seok C, Freed KF, Szleifer I. J Chem Phys 2004;120:7174e82. Kriksin YA, Khalatur PG, Khokhlov AR. J Chem Phys 2005;122:114703. McNamara J, Kong CY, Muthukumar M. J Chem Phys 2002;117:5354e60. Semler JJ, Genzer J. J Chem Phys 2003;119:5274e80. Cerda JJ, Sintes T. Biophys Chem 2005;115:277e83. Jayaraman A, Hall CK, Genzer J. J Chem Phys 2005;123:124702. Patra M, Linse P. Macromolecules 2006;39:4540e6. Chen H, Peng C, Ye Z, Liu H, Hu Y, Jiang J. Langmuir 2007;23:2430e6. AlSunaidi A. Macromol Theor Simul 2007;16:86e92. Hoda N, Kumar S. Langmuir 2007;23:1741e51. Chen H, Peng C, Sun L, Liu H, Hu Y, Jiang J. Langmuir 2007;23:11112e9. Al Sunaidi A. Macromol Theor Simul 2008;17:319e24. Koutsioubas AG, Vanakaras AG. Langmuir 2008;24:13717e22. Kolinski A, Skolnick J. Adv Chem Phys 1990;78:223e78. Sikorski A. Macromol Theor Simul 1993;2:309e18. Binder K, Müller M, Baschnagel J. In: Kotelyanskii MJ, Theodorou DN, editors. Simulation methods for polymers. New York: M. Dekker; 2004. p. 25e47.