## Introduction

Polymers can be adsorbed on surfaces/interfaces through physical adsorption or chemical bonding. Adsorption of flexible polymers can significantly and permanently modify the properties of surfaces1. Simultaneously the attractive surface also changes the conformational and dynamic properties of polymers2,3,4. The properties of adsorbed polymers play important roles in many chemical and biological applications1,5,6,7, such as polymer nanocomposite materials8, compatibilization by copolymers9,10,11, colloid stabilization by polymeric surfactants12,13, coating and lubrication14, and DNA segregating in bacteria15 and packaging in viruses16. The adsorption of polymers on surfaces/interfaces and the dynamical properties of adsorbed polymers have motivated extensive studies in experiment3,17,18,19,20,21,22,23,24,25, theory and simulation2,4,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49.

The self-avoiding walk (SAW) model polymer on the simple cubic (sc) lattice was extensively used for studying the adsorption of polymers in theory as well as in simulations27. The surface is usually simply assumed to be infinitely large, flat, and homogeneous. In this model system, every walk contacting with the surface is assigned a polymer-surface interacting energy –EPS. With increasing the surface attraction strength EPS (the same as decreasing temperature), the polymer exhibits a phase transition from a desorbed state to an adsorbed state at the critical attraction (temperature). The critical attraction (temperature) was usually named as the critical adsorption point (CAP)27,31,35,41, at which the free energy of an adsorbed polymer is equal to that of a free polymer in solution26,37. At temperature below the CAP, the polymer is partially adsorbed on the surface and the dynamics of the polymer is confined. While at very low temperature far below the CAP, the polymer can be regarded as a fully adsorbed polymer30. The fully adsorbed polymer can be simply regarded as a two-dimensional (2D) polymer, while the desorbed polymer at temperature above the CAP can be regarded as a free polymer in three-dimensional (3D) dilute solution.

It was found that the dynamic properties of a polymer, such as self-diffusion coefficient and rotational relaxation time, were significantly influenced by the attractive surfaces. The translational diffusion parallel to the surface was slowed down obviously after the polymer was adsorbed on the surface25,30,31,32,50. However, there were contradictory results about the scaling exponent α of the translational diffusion coefficient Dxy with polymer length N, Dxy ~ N−α. It was pointed out that the exponent α was dependent on system temperature for the adsorbed SAW lattice polymer in the absence of intra-polymer attraction, specifically α was changed from 1 for partially adsorbed polymers to 2 for fully adsorbed ones30. However, simulation on an off-lattice polymer model showed that α was roughly independent of the attraction strength of the surface for the polymer in the absence of intra-polymer attraction31. In that case α ≈ 1.1 was estimated for adsorbed polymers31. While α ≈ 1.1 was also found for both desorbed polymer and fully adsorbed polymer using lattice polymer model32. But a small value of α ≈ 0.8 was estimated for a partially adsorbed SAW polymer at temperature near CAP32. Experiment on surface diffusion of Poly(ethylene glycol) found that the diffusion of polymers was slowed down and the value of α was increased obviously after adsorption22. Similarly, experiment showed that DNA diffused slower in agarose gels than in solution, and α was increased obviously by confinements51.

On the other hand, studies have revealed that the rotational relaxation time τR of a polymer became larger when the polymer was end-grafted33,52 or adsorbed on surfaces25,30,31,32,43. It was found that τR increased with the attraction strength of surface for the partially adsorbed polymers30,31,32,43. In dilute solution the SAW polymer chain follows Rouse dynamics as τR ~ Nβ with the exponent β = 1 + 2ν = 2.2. Hahn and Kovac found that the presence of an impenetrable surface did not change the scaling of the rotational relaxation times with chain length for end-grafted polymers52. Although τR was increased with a decrease in the temperature, β was found to be roughly independent of temperature30. While simulation on BF polymer showed that for a partially adsorbed polymer β lied in the range 2.2 < β < 2.5, that is, as the temperature fell through the adsorption transition, a smooth crossover from 3D to 2D dynamics took place32. Off-lattice Monte Carlo (MC) simulation found that the relaxation time of the partially adsorbed polymer could be scaled with a large β ≈ 2.6531. However, there is no experimental data about the behavior of β for adsorbed polymers.

In addition to the polymer-surface attraction EPS, the intra-polymer attraction EPP is usually taken into account for more realistic polymer model. When two non-bonded monomers locate at nearest neighbor (NN) sites on the lattice, an attractive energy −EPP is assigned. With an increase in the intra-polymer attraction strength EPP, polymer exhibits a phase transition from an extended coil conformation with large statistical size to a compact globule with small statistical size. Therefore rich phases were found for the adsorbed polymer on surfaces after EPP was taken into account41,53,54. However, the dynamics of the adsorbed polymer is not fully understood, especially when the intra-polymer interaction is taken into account.

In this work, we present our study on the dynamics of a physically adsorbed polymer on an attractive, homogeneous surface using dynamic MC simulation. We mainly study the influence of EPS and EPP on the translational diffusion and rotational relaxation of the adsorbed polymer. Results show that both the translational diffusion coefficient Dxy parallel to the surface and the rotational relaxation time τR are dependent on EPS and EPP. We find that Dxy decreases with increasing EPS or EPP. The scaling exponent α in Dxy ~ N−α is roughly independent of EPS for long polymers, but increases with EPP. However the behavior of τR is much more complicated. Although τR always increases with EPS, it decreases with increasing EPP for the partially adsorbed polymer and increases with EPP for the fully adsorbed polymer. And the scaling exponent β in τR ~ Nβ always decreases with increasing EPP. We find that β is roughly independent of EPS for the adsorbed polymer at EPP = 0, but β increases with EPS at EPP > 0.

## Simulation model and method

### Model

We consider a coarse-grained linear polymer model on the sc lattice. The simulation box is a cuboid with the length Lx, Ly, and Lz in x, y, and z directions, respectively. The polymer chain of length N is composed of N identical monomers numbered sequentially from 1 to N. Here, a monomer corresponds to a small group of atoms instead of a specific atom in the polymer, and a bond represents the linkage between two monomers rather than a specific covalent bond between two atoms. Each monomer occupies one lattice site and each lattice site cannot be occupied simultaneously by two or more monomers. The bond length can be varied among 1, , and in the present model. In addition to the self-avoiding on the sc lattice, the intra-polymer NN monomer-monomer attraction −EPP is considered. In this work EPP = 0 means only self-avoiding effect is taken into account for the polymer chain, that corresponding to an ordinary SAW polymer. An additional attraction is introduced in the polymer when EPP > 0. And the statistical size of the polymer, e.g. the mean-square radius of gyration 〈RG2〉, is decreased with increasing EPP.

For the polymer in 3D dilute solution, periodic boundary conditions (PBCs) are adopted in all the three directions. While for the adsorbed polymer, PBCs are only adopted in the x and y directions. An infinitely large, impenetrable, homogeneous, flat surface is placed at z = 0, which produces a homogeneous attraction to the polymer. When monomers are at the NN layer of the surface, i.e. at z = 1, an attractive polymer-surface attraction −EPS is assigned. Therefore the energy of the polymer can be described as

Here nPP is the number of intra-polymer NN pairs and nPS is the surface contact number of polymer.

### Simulation method

The dynamics of polymer chains is achieved by adopting bond fluctuation method invented by Carmesin and Kremer55 and Metropolis algorithm. The global dynamics of polymer chains is achieved by a huge number of local displacements of monomers resulting from random collisions between chain monomers and solvent molecules. For each local displacement, a monomer is chosen randomly and attempted to move one lattice spacing selected randomly from its six NN sites. If the move satisfies: (i) the new site is empty, (ii) bonds don’t intersect, and (iii) bond lengths vary among 1, , and , it will be accepted with a probability p = min[1, exp(−ΔE/kBT)], where ΔE is the energy shift due to the change of monomer’s site. In simulation, we count the change of numbers nPP and nPS for every move, and thus obtain the energy shift ΔE. During one MC step (MCS), every monomer attempts to move once on average. In this work, MCS is the unit of time. And the unit of energy is kBT with kB the Boltzmann constant and T the temperature.

At the beginning of every simulation run, a polymer is generated and equilibrated for about N2.2 (≈N1+2ν) MCS between a virtual surface at z = Lz/4 and a top surface at z = Lz. The virtual surface is repulsive to polymer. With the virtual surface, polymer is equilibrated before adsorption. Then we remove the virtual surface and let the polymer undergo trial moves until it contacts with the bottom adsorbing surface and settles into the adsorbed equilibrium state. At last we let the polymer random diffuse for sufficiently long time which is used for recording the polymer conformational properties. We perform 5000 independent runs with different initial configurations and random number series. Simulation results are averaged over these 5000 independent samples and the simulation errors in the measured parameters are small.

## Results and Discussions

We study the dynamics of a single adsorbed polymer on a flat surface. Simulations are carried out at kBT = 1 below the CAP of the SAW polymer at EPP = 0 and EPS = 1, which was estimated to be about 1.62556. And it was pointed out that the CAP was increased with an increase in EPP48,53. Our simulations are performed at EPP ≥ 0 and EPS ≥ 1, therefore the polymer is always adsorbed during the whole statistical time46. However, at such low temperature, the polymer chain can still diffuse and adjust its configuration50. Since the attraction is not strong enough in this work, the number of adsorbed monomers is less than the polymer length, therefore the adsorbed polymer is called as a partially adsorbed polymer.

In addition to the partially adsorbed polymer, there are two other states for polymers, i.e. a fully adsorbed polymer and a desorbed polymer. A fully adsorbed polymer would appear at high EPS. In the present work a 2D polymer model on surface is adopted as the fully adsorbed polymer. While a desorbed polymer appears at low EPS, and thus a 3D polymer in dilute solution is adopted for the desorbed polymer.

The polymer also exhibits a coil-globule transition in solution as well as on adsorbing surface57,58. For the present model, the coil-globule transition is occurred at about EPP = 0.5, although the transition point increases slightly with EPS48,53,57.

### Translational diffusion of partially adsorbed polymer

We have measured the translational diffusion for the adsorbed polymer chain. The diffusion is characterized by the mean-square displacement (MSD) of the center of mass

Here < > denotes an ensemble average over independent samples, and rcm(t) is the position vector of the center of mass of polymer chain at time t and rcm(0) at initial time t = 0. For the partially adsorbed polymers, the diffusion in the surface normal direction can be neglected and thus the diffusion is mainly confined parallel to the surface. Therefore, we have the parallel component 〈Δr2(t)〉xy ≈ 〈Δr2(t)〉 for the partially adsorbed polymers. In this sense, the partially adsorbed polymer can be regarded as a quasi-2D one.

Simulation results show that MSD increases linearly with time for the partially adsorbed polymer, analogous to the free polymer in 3D dilute solution. As an example, we show the time dependence of MSD in the inset of Fig. 1. Thus, the translational diffusion coefficient Dxy for such a quasi-2D system can be obtained through

While for the polymer in 3D dilute solution, we have Dxy = D due to the isotropy of our simulation system. We find that the translational diffusion coefficient Dxy is dependent on the simulation parameters, such as intra-polymer attraction EPP, polymer-surface attraction EPS, and polymer length N.

We have studied the influence of the intra-polymer attraction EPP on the translational diffusion coefficient Dxy. Figure 1 shows the dependence of Dxy on EPP for the polymer adsorbed on surface at EPS = 1, 1.5, and 2. We find that the simulated data for N = 8, 16, 32, 64, 96, 128 are qualitatively similar, so here we only present the results for N = 64. It is clear that Dxy decreases smoothly with increasing EPP. It is known that polymer compacts itself or monomers aggregate with the increase in EPP. This decreases the mobility of monomers and thus decreases the diffusion coefficient of the whole polymer chain.

On the other hand, from Fig. 1 we note that Dxy also decreases with an increase in the polymer-surface attraction strength EPS for all EPPs. Since the number of adsorbed monomers increases with EPS, the results indicate that the translational diffusion is obviously slowed down with an increase in the number of adsorbed monomers. Because of confinement introduced by the surface, the mobility of adsorbed monomers is lower than that of desorbed monomers, thus the diffusion coefficient decreases with increasing EPS.

To further understand the diffusion behavior of the partially adsorbed polymers, the diffusion constants of the polymer in 3D dilute solution and on 2D surface are also calculated. The polymer in the 3D dilute solution corresponds to the utmost limit of weak attraction of the surface and the polymer on the 2D surface corresponds to the utmost limit of strong attraction of the surface. The results of the 3D and 2D diffusions are also presented in Fig. 1. We find that 3D diffusion is faster than 2D one. The reason is that the 2D polymer suffers stronger excluded volume (EV) effect, thus the mobility of 2D polymer is lowered down.

It is clear to see that the diffusion of adsorbed polymer locates between the 3D and 2D cases. That is, the adsorbed polymer diffuses slower than the 3D one in dilute solution, but faster than the 2D one on surface. In other words, the adsorbing surface prevents the diffusion of polymers, and the diffusion becomes slow as the adsorption strength EPS increases. However, such a difference in Dxy dies away with the increase in EPP. For example, at EPP = 1, one can see that Dxy is roughly independent of EPS.

It was pointed out that the polymer length N dependence of D scales as D ~ N−1 for the Rouse model when hydrodynamic interactions are ignored for ideal random walk chains59. We have checked the scaling relation

for polymers in different situations. Figure 2 presents the dependence of Dxy on N for polymers in dilute solution, partially adsorbed on surface, and fully adsorbed on surface with different intra-polymer interactions.

For the SAW polymers without intra-polymer attraction (EPP = 0), we find that Dxy can be scaled with N in the power law relation Dxy ~ N−α and the apparent exponent is found to be α ≈ 1.06 for all cases, i.e., it is roughly independent of EPS. The value α ≈ 1.06 indicates that the dynamics of SAW polymers is of Rouse type. It is also in agreement with earlier simulation results for adsorbed polymer chains in good solvent31.

When the intra-polymer attraction is considered, such as EPP = 1, the apparent exponent α becomes larger than that at EPP = 0. At EPP = 1, polymer becomes compact and the degree of compact increases with the chain length N. Therefore, the diffusion coefficient is further decreased with increasing N, resulting in a larger exponent α at larger EPP. For polymer on 2D surface, we find the simulation results can be well expressed by the scaling law Dxy ~ N−α with α ≈ 1.66.

Whereas for the desorbed polymer in 3D dilute solution, the dependence of Dxy on N is of polymer length dependence at EPP = 1 and 2. We find that Dxy decreases faster for short polymer length, i.e. the apparent exponent α estimated from short polymer region is larger. However, for long polymer region, it is interesting to see that the apparent exponent α is roughly the same as that of polymer on 2D surface. For partially adsorbed polymers, the diffusion coefficient lies between that of 3D and 2D cases. At EPS = 1, we find the diffusion coefficient is close to the 3D one for short polymer lengths while that is close to the 2D one for long polymer lengths. While at EPS = 2, the diffusion coefficient is close to 2D one even for short polymer lengths. Therefore we conclude that the apparent exponent α of the partially adsorbed long polymers is the same as that of polymer on 2D surface. A big value α ≈ 2.25 is estimated at EPP = 2.

Figure 2 shows that the apparent exponent α of long polymer is only dependent on EPP but is independent of EPS. Figure 3 presents the apparent exponent α for the diffusion of long polymer. We find that α increases smoothly with EPP. As already shown in the above discussion, the decrease of Dxy with increasing EPP is due to the fact that more and more monomers become close to each other. The longer the chain length is, the stronger the effect is. So the apparent exponent α increases with EPP.

In short, the diffusion of adsorbed polymer on the homogeneous surface is obviously slowed down by the attraction of surface. This is in agreement with experimental observations22,25,51 and other simulations30,31,32. Our results show that the exponent α in the scaling law Dxy ~ N−α is independent of the surface attraction, however, this is inconsistent with experimental results22. We also observe a new result that α increases with the intra-polymer attraction.

We notice that the experiment on the surface diffusion of Poly(ethylene glycol) found that the scaling exponent α was increased obviously after adsorption22. The reason for such an increase is complex and is not clear22. Since our simulation results show that the homogeneous surface does not affect the scaling exponent α, one of the possible reasons for the change of α is that the surface in experiment is heterogeneous. A heterogeneous surface with non-uniformly distributed attraction points will produce a parallel barrier for the diffusion, and that may change the diffusion behavior as that of polymer in crowded environment. It was found that value α for the diffusion of DNA in agarose gels was increased obviously51. The diffusion of polymer on heterogeneous surface deserves further study. Another possible reason is that the hydrodynamic interaction (HI) effect, which was included in experiment, is not considered in the present work. A surface can change the HI effect because of less solvent molecules near surface. It will be also interesting to simulate the HI effect on the diffusion of adsorbed polymers in future.

### Rotational relaxation of partially adsorbed polymer

In this subsection, we study the rotational relaxation of the partially adsorbed polymer chain on homogeneous surface. We have calculated the end-to-end vector rotational autocorrelation function of the polymer29,30,52,60

where R(t) is the end-to-end vector of a polymer chain at time t and < > also means an average over the ensemble of independent samples. Here ρ(t) describes the degree of rotation for polymer. For the partially adsorbed polymer chain at temperature below CAP, the end-to-end distance vector R is roughly parallel to the surface, thus ρ(t) describes the rotation parallel to surface while the contribution from the surface normal direction can be neglected. It is expected that ρ(t) decays as

thus one can estimate the rotational relaxation time τR by fitting the decay of ρ(t) to an exponential decay function. Figure 4 presents the simulation results of ρ(t) for N = 64 at EPS = 1 and EPP = 0, 0.3, and 0.5. One can see that ρ(t) decays exponentially in a relatively long time region. However, the width of the exponentially decay region is dependent on the interaction parameters. In this work, τR is estimated from the decay of ρ(t) in the region 0.06 < ρ(t) < 0.30 where the best fitting can be always obtained29.

It is interesting to see that the rotational relaxation time τR of the partially adsorbed polymer decreases as the intra-polymer attraction EPP increases. This is possibly because the intra-polymer attraction reduces the size of polymer. Then the moment of inertia of polymer for rotation, which is in proportion to the size of polymer, decreases with increasing EPP, too. Therefore the intra-polymer attraction accelerates the rotation of polymer. However, with further increase in EPP, the conformation of polymer chain will be changed from a random coil state to a compact globule state53. We find that it is difficult to estimate τR for polymer in the globule state at EPP > 0.5. Therefore we restrict EPP < 0.5 in this work.

We have investigated the influence of surface attraction and intra-polymer attraction on τR for the partially adsorbed polymer. The values of τR at different EPP and EPS are simulated. Figure 5 presents the dependence of τR on EPP for N = 64 at EPS = 1, 1.5, 2, and 2.5. For comparison, results for polymer chains in dilute solution (desorbed polymer) and on 2D surface (fully adsorbed polymer) are also presented. We find that τR increases with EPS for the partially adsorbed polymer, in agreement with previous simulation results30,31,32,43 and experiment25. The results show that the attractive surface obviously slows down the rotation of polymer.

It is interesting to notice that different behaviors of τR are observed for polymer in 3D dilute solution and on 2D surface, respectively. We find that τR decreases with an increase in EPP in 3D dilute solution, but increases with EPP on 2D surface. The result implies that the rotational behavior of polymer on surface is different from that in solution.

For the partially adsorbed polymer chain, τR is also dependent on EPP. Figure 5a shows that τR decreases with increasing EPP at small EPS but it increases with increasing EPP at EPS = 2.5. It is clear that the behavior of the partially adsorbed polymer changes from 3D like to 2D like with the increase in EPS. On the other hand, τR increases with EPS, the underlying reason could be explained as follows: the polymer-surface attraction keeps monomers close to the surface and makes it hard to relax rotationally. Figure 5b presents the ratio τR/τR,0 for different EPS, where τR,0 is the rotational relaxation time at EPP = 0. It is clear to see that the effect of surface on the polymer rotation increases with EPS.

We have calculated τR for other polymer lengths at different EPP and EPS. For all polymer lengths, we observe that τR of the adsorbed polymer is larger than that in dilute solution, but smaller than the one on 2D surface. And for all EPP, τR increases monotonically with EPS. So the adsorbing surface not only reduces diffusion as discussed in the previous subsection, but also prevents the polymer from relaxing rotationally.

From Fig. 5 we see that the behavior of the partially adsorbed polymer changes from 3D like to 2D like as the surface attraction strength EPS increases. We have also simulated the rotational relaxation times for different polymer lengths from 8 to 128. For the partially adsorbed polymers, we find that the relationship between τR and EPP is also dependent on the polymer length. Figure 6 presents the dependence of τR on EPP for N = 8, 16, 64, and 96 at EPS = 1. We find that τR decreases with increasing EPP for long polymers but it increases with increasing EPP for short polymers. It is known that, near CAP, the number of adsorbed monomers M increases with N as M ~ Nϕ with the crossover exponent ϕ near 0.527,35. Since the fraction of adsorbed monomers M/N ~ Nϕ−1 decreases with increasing polymer length, it is reasonable to find that long adsorbed polymers behave more like the polymer in dilute solution whereas short ones behave more like that on 2D surface. Shorter polymers are significantly influenced by the adsorbing surface and the corresponding τR increases with EPP.

Figure 7 presents the polymer length dependence of the rotational relaxation time τR for polymers in different situations. We find that the relaxation time can be always scaled as

We estimate β = 2.3 for the simple SAW polymer (i.e. EPP = 0) in dilute solution, indicating that the SAW polymer in dilute solution roughly follows Rouse dynamics with the exponent β = 1 + 2ν3D = 2.2 with the Flory exponent ν3D = 0.6. While we find β = 2.7 for the SAW polymer chain on 2D surface, also close to the expected value 1 + 2ν2D = 2.5 with ν2D = 0.75 for the dynamics of a SAW polymer on 2D surface. It is interesting to find that, for the partially adsorbed SAW polymer in the absence of intra-polymer attraction, i.e. EPP = 0, the exponent β is roughly independent of the surface attraction and is roughly equal to that of a 2D SAW polymer, that is in agreement with earlier finding that β of a partially adsorbed polymer was independent of temperature30. Our value β = 2.7 for the adsorbed polymer is larger than 2.56 for a BF polymer model32 but close to β = 2.65 for an off-lattice polymer model31.

When the intra-polymer attraction is taken into account, for example at EPP = 0.5, the scaling relation (Eq. 7) is still hold, but the exponent β is dependent on the surface attraction. We find β increases with EPS.

We find that the exponent β is dependent on both the surface attraction and the intra-polymer attraction. The exponent β at different EPP and EPS is calculated and the results are presented in Fig. 8. We find that β decreases with increasing EPP. The intra-polymer attraction attracts monomers together, that reduces polymer’s statistical size and accelerates the rotation. As the effect of the intra-polymer attraction on the polymer increases with polymer length, it is reasonable to find β decreases with increasing EPP. On the other hand, the surface will retard the rotation of polymer. Thus we find β increases with increasing EPS.

## Conclusions

The dynamics of a bond fluctuation polymer chain adsorbed on an attractive, homogeneous surface is simulated by using dynamic Monte Carlo simulations. We have investigated the translational diffusion parallel to surface and the rotational relaxation of the partially adsorbed polymer. For comparison, we have also simulated the dynamics for a desorbed polymer chain and for a fully adsorbed one. Results show that the dynamics of polymer is strongly dependent on the intra-polymer attraction EPP and the polymer-surface attraction EPS. However, the effects of the attractive surface on the diffusion and rotation of polymers are different. The main results are summarized in Fig. 9.

For the desorbed polymer in dilute solution, the translational diffusion is reduced whereas the rotation is speeded up with an increase in the intra-polymer attraction EPP.

The translational diffusion of polymer is reduced when it is adsorbed on surface. The translational diffusion coefficient Dxy of the adsorbed polymer also decreases with increasing EPP. The scaling exponent α of Dxy for long polymer length N is independent of EPS, i.e., it is the same for the desorbed polymer, partially adsorbed polymer, and fully adsorbed polymer. However α increases with EPP from α = 1.06 at EPP = 0.

The rotational relaxation time τR of the adsorbed polymer always increases with EPS, indicating that the surface attraction reduces the rotation of the adsorbed polymer. However, the dependence of τR on EPP is dependent on the state of the adsorbed polymer. We find that, with an increase in EPP, τR decreases for the partially adsorbed polymer but increases for the fully adsorbed polymer. The scaling exponent β of τR with N is also dependent on EPP and EPS. At EPP = 0, we find β ≈ 2.7 independent of EPS for the partially adsorbed polymer. But at EPP > 0, we find β increases with EPS. However, β always decreases with increasing EPP.