Untangling the Effects of Chain Rigidity on the Structure and Dynamics

Article pubs.acs.org/Macromolecules

Untangling the Effects of Chain Rigidity on the Structure and Dynamics of Strongly Adsorbed Polymer Melts Jan-Michael Y. Carrillo,*,†,‡ Shiwang Cheng,§ Rajeev Kumar,†,‡ Monojoy Goswami,†,‡ Alexei P. Sokolov,∥,§ and Bobby G. Sumpter†,‡ †

Center for Nanophase Materials Sciences, ‡Computer Science and Mathematics Division, and §Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States ∥ Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, United States ABSTRACT: We present a detailed analysis of coarse-grained molecular dynamics simulations of semiflexible polymer melts in contact with a strongly adsorbing substrate. We have characterized the segments in the interfacial layer by counting the number of trains, loops, tails, and unadsorbed segments. For more rigid chains, a tail and an adsorbed segment (a train) dominate while loops are more prevalent in more flexible chains. The tails exhibit a nonuniformly stretched conformation akin to the “polydisperse pseudobrush” originally envisioned by Guiselin. To probe the dynamics of the segments, we computed the layer z-resolved collective intermediate dynamic structure factor, S(q,t,z), mean-square displacement of segments, and the second Legendre polynomial of the time autocorrelation of unit bond vectors, ⟨P2[ni⃗ (t,z)·n⃗i(0,z)]⟩. Our results show that segmental dynamics is slower for stiffer chains, and there is a strong correlation between the structure and dynamics in the interfacial layer. There is no “glassy layer”, and the slowing down in dynamics of stiffer chains in the adsorbed region can be attributed to the densification and a more persistent layering of segments.

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INTRODUCTION The adsorption of polymers on surfaces is a fundamental problem in polymer physics1,2 which has been extensively examined experimentally,3−12 theoretically,13−22 and through computer simulations.23−35 Particularly of technological interest are systems involving polymer nanocomposites and polymer− electrolyte systems for organic electronic applications. The importance of a thorough understanding of polymer chain behavior in contact with a substrate cannot be overstated. It is known that in the presence of interfaces the structure and dynamics of polymers are influenced by the interface where dynamics differ from bulk and polymer conformations are perturbed within an interfacial layer.7,8,36−47 There is now growing experimental evidence of the formation of the “reduced mobility interface” (RMI) layer in between unadsorbed chains in a matrix and the adsorbed layer where the dynamics is intermediate and distinguishable from those of the bulk and the adsorbed layer.6,37,39,47,48 This leads in particular to the observed shift in the glass transition temperature, Tg.49−51 However, there are still considerable gaps in the understanding of the underlying physics and dependencies on polymer properties and its interaction with the substrate. In general, a fully flexible chain model is inadequate in describing experimental results52 since the backbone of a real polymer chain is not completely flexible. A more accurate description of experimental results can be achieved when another length scale, representing the stiffness of the chain along the molecular backbone, is introduced.24,52−54 This © XXXX American Chemical Society

highlights the importance of chain stiffness in polymer− substrate contacts and interactions. In this article we attempt to describe the structure and dynamics of semiflexible polymer melts at the segment level. We believe that in this framework, coarse-grained molecular dynamics simulations provide considerable insight for understanding the structure and dynamics of polymer melts by explicitly incorporating the substrate and chain rigidity into the model. The article is organized as follows: In the succeeding section we describe the simulation results and discuss the behavior of polymer chains having different degrees of rigidities and compare the properties relative to bulk properties. To accomplish this, we performed z-resolved analyses of the structure and dynamic properties that were investigated in the bulk configuration (z is the distance from the substrate surface). In a recent study by Harton et al.55 it was shown that the interfacial layer thickness around spherical nanoparticles could be smaller than at flat surfaces, suggesting that the thickness of the interface can be tuned by changing the curvature of the surface (e.g., changing nanoparticle size). In this study we show that the same effect can be achieved by changing the rigidity of the polymer. Hence, we also looked into the size of the perturbed or interfacial layer as a function of persistence length. Finally, we summarize our results in the Conclusions section. Received: March 25, 2015 Revised: May 8, 2015

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DOI: 10.1021/acs.macromol.5b00624 Macromolecules XXXX, XXX, XXX−XXX

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SIMULATION METHOD

and its relative size to the persistence length and segment size. In this work, we are investigating the limit where lp ≫ d. More details of the simulations are provided in Appendix A. Prior to simulating the slab configuration, we performed simulations of polymers in the bulk configuration (without the adsorbing substrate and with periodic boundaries in all directions). This set of simulations serves as a reference for comparison between the two simulation sets. We show that the simulation results in the bulk configuration for both static and dynamic properties follow qualitatively known theoretical and simulation results for flexible and semiflexible polymer chains. These simulation results are provided in Appendix B, which includes the calculation of bulk structural properties such as the mean square end-to-end distance, ⟨R2⟩, mean-square radius of gyration, ⟨Rg2⟩, and the single chain scattering form factor, P(q). Thereafter, we characterized the bulk dynamic properties by computing the intermediate single chain and collective dynamic structure factors (Ssc(q,t) and Scol(q,t), respectively), the segment mean-square displacement, ⟨|r(⃗ t) − r(⃗ 0)|2⟩ (MSD), and the time-autocorrelation function of the second Legendre polynomial of unit bond vectors, ⟨P2[n⃗i(t)·n⃗i(0)]⟩.

In this work we have performed molecular dynamics simulations of polymer melts based on the work of Kremer and Grest (KG)56 with chains having a degree of polymerization, N = 50, and bead number density, ρb = 0.85σ−3, as illustrated in Figure 1.

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RESULTS AND DISCUSSION Structure in the Interfacial Layer. We begin our discussion of the structure of the adsorbed polymers by first defining the interface layer as the region between the hard interface (substrate) and the sof t interface (bulk polymers), where the polymer structure and dynamics deviate from its bulk properties. The polymers that belong to this region could be classified as either adsorbed or unadsorbed. We identified adsorbed segments as segments having a z coordinate that is less than 1.161σ. This value is a little over 21/6σ, which is the minimum of the LJ 6−12 substrate potential. A polymer is considered adsorbed if any of its segments falls in this z coordinate range. Overall, we found that the fraction of segments adsorbed on the substrate per chain decreases as a function of the z distance for the center-of-mass of the chain. Initially this function follows an exponential decay in the vicinity of the substrate and abruptly decreases at z values near the region where chain properties are bulk-like. Further classifications of the beads in this region include loops, tails, and f ree segments. Loops are connected and unadsorbed segments where both ends are adsorbed. An adsorbed bond is a loop with a degree of polymerization equal to 1 and is classified as a train. Tails are connected and unadsorbed segments where one end is free and the other is adsorbed, while a free segment is a segment that belongs to an unadsorbed chain. The definitions for loops, trains, and tails are consistent with the definitions in the reflectometry study by Léger et al.5 and theoretical work by Guiselin17 for which our simulation results were compared. We have found that with increasing lp at a constant εw more beads were adsorbed on the surface, as indicated by the increasing peak amplitude of the oscillating segment density, ρmon(z), shown in Figure 2a. This result is in agreement with previous simulations.32,33 The period of oscillation in ρmon(z) also increases as lp increases, which is indicative of the prominence of the second characteristic length scale lp over the bond length b. And, this prominence manifests in the packing of segments and bonds in the direction normal to the substrate. Although more beads are adsorbed on the substrate, this is compensated with an increase in the density of unadsorbed or free beads, ρfree(z), as seen in Figure 2b. For the flexible chain, the peaks of the oscillations in ρfree(z) are dampened and follow

Figure 1. (left) Cross sections of the simulation box showing polymers that are bulk-like in blue and adsorbed polymers in red; other space filling polymers are excluded in the image. (right) A bending potential ((black ●) Kbend = 0; (red ■)Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; (brown ▽) Kbend = 6) is applied to neighboring unit bond vectors, n⃗i, increasing the rigidity of the polymer chain and the persistence length, lp, which is proportional to the exponential decay constant of the bond−bond correlation function.

The simulations were performed using LAMMPS57 with GPU acceleration58 at the Oak Ridge Leadership Computing Facility (OLCF). The KG model has been widely used to simulate polymer melts above the glass transition temperature, Tg, and is known to have an entanglement length, Ne = 35. We have introduced chain rigidity to the model by applying a bending bend potential energy to adjacent polymer bonds, Ui,i+1 = kTKbend(1 − (n⃗i·ni⃗ +1)), where ni⃗ is a unit bond vector along the polymer backbone determined by eq A4 in Appendix A. Rigidity is increased by increasing the value of Kbend, and the persistence length, lp, is evaluated by fitting the bond−bond correlation function of the unit bond vectors with an exponential decay, ⟨n⃗i·n⃗i+l⟩ = exp(−lb/lp), where b is the average bond length.1 This scheme is similar to the works of Faller59 and Riggleman. 60 In the simulations, we applied periodic boundaries in the x and y directions while attractive shortranged (cutoff = 2.5σ) 12−6 Lennard-Jones substrates are located at the top and bottom of the simulation box. The substrate to segment interactions, εw, ranges from 1.0 to 8.0 kT. We have ensured that the z dimension is large enough such that the chains situated at the middle of the slab are bulk-like. Despite employing a smooth analytical substrate, the results obtained in our MD simulations, such as the layering of the segments in the vicinity of the substrate, qualitatively agree with the results of the MD simulations by Vilges et al.23,29 where the substrate was setup as discrete Lennard-Jones beads. Furthermore, we believe that real systems have discrete adsorption points (i.e., hydrogen-bonding points); hence, other parameters are critical in determining the adsorbed polymer structure.61 These include the distance between the adsorption points, d, B


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Figure 2. Bead number densities for total number of segments (a), free beads (b), trains and loop (c), and tails (d) for adsorbed polymers with εw = 8.0 kT and different values of Kbend (Kbend = 0 (black); Kbend = 1 (red); Kbend = 2 (blue); Kbend = 3 (gray); Kbend = 4 (green); Kbend = 5 (magenta); Kbend = 6 (brown). The black dotted line is the bulk density, ρb, and the dotted-cyan-line at (b) corresponds to eq 1. The inset in (a) are the monomer densities for z > 1σ to z < 3σ, and the inset in (c) shows the loop densities for z > 1σ. The red arrow on each plot indicates the direction of the change in density as Kbend increases.

the description proposed by de Gennes2 (see cyan-dotted line in Figure 2b) ⎞ ⎛ z ⎟ ρfree (z) = ρb tanh2⎜⎜ 2 1/2 ⎟ ⎝ ⟨R g ⟩ ⎠

to be around one, and a peak is found for chains with a centerof-mass near the substrate having low values lp. Hence, the prevalent conformation of the chains is that of many short adsorbed loops, comprising a train, and a single long tail for chains with high values of lp, while loops are only observed near the substrate for chains with low values of lp. In Appendix C, we demonstrate that a semiflexible chain in the melt can be rescaled to a flexible polymer by increasing the statistical segment length. We were able to show that the chain center-of-mass distribution and chain-start (where chain “start” is defined as the chain end closest to the surface) probability distribution for different values of Kbend collapses to a single curve (see Figure 12) and probability distributions of the number of Nloop and Ntail for different values of Kbend follows the same power law scaling (see Figure 13). This rescaling procedure corroborates with the results found in selfconsistent field models16,19 with the caveat that this procedure is not applicable for dilute systems because the characteristic ratio is not conserved during rescaling as shown by van der Linden et al.19 Further description of the adsorbed layer showing the structure of the layer as a polydisperse pseudobrush (see Figure 14) and the adsorbed chain as having tails that are nonuniformly stretched (see Figure 15) are also presented in Appendix C. Nematic ordering of the trains in the adsorbed layer is prevalent for systems with high values of lp; hence, the persistent segments of chains in these layers have higher persistence length in comparison to the bulk lp34 (see Figure 4a). Also, this is indicative that there is improvement of segments packing25 on the substrate, and more segments are adsorbed or belong to trains. To determine the orientation of the bond vectors along the polymer chain, we considered the second Legendre polynomial of the unit bond vector to the positive z-axis, ⟨P2(n⃗i(z)·z+)⟩ (see Figure 4b), and defined as

(1)

The beads belonging to trains (Nloop = 1) are found in the first adsorption layer, while the density of beads belonging to loops, ρloop(z), found in the next adsorption layers, has lower peak amplitudes as lp increases. This indicates that loops are less favored and the tail density, ρtail(z), is lower for stiffer chains (see Figure 2c,d). Despite having lower ρtail(z), tails extend farther from the substrate for stiffer chains. Because of the presence of the loops and tails, we can envision the structure of the interfacial layer as a polydisperse pseudobrush, which was first proposed by Guiselin.17 The average number of loops per chain in Figure 3, ⟨Nloop⟩, is found

Figure 3. Average number of segments in a loop, ⟨Nloop⟩ or a tail, ⟨Ntail⟩ of a chain with the center-of-mass located at z distance away from the substrate for adsorbed polymers with εw = 8.0 kT and different values of Kbend ((black ●) Kbend = 0; (red ■) Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; (brown ▽) Kbend = 6). The snapshots are typical configurations of adsorbed polymer chains with Kbend = 2, εw = 8.0 kT and with z center-of-mass location near the substrate (z ∼ 0) and far from the substrate (z > 0).

⟨P2[ni⃗ (z) ·z+]⟩ =

1 [3(n ⃗(z)·z)̂ 2 − 1] 2

(2)

where z pertains to the normal distance of the midpoint of ri⃗ − ri⃗ −1. It is found that there is layering of the bond orientation C


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Figure 4. (a) Top view of the adsorbed polymers with Kbend = 3, εw = 8.0 kT, and z center-of-mass location that is less than 4σ away from the substrate (z ≤ 4σ). The inset shows the same viewpoint featuring beads of the same color belong to the same chain. (b) Average second Legendre polynomial of the dot product of a unit bond vector to the positive z-axis, ⟨P2n⃗i(z)·z+⟩ for εw = 8.0 kT and different values of Kbend (Kbend = 0 (black); Kbend = 1 (red); Kbend = 2 (blue); Kbend = 3 (gray); Kbend = 4 (green); Kbend = 5 (magenta); Kbend = 6 (brown)). The data for Kbend = 6 was truncated where the plot reaches zero at z = 25σ. The red arrow indicates the direction of the change in ⟨P2(n⃗i(z)·z+)⟩ as Kbend increases.

Figure 5. Dynamic properties of adsorbed semiflexible chains within z > 1σ to z < 2σ interval away from a substrate having εw = 8 kT and chains with different Kbend ((black ●) Kbend = 0; (red ■) Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; (brown ▽) Kbend = 6. (a) Normalized collective intermediate dynamic structure factor at q = 6.51σ−1. (b) Mean-square-displacement of segments. (c) Time-autocorrelation function of the second Legendre polynomial of unit bond vectors. 1τ is approximately 1 ns (τ ≈ ns).

along the axis perpendicular to the plane where the bond vector orientation oscillates from being preferentially parallel (⟨P2(ni⃗ (z)·z+)⟩ = −0.5) to perpendicular (⟨P2(n⃗i(z)·z+)⟩ = 1.0) to the substrate plane. As lp increases, the bond vector prefers the orientation that is parallel to the substrate plane and the range of the region in between the adsorbed orientation and that of the isotropic orientation (⟨P2(n⃗i(z)·z+)⟩ = 0) becomes longer as lp increases. Adsorbed Dynamic Properties. We are interested in understanding the gradient of the polymer segmental dynamics as a function of distance from the surface. To achieve this, we investigated the dynamics of the beads and the rotation of the bonds at the interfacial layer by calculating the z-resolved analogues of Figure 11(b−d). In calculating the z-resolved collective intermediate dynamic structure factor in Figure 5a, we calculated the spectral density of a slab with the thickness, Δz = 1σ, as a function of z and t using eq B4 in Appendix B and then calculated the time autocorrelation function of the spectral density. The behavior in Figure 5a, which is the normalized Scol(q,t,z) at q = 6.51σ−1 and z inside 1σ to 2σ interval, was qualitatively similar to the bulk behavior albeit with a much longer relaxation time. An apparent decrease in the decay of Scol(q,t,z) is seen for segments belonging to stiffer chains on the time scale of 1τ (1τ is approximately 1 ns). This slowing down can be attributed to the bending modes in stiffer

chains and is visible in the MSD (see Figure 5b), showing stiffer chains transitioning differently than flexible chains where there is an immediate transition from the ballistic to the Rouse regime. In calculating the z-resolved segment MSD and ⟨P2[ni⃗ (t,z)· n⃗i(0,z)]⟩ in Figures 5b,c, we imposed locality by considering only segments or bonds which remain inside the Δz thickness within the time interval Δt.23,29,50,62 Also, we found the interfacial behaviors were qualitatively similar to the bulk behavior (e.g., Scol(q,t,z)/Scol(q,0,z) can be described by eq B7 in Appendix B and ⟨P2[ni⃗ (t,z)·n⃗i(0,z)]⟩ by a power law) albeit with a much longer relaxation time. This means that an effective relaxation time can be determined by obtaining the value of t at Scol(q,t,z)/Scol(q,0,z) = 1/e and ⟨P2[ni⃗ (t,z)·n⃗i(0,z)]⟩ = 1/e, where e ≈ 2.718 28. And as expected, the systems with higher lp tend to have higher t at 1/e; hence the dynamics is slower.63,64 We wanted to know which has faster dynamics, segments belonging to loops, tails, or free segments? In Figure 6 we show the segment MSD in the interval of 1−2σ in the direction parallel to the substrate. In considering the averages, aside from maintaining locality (a segment stays within Δz), we only D


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Figure 6. Mean-square-displacement parallel to the substrate of the segments of adsorbed semiflexible chains with Kbend = 6 and within z > 1σ to z < 2σ interval away from a substrate having εw = 8 kT for free, tail, and loop segments. The trains are found at z < 1.161 σ away from the substrate.

counted instances when a segment that is classified as either a loop, tail, or free remains in the same classification in the time interval, Δt. As expected, the dynamics of the free segments were the fastest, followed by tails and finally loops.65 This trend in the dynamics results mainly because loops are connected to two tethering points, tails to one, while there are none for free segments. The trains are only found at z < 1σ interval and are mobile as shown in the MSD in Figure 6. The train dynamics is about 1 order of magnitude slower compared to the bulk based on MSD values, indicating the absence of a “glassy layer” for our model which has smooth surfaces. This is unlike the simulations by Gao et al. which have discrete substrate beads and where they observed a glassy layer.66 We expect the train dynamics to be highly dependent on the type (e.g., discrete vs smooth walls) and interaction strength of the substrate and the persistence length of the polymer. For our system, increasing εw (εw = 1−8 kT) at constant Kbend (Kbend = 0 or Kbend = 2) in the layer interval z < 1σ results in an increase of the static properties, ρmon and ⟨P2(n⃗i(z)·z+)⟩, and an increase in the dynamic properties, Scol(q,t)/Scol(q,0) and ⟨P2[ni⃗ (t,z)·n⃗i(0,z)]⟩. There is a general slowing down of the dynamics at the z < 1σ layer for higher values of εw. However, the layer is not glassy (relaxation time > 100 s or 1011 τ) in the direction parallel to the substrate. Extent of the Interfacial Layer. To measure the extent of the effect of the interface on the density of the segments in the vicinity of the substrate, we calculated the variance, Δ2ρ(z), as a function z. The variance in density is Δ2ρ(z) = (ρ(z) − ρb) 2 where ρb is the bulk density. The dynamics counterpart of Δ2ρ(z) was calculated by comparing the time-resolved data points of the collective intermediate dynamic structure factor, Scol(q,t,z), for 1σ slices with the bulk values such that Δ2s (z) =

1 nt

2 ⎛ S (q , t , z ) Scol,b(q , ti) ⎞ col i ⎜ ⎟ − ∑⎜ ⎟ S (q , 0, z) Scol,b(q , 0) ⎠ i ⎝ col

Figure 7. (a) Variance of the segment density and (b) intermediate collective dynamic structure factor at q = 6.15 σ−1 for semiflexible polymers with Kbend = 1 (red ■) and 6 (brown ▽) and within z distances away from a substrate having εw = 8 kT. (c) Relation of the transition point derived from the variances, Δ2ρ(z) and Δ2s (z), for different values of Kbend ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6.

for Kbend equal to 1 and 6, and both zρ* and zs* are higher for higher values of Kbend. In Figure 7c, we show the strong variations of both z*ρ with z*s with chain stiffness. The most important observation is a strong correlation between these two parameters, which indicates that segmental dynamics is dependent on density and the layering of the segments near the substrate. Stiffer chains have a thicker interfacial layer with perturbed structure, and this reflects in a thicker layer with perturbed dynamics. In the recent experiments by Napolitano et al.,7,8 it was shown that the depression in Tg was related to the structure and the growth of the irreversibly adsorbed polymer layer. According to their interpretation, the growth of the Guiselin brush results in a better packing and densification of the layer, resulting in the decreased extent of the Tg depression. Their observation was explained through the free volume holes diffusion model.46,67 In this regard, our simulation results agree with Napolitano’s experiments, which directly relate the segment dynamics to the layer density. Furthermore, we observed that the dynamic thickness is larger than the structural thickness (z*s > z*ρ ) supporting the concept of a “reduced mobility interface” where segments that are structurally bulk-like are being dragged and slowed down.6

nt

Δ2ρ(z)

(3)

Δ2s (z)

We plotted and in Figures 7a and 7b, respectively. In these plots we determined the z value where the variance begins to saturate, and the values of ρ(z) and Scol(q,t,z) are statistically indistinguishable from the bulk values. These points mark the transition from perturbed to bulk-like behavior. We designated these points as zρ* and zs* for ρ(z) and Scol(q,t,z), respectively. In both Figures 7a and 7b we plotted data points E


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Figure 8. Variance of the time-autocorrelation function of the second Legendre polynomial of the unit bond vectors as a function of z/lp for semiflexible polymers within z distances away from a substrate having εw = 8 kT and with different values of Kbend ((black ●) Kbend = 0; (red ■) Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; (brown ▽) Kbend = 6. Black line is a guide to the eye.

Figure 10. Structural properties of semiflexible chains in the bulk. (a) Mean-square end-to-end distance ⟨R2⟩ as a function of persistence length lp. The black solid line is from eq B1. (b) Single chain form factor P(q) for chains of different persistence length. The gray-dashed line is the Debye function (eq B2) with ⟨Rg2⟩ obtained from Kbend = 0, and the brown solid line is the form factor of an infinitely thin rod (eq B3) with length L = Nb. (c) Collective static structure factor S(q) for chains of different persistence length (the intensities are shifted for clarity). The different symbols in (a) and (b) pertain to systems with different values of Kbend ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6). The different line colors in (c) pertain to systems with different values of Kbend (Kbend = 0 (black); Kbend = 1 (red); Kbend = 2 (blue); Kbend = 3 (gray); Kbend = 4 (green); Kbend = 5 (magenta); Kbend = 6 (brown)).

We performed a similar analysis for the rotational dynamics of bonds in the vicinity of the substrate and calculated the variance of the average of the second Legendre polynomial of the time-autocorrelation of the unit bond vectors, ΔP22(z) where ΔP2 2(z) =

Figure 9. Parallel, ⟨Rxy2⟩ (a), and perpendicular, ⟨Rz2⟩ (b), components of the mean-square end-to-end distance of a polymer melt with different Kbend values ((black ●) Kbend = 0; (red ■) Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; (brown ▽) Kbend = 6) and in contact with an adsorbing substrate with εw = 8 kT. The lines are the bulk values. In (c), each component is normalized by their respective bulk values, and their location is indexed by the location of the chain center-of-mass divided by the unperturbed radius of gyration, z/⟨Rg2⟩b1/2. Filled symbols are the xy components while open symbols are the z components.

1 nt

nt

∑ (⟨P2[nk⃗ (ti , z)nk⃗ (0, z)]⟩ − ⟨P2,b[nk⃗ (ti)nk⃗ (0)]⟩)2 i

(4)

ΔP22(z)

In Figure 8 we show that the behavior of for different values of Kbend or lp is universal and is dependent on lp as the transition region of the different curves collapses after normalizing z with lp. This indicates that segment rotational dynamics, as measured by the rotation of the bond vectors, is dependent on lp. In Figures 9a and 9b we plotted the parallel and perpendicular components of the mean-square end-to-end distance of the polymer chain binned along the z direction where the F


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Figure 11. Dynamic properties of semiflexible chains in the bulk. (a) Normalized single chain intermediate dynamic structure factor. (b) Normalized collective intermediate dynamic structure factor. (c) Mean-square-displacement, MSD, showing scaling laws that indicates Rouse (1/2) and semiflexible (3/4) regimes. (d) Time autocorrelation of the second Legendre polynomial of local unit bond vectors. The different symbols in (a) and (b) pertain to systems with different values of Kbend ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6). The different line colors in (c) and (d) pertain to systems with different values of Kbend (Kbend = 0 (black); Kbend = 1 (red); Kbend = 2 (blue); Kbend = 3 (gray); Kbend = 4 (green); Kbend = 5 (magenta); Kbend = 6 (brown)). 1τ is approximately 1 ns (τ ≈ ns).

Figure 12. (a) Normalized center-of-mass density ρcm(z)/ρcm. The solid black line is eq C1, and the red dashed line is z0 = (2⟨Rgz2⟩b/π)1/2. (b) Fraction of total number of chains which have a chain “start” at z where chain “start” is defined as the chain end closest to the surface. The solid black line is eq C2.The different symbols pertain to systems with different values of Kbend ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6).

Figure 13. Probability distribution of Nloop (a) and Ntail (b) for adsorbed polymers with εw = 8.0 kT and different values of Kbend. ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6).

of influence of the substrate which ranged from 1 to 3Rg. Previous computer simulations for flexible chains showed a range from 1 to 2Rg.24,26,29,68−70 However, here we observed a systematic increase in the range of the perturbed region as lp increases. We also noticed that the same effect is not achieved if we increase the segment−substrate interaction parameter from εw = 1.0 to 8.0 kT for Kbend values of 0 and 2. This behavior has also been seen in simulations23,29 and self-consistent field

indexing in z pertains to the z coordinate of the center-of-mass of a chain. The polymer chains within the interfacial layer are oriented parallel to the substrate; hence, the z component of ⟨R⟩ is lower than the unperturbed size while it is higher for the x and y directions. In Figure 9c we plotted the z resolved ratio of the polymer mean-square end-to-end distance to its unperturbed value, ⟨R2(z)⟩/⟨R2⟩b. This quantity shows the extent G


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structures using polymers with widely different rigidities such as polymethacrylates and polydienes.73

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CONCLUSIONS We have performed molecular dynamics simulations of adsorbed polymer chains with different rigidities. We have been particularly interested in the structure and dynamic properties of the chains and their segments in the presence of a strongly adsorbing substrate that has an interaction range on the order of a segment size and the distance between adsorption points that is infinitesimally small. Within the interfacial layer we determined the statistics of free, loop, tail, and train segments. The results revealed that loops are more prevalent for more flexible chains at constant degree of polymerization, N. For an adsorbed chain, its size in the direction perpendicular to the plane of the substrate is determined by its tail height. The tails were found to be nonuniformly stretched. Thus, we could envision the structure of the adsorbed polymers as a highly polydisperse pseudobrush or a Guesilin brush17 where the adsorbed chains are nonuniformly stretched. For more rigid chains, nematic ordering of the trains parallel to the substrate was observed, and an enhancement of lp in the segments belonging to the trains is observed. Also, more segments are adsorbed in the first adsorption layer for stiffer chains. The orientation of the bonds in the interfacial layer is preferentially parallel to the substrate for more rigid chains. The dynamics of the segments in the second adsorption layer, where we can find mostly tails, loops, and free segments, showed dynamic behavior similar to the bulk, albeit with a much longer relaxation time. Similar to the bulk system, the chains that are more rigid have slower dynamics. The dynamics of segments belonging to free chains in the perturbed region is the fastest followed by tails and then loops. This is because loops are tethered at two points, tail at one and none for free segments. The range of the interfacial layer for the dynamic properties of segments increases as persistence length is increased. This

Figure 14. Grafting density of the Guiselin brush, ρT, for systems with εw = 8.0 kT and different values of Kbend ((●) Kbend = 0; (■) Kbend = 1; (◆) Kbend = 2; (▼) Kbend = 3; (▲) Kbend = 4; (◇) Kbend = 5; (▽) Kbend = 6).

(SCF)22 calculations for flexible chains in polymer melts which asserts that polymer chain conformations do not depend on the substrate−polymer interactions. It also agrees with the conclusion by Fleer et al.22 that the conformations are similar to a single end-grafted chain under critical conditions. This is confirmed when we also observed that the substrate serves as a reflective boundary and the chain conformation agrees with Silberberg’s hypothesis14,15 for all considered chain rigidities (see Appendix C). We also expect that in capped polymer thin films the extent of the effect of the substrate will be enhanced for more rigid polymers. For example, in the case of polystyrene,71 the conformation is perturbed for polymer films having film thickness less than 6 Rg. We expect this limit to be lower for polydimethylsiloxane which is a more flexible polymer. Indeed, dynamic force measurements seem to suggest this value to be around 1Rg.72 In computer simulations of flexible chains this upper limit is 4Rg.70 In one of our systems (Kbend = 5) we still observed oscillations in ⟨R2z⟩ in locations near the center of the slab for slab thickness of 17.5Rg. On the basis of these results, we encourage experimentalists to explore polymer thin film

Figure 15. (a) ⟨Rz2⟩ of adsorbed polymers as a function of their corresponding tail length, ⟨Rz2⟩tail. (b) Tail size, ⟨R2⟩tail, as a function of its unperturbed chain size described by eq B1 (Appendix B) and Ntail. The line is f(x) =x. (c) Typical configuration of an adsorbed polymer chain with Kbend = 2, εw = 8.0 kT, and ⟨Rz2⟩tail1/2 ≈ L with z center-of-mass location near the substrate (left) and far from the substrate (right). D is the size of the confining tube. (d) Dependence of ⟨Rz2⟩ on Ntail. The line with power law of 2 is (Ntailb) 2 or the size of a fully stretched tail. The different values of Kbend are represented as (black ●) Kbend = 0; (red ■) Kbend = 1; (blue ◆) Kbend = 2; (gray ▼) Kbend = 3; (green ▲) Kbend = 4; (magenta ◇) Kbend = 5; and (brown ▽) Kbend = 6). H


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Macromolecules range can be directly correlated to the densification and a more persistent layering in the perpendicular direction of the segments for systems having stiffer chains. The orientational dynamics of the bond vectors is slowed down for stiffer chains and is dependent on the persistence length. We further determined the size of the chain in both parallel and perpendicular directions to the plane of the substrate. The size is perturbed up to 3Rg, and we observed a systematic increase in the range of the perturbed region as chain rigidity was increased.

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Table 1. System Sizes and Chain Properties for Bulk Simulations Kbend

Lx [σ]

Ly [σ]

Lz [σ]

m

lp [σ]

b [σ]

0 1 2 3 4 5 6

43.68 43.68 43.68 43.68 43.68 43.68 43.68

43.68 43.68 43.68 43.68 43.68 43.68 43.68

43.68 43.68 43.68 43.68 43.68 43.68 43.68

1417 1417 1417 1417 1417 1417 1417

0.67 1.06 1.65 2.44 3.41 4.61 5.94

0.965 0.964 0.964 0.964 0.964 0.964 0.964

APPENDIX A where mi = 1 is the bead mass, vi⃗ (t) is the bead velocity, and F⃗i(t) denotes the net deterministic force acting on the ith bead. The stochastic force F⃗Ri (t) has a zero average value ⟨F⃗Ri (t)⟩ = 0 and δ-functional correlations ⟨F⃗Ri (t) FRi (t′)⟩ = 6kTξδ(t − t′). The bead friction coefficient ξ was set to ξ = 1/7.0mi/τ, where τ is the reduced time unit τ = σ(mi/kT)1/2. The velocity-Verlet algorithm implemented in LAMMPS57 with GPU acceleration58 with a time step of Δt = 0.01τ was used for integrating the equations of motion in eq A6. Bulk Simulations Protocol. Similar to the procedure described in the simulations by Carrillo and Sumpter,38 chains in random walk configurations were initially distributed in the simulation box with a number density of ρb = 0.0025σ−3. Then the system was gradually compressed to a final density of ρb = 0.85σ−3, after which the pairwise interaction potential was turned off, thereby removing excluded volume interactions for an equilibration period of 105τ. With the excluded volume interactions turned off, the chains behave like phantom chains and can equilibrate at a much faster rate. This step was followed with another 105τ equilibration period where the pairwise interactions are turned on. Following this process, the simulations were run for another 4 × 104τ for the production step. Simulations with Substrates Protocol. Similar to the “bulk” procedure, chains in random walk configurations were initially distributed in the simulation box with a number density of ρb = 0.0025σ−3. However, unlike the bulk simulations, a substrate described by eq A7 is positioned at the top and bottom of the simulation box such that the system is no longer periodic in the z direction. In eq A7, rwall is the |z| distance of a bead from a wall, rc is the cutoff of the wall potential, and εw is the well depth of the LJ potential wall. Initially, the substrate parameters were set to εw = 1 kT and rc = 21/6σ such that none of the segments are adsorbed during the equilibration period. The system was gradually compressed in the z direction to a final density of ρb = 0.85σ−3

Simulation Details

We performed coarse-grained molecular dynamics simulations of semiflexible polymer chains composed of Lennard-Jones beads with degrees of polymerization, N = 50. The LJ beads interact with other beads through the truncated and shifted Lennard-Jones potential where well depth, ε = 1.0 kT, size of the bead, σLJ = 1.0σ, and LJ cutoff of 21/6σ. ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ 1/6 ⎪ ⎪ 4ε⎢⎝⎜ ⎠⎟ − ⎝⎜ ⎠⎟ ⎥ + ε , r < 2 σ r ⎦ UWCA(r ) = ⎨ ⎣ r ⎪ ⎪ 0, r ≥ 21/6σ ⎩

(A1)

The cutoff parameter sets the potential, which is also known as the Weeks−Chandler−Anderson (WCA),74 to take into account only the repulsive portion of the potential. The segments forming the polymers are connected to other segments by finite extensible nonlinear elastic (FENE)56 bonds where R0 = 1.5σ and KFENE = 30.0 kT. ⎡ ⎛ r ⎞2 ⎤ UFENE = −0.5KFENER 02 ln⎢1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ R 0 ⎠ ⎥⎦

(A2)

Chain rigidity was imposed by applying a bending potential energy to adjacent polymer bonds Uibend , i + 1 = kTK bend(1 − (ni⃗ · ni⃗ + 1))

(A3)

where n⃗i is a unit vector obtained from the position vectors of adjacent monomers and defined as

ni⃗ =

ri ⃗ − ri −⃗ 1 | ri ⃗ − ri −⃗ 1|

(A4)

Rigidity is increased by increasing the value of Kbend, and the persistence length, lp, is evaluated by fitting the bond−bond correlation function with an exponential decay ⟨ni⃗ ·ni⃗ + l⟩ = exp( −lb/lp)

⎧ ⎡⎛ 12 ⎛ σ ⎞6 ⎤ σ ⎞ ⎪ ⎢ ⎪ 4εw ⎜ ⎟ −⎜ ⎟ ⎥ , rc < 2.5σ ⎝ rwall ⎠ ⎥⎦ Uwall(rc) = ⎨ ⎢⎣⎝ rwall ⎠ ⎪ ⎪ 0, rc ⩾ 2.5σ ⎩

(A5)

where b is the average bond length (see Figure 1). The parameters obtained from the use of eq A5 are listed in Table 1 for different values of Kbend. The simulation box dimensions (Lx, Ly, and Lz) and the number of chains, m, for the simulations in the bulk are listed in Table 1. The number density is kept at ρb = 0.85σ−3. The simulations were performed at a constant temperature, which was maintained by coupling the system to the Langevin thermostat such that the motion of the beads can be described by the equation mi

dvi⃗(t ) R = Fi ⃗(t ) − ξvi⃗(t ) + Fi⃗ (t ) dt

(A7)

This simulation box compression step was followed with a simulation protocol which had three equilibration steps. The first equilibration step, which lasted for 104 τ, sets εw = 1.0 kT and the substrate interaction cutoff to 21/6σ in eq A7 to prevent the segments from adsorbing to the substrate while excluded volume interactions was turned off. The second equilibration step consisted of a run lasting for 104τ where the substrate interaction cutoff in eq A7 remained at 21/6σ, εw =1.0 kT, and excluded volume interactions was turned on. The third equilibration step, which lasted for 4 × 104τ, sets the substrate

(A6) I


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Macromolecules cutoff in eq A7 to 2.5σ, and varying εw ranging from 1.0 to 8.0 kT. Finally, this was followed by a production run lasting 8 × 104τ. The system sizes for the simulations with substrates are tabulated in Table 2.

and the other by the bead size. This second peak is at q = 6.3σ−1 for Kbend = 6. Bulk Dynamic Properties. The dynamics of semiflexible polymers in melts is typically slower because translational motion perpendicular to the direction of the persistent segment is severely restricted and motion is directed toward the persistent length axis.78 This is the general trend observed in our MD simulations as seen in Figure 11 where we plotted the normalized single chain and collective intermediate dynamic structure factors, the segment mean-square displacement, and the time-autocorrelation of the second Legendre polynomial of unit bond vectors. We also calculated the analogues of the static or structural parameters and for the description at the chain level, the single chain dynamic structure factor, Ssc(q,t), was calculated using the same procedure as in the work by Carrillo and Sumpter38 and compared to the Rouse model for the ideal chain case:64

Table 2. System Sizes for Simulations with Substrates

■

Kbend

Lx [σ]

Ly [σ]

Lz [σ]

m

0 1 2 3 4 5 6

43.68 43.68 43.68 43.68 43.68 43.68 43.68

43.68 43.68 43.68 43.68 43.68 43.68 43.68

44.81 44.81 44.81 132.17 132.17 219.53 219.53

1417 1417 1417 4251 4251 7085 7085

APPENDIX B 1 Ssc(q , t ) = N

Bulk Structural Properties

First we characterized the average parameters of a single polymer chain in a polymer melt. This includes the meansquare end-to-end distance ⟨R2⟩, mean-square radius of gyration ⟨Rg2⟩, and single chain scattering form factor P(q). We evaluated the mean-square end-to-end distance of the chains in the bulk polymer melt (see Figure 10a), and we found that the chains behave like a Kratky−Porod75 worm-like chain where ⎛ ⎛ Nb ⎞⎞ ⎟⎟⎟ ⟨R2⟩ = 2lpNb − 2lp2⎜⎜1 − exp⎜⎜ − ⎟ ⎝ l p ⎠⎠ ⎝

N

×

p=1

where DR is the diffusion coefficient of a Rouse chain (DR = kT/Nξf), τr is the largest mode relaxation time (τr = ξfb2N2/ 3π2kT), and ξf is the segment friction coefficient. It is clear that the ideal chain follows Rouse dynamics as shown in the black line in Figure 11a. As the chain becomes stiffer, the chain relaxation time becomes longer than the Rouse relaxation time, which is in agreement with the observations of Faller et al.59 For the collective behavior, we calculated the collective intermediate structure factor, Scol(q,t), from the time correlation of the Fourier transform of the bead density distribution, ρ(q,⃗ t) = ∑ieiq⃗ri⃗ (t) such that 1 Scol(q ⃗ , t ) = ⟨ρ(q ⃗ , t ) ·ρ( −q ⃗ , 0)⟩ (B6) Nm

(B1)

This quantity was evaluated at q = 2π/b = 6.51σ−1 which describes the length scale of a bond and is near the vicinity of the peak of the collective static structure factor, S(q), which is at 7.08σ−1 for flexible chains, and a shoulder emerges at 6.3σ−1 as lp increases. Harnau et al.79 noted that the this quantity cannot be expressed by a single exponential function for a semiflexible chain. Hence, we followed the procedure by Moe and Ediger80 where we fitted the dynamic structure with a sum of an exponential and a stretched exponential of the form

(B2)

with a Porod slope of −2. In calculating the P(q) for the ideal chain, the ⟨Rg2⟩ of system with Kbend = 0 was used. As lp increases, the Porod slope at higher q increases from −2 and approaches −1. A Porod slope of −1 describes the scattering form factor of thin rods with length L and whose scattering form factor, Prod(q), is given by the equation 2 qL

∫0

qL

sin 2(qL /2) sin(x) dx − x (qL /2)2

(B3)

Scol(q , t ) = ae−t / τR1 + (1 − a)e−(t / τR 2)

We also calculated the collective static structure factor for all the beads in the simulation box for systems with different lp (see Figure 10c) by taking the Fourier transform of the density distribution of the simulation box:76 S(q ⃗ ) =

1 Nm

β

(B7)

The use of a stretched exponential indicates the presence of multiple relaxation modes. We were able to fit the data for the flexible chain with only a single exponential (a = 1) while the fitting for the stiffest chain (Kbend = 6) had the parameters a = 0.536 and β = 0.538 (see Figure 11b). The motion of the segment is further quantified with the calculation of the segment mean-square-displacement, MSD, defined as

Nm Nm

∑ ∑ ⟨ exp[−iq ⃗ ·( rj⃗ − rk⃗ )]⟩ j=1 k=1

⎛ tp2 ⎞⎤⎤ ⎛ p π n ⎞ ⎛ pπ m ⎞ ⎡ 1 ⎜ ⎟ cos⎜ ⎟⎢1 − exp⎜ − ⎟⎥⎥ cos ⎝ N ⎠ ⎝ N ⎠⎢⎣ p2 ⎝ τr ⎠⎥⎦⎥⎦ (B5)

and b is the average bond length of the polymer chains. In Figure 10b we calculated the form factor of a single chain in the melt, P(q). For an ideal chain, which is best represented by the system with Kbend = 0, P(q) is described by the Debye function 2 PD(q) = 2 2 2 (exp( −q2⟨R g 2⟩) − 1 + q2⟨R g 2⟩) (q ⟨R g ⟩)

Prod(q) =

∑

⎡ 2Nb2q2 ∑ exp⎢⎢−q2DR t − 1 |n − m|q2b2 − 2 6 3π n,m ⎣ N

(B4)

where m is the number of chains in the simulation box. We noted that for flexible chains a peak77 occurs at q = 7.08σ−1 and starts to split at a lower q for chains with higher lp. This indicates the two distances in the bead arrangements where one pertains to the arrangement dictated by the persistent segments

2 MSD = ⟨| r (⃗ t ) − r (0) ⃗ |⟩

(B8)

where r(⃗ t) is the location of the ith atom after an elapse time of t from its original location r(⃗ 0). For the flexible chains, the J


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Macromolecules Rouse regime (MSD ∼ t1/2) can be clearly seen in Figure 11c. For the semiflexible chains, a power law of 3/4 is expected because of the dominance of the bending modes.81 This is clearly observed in our simulations and was perviously observed by Steinhasuer et al.82,83 in single polymer chains simulations but were only approached in their melt simulations owing to entanglement effects because they simulated longer chains. It was also noted by Faller et al. that the entanglement length decreases as the persistence length is increased.59 It is to be noted that in Figure 11c the expected diffusive regime (MSD ∼ t) at long time scales was not reached in both flexible and semiflexible cases. Furthermore, we calculated the time-autocorrelation function of the second Legendre polynomial of the unit bond vectors, ⟨P2[ni⃗ (t) ·n⃗i(0)]⟩, defined as 1 [3(ni⃗ (t )·ni⃗ (0))2 − 1] 2

⟨P2[ni⃗ (t ) ·ni⃗ (0)]⟩ =

Number of Loops and Tails Distribution of Adsorbed Chains. In Figure 13a we considered the probability distribution of the number of loops, P(Nloop) and we were able to recover the scaling laws (power law of −3/2) for the loop distribution as predicted by the self-consistent field model by Scheutjens and Fleer13 and the MD simulations of flexible polymers by Vilgis et al.29 The system with higher values of lp follows this scaling law, however oscillations are seen and can be attributed to finite N effects. Similar behavior is observed for the probability distribution of tails, P(Ntail) (see Figure 13b) where we found a power scaling law of −1/2, which follows theoretical predictions22,87 and is in agreement with MD simulations of flexible polymers by Vilgis et al.29 Stretched Tails of Adsorbed Chains and Polydisperse Pseudobrush Structure of the Adsorbed Layer. The number of adsorbed beads (trains) increases as lp increases, and we expect the grafting density of the Guiselin brush, ρT, to decrease because the Guiselin brush now has more beads in the anchor points at the expense of the number of pseudo tails. To confirm this, we counted the number of pseudo tails by considering all tails and all loops having Nloop > 1 (note that Nloop = 1 are segments that belong to trains). The Guiselin brush grafting density is therefore ρT = (ntail + 2nloop)/LxLy, where LxLy is the area of the substrate and ntail and nloop are the number of tails and loops of the adsorbed polymers, respectively. In Figure 14, it is observed that the value of ρT monotonically decreases with respect to lp. The height of an adsorbed chain is dictated by the height of its tail (see Figure 15a), and this behavior is independent of lp as all data points collapse to a single curve. We compared the size of a tail, ⟨R2⟩tail, with its predicted unperturbed size by plugging Ntail into eq B1, from which we observed that this value is greater than that of the unperturbed size, indicating that these tails are extended (see Figure 15b). Guiselin17 remarked that the adsorbed layer can be considered as a highly polydisperse pseudobrush where the chains are nonuniformly stretched. Our simulations agree with his remark, and as shown in Figure 15d, the height of an adsorbed chain has a power law that crosses over from 1 for short Ntail and 2 for long Ntail and a crossover region that is dependent on lp. A typical structure of an adsorbed polymer is shown in Figure 15c where a polymer with z center-of-mass location near the substrate is loop dominated and ⟨Rz2⟩1/2 ≈ L is determined by the height of loops which is of the same order as that of a tail (Nloop/2 ≈ Ntail). For an adsorbed polymer with z center-of-mass location far from the substrate, the height of the polymer is determined by the tail. Using the scaling argument, we can explain the power laws observed in Figure 15d. The structure of a single adsorbed polymer chain is brought by the interplay between the elastic (stretching), confinement, and adsorption free energies:

(B9)

to characterize the relaxation of local bond orientations similar to the procedure used by Faller et al.59 This quantity can be compared to NMR measurements. We observe strong slowing down of the reorientation of the bond vector as persistence length is increased in agreement with Faller’s observations. It was also observed in both Faller’s and our simulations that this quantity decays algebraically (power law) at short time scales and exponentially at long time scales. In Figure 11d, for Kbend = 0 (black), the decay is a power law with slope of −1.1 while for the Kbend = 2 with lp = 1.64σ (blue) the exponential decay at long time scales is already apparent.

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APPENDIX C

Adsorbed Polymer Chain Conformations

Reflected Random Walk Conformation of Adsorbed Chains. Silberberg conjectured that for a polymer melt in the presence of a substrate the conformation of the polymer chains near the substrate is that of a reflected ideal chain where the substrate serves as a “reflective boundary”.14,15 This has been confirmed in experiments of polystyrene films on silicon wafers12,84 as well as observed in theoretical results and computer simulations.22,23,28,35,85,86 Under the condition of random walks with reflective boundaries an enrichment of the number density distribution of the chain center of mass is expected at z0 = (2⟨Rgz2⟩b/π)1/2, and for z > z0 the distribution is described by the equation ρcm (z) =

1 erf(z / 2⟨R gz 2⟩b )

(C1)

We also enumerated the probability of a chain starting at distance z from the substrate, ρstart(z), where chain “start” is defined as the chain end closest to the surface. This probability is analytically solved as ρstart (z) = 1 −

⎞ 1 ⎛ z ⎟⎟ erf⎜⎜ 2 ⎝ (2⟨R z 2⟩b )1/2 ⎠

lpNtailb L2 1 + (Felast + Fconf + Fads) ≈ kT lpNtailb D2 − εw (N − Ntail)

(C3)

In writing eq C3, we have neglected prefactors. D is the size of the confining tube due to the crowding caused by neighboring pseudotails and is related to the grafting density as ρT ∼ D−2. Also, we are considering the case for melts where excluded volume interactions are screened. Minimizing eq C3 with respect to Ntail and solving for Ntail results in the relation L ∼ Ntail or ⟨Rz2⟩ ∼ Ntail2. In the case of polymers adsorbed near the substrate and which have tail lengths of the scale smaller than D,

(C2)

Both ρcm(z) and ρstart(z) in the simulations are plotted in Figure 12 and are found to be in agreement with Silberberg’s hypothesis for all considered chain rigidities. In Figure 12a the peak of ρcm decreases as lp increases. This effect is also seen in lattice Monte Carlo simulations of flexible chains when the number of statistical segments are decreased.28 K


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the chain segment is unperturbed in the z direction and gives the relation L2 ∼ Ntail or ⟨Rz2⟩ ∼ Ntail.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (J.-M.Y.C.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank A. V. Dobrynin for useful discussions. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The work was performed at the Center for Nanophase Materials Sciences, a DOE Office of Science User Facility. This research used resources of the Leadership Computing Facility at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC05-00OR22725 with UT-Battelle, LLC.

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