Entanglement Reduction and Anisotropic Chain and Primitive Path

Article pubs.acs.org/Macromolecules

Entanglement Reduction and Anisotropic Chain and Primitive Path Conformations in Polymer Melts under Thin Film and Cylindrical Confinement Daniel M. Sussman,*,† Wei-Shao Tung,‡ Karen I. Winey,‡,§ Kenneth S. Schweizer,∥,⊥ and Robert A. Riggleman*,§ †

Department of Physics and Astronomy, ‡Department of Materials Science and Engineering, and §Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States ∥ Departments of Materials Science, Chemistry, and Chemical & Biomolecular Engineering and ⊥Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, United States ABSTRACT: We simulate and theoretically analyze the properties of entangled polymer melts confined in thin film and cylindrical geometries. Macromolecular-scale conformational changes are observed in our simulations: the average end-to-end vector is reduced normal to the confining surfaces and slightly extended parallel to them, and we find that the orientational distribution of the chain end-to-end vectors is transmitted to the primitive path entanglement strand level. Treating the chains as ideal random walks and the surfaces via a reflecting boundary condition we are able to accurately theoretically predict the anisotropic global and primitive-path level conformational changes. Combining this result with a recently developed microscopic theory for the dependence of the tube diameter on orientational order allows a priori predictions of how the number of entanglements decreases with confinement in a geometry-dependent manner. The theoretical results are in excellent agreement with our simulations.

I. INTRODUCTION The structure and dynamics of polymer melts under strong confinement have attracted intense interest since large thickness-induced shifts in the glass transition temperature were reported two decades ago.1,2 While such changes in the glassy behavior have been hotly debated, extensional measurements of thin glassy films3 also indicate that there is an increase in the entanglement molecular weight Me under nanoscale confinement. Separately, experiments on polymer nanocomposites, where nanoparticles provide internal confining surfaces,4−9 have also indicated a modification of both the melt diffusion and the rheological properties in the rubbery regime. The interpretation of all of these experiments is challenged by our lack of a microscopic picture for the changes in the entanglement network near interfaces, either under nanoscale confinement or in the vicinity of nanoparticle surfaces. As polymer nanocomposites find broader application and nanofabrication technologies mature, shrinking the sizes of devices, it will be critical to understand the modifications of the entanglement network of confined polymer systems.10−12 Changes in thin-film Me and in the polymer mean-squared end-to-end distance, Ree2, have been previously investigated in both experiments3 and simulations.13−16 Recently, experiments have been performed under cylindrical confinement,17 but to the best of our knowledge no corresponding simulations have © 2014 American Chemical Society

been systematically performed. Crucially, the precise link between changes in polymer conformation induced by confinement and changes in the entanglement properties is not understood. This is part of a broader conversation on the nature of entanglements in polymer melts, where even questions as simple as “does the tube diameter increase or decrease” in response to chain stretch or orientation are as yet unresolved.18 A growing consensus, however, suggests that both confinement and continuous shear deformations lead to a dilution of the entanglement network and a larger tube diameter.3,13−17,19 In this article, we first report the results of molecular dynamics simulations of entangled chains under both thin film and cylindrical confinement. We find that entanglement loss is accompanied by systematic changes in not just the global chain orientational order but also the orientational distribution of the primitive path steps themselves. By probing both thin films and cylinders, we expect that, in contrast to an existing comparison between thin film simulations15 and cylindrical experiments,17 the entanglement network is quite sensitive to the difference between 1D and 2D confinement. We then attempt to Received: June 9, 2014 Revised: August 26, 2014 Published: September 11, 2014 6462

dx.doi.org/10.1021/ma501193f | Macromolecules 2014, 47, 6462−6472

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cavity with the required geometric shape. We then insert the polymer chains that were equilibrated with smooth confining walls and re-equilibrate them. The length of the simulation box in the unconfined directions is then scaled to reach a polymer bead density of ρs = 0.85/σ3. The simulation box confined using the rescaled walls is then re-equilibrated a final time, and the accessible volume is re-evaluated to confirm that ρs = 0.85/σ3 (to within ∼0.5%). We studied N = 50, 350, and 500 for cylindrical confinement and N = 350 for thin film confinement. We first quantify the degree of confinement by the accessible volume of the system. For confined polymer chains, the closest possible separation (rmin) between a polymer bead and a wall bead is assumed to be the same as the closest distance between two polymer beads in a bulk condition. A random point is generated within the simulation box and is then determined as inaccessible if the point is either inside the wall region or the minimum distance between the point and any wall bead is less than rmin. A large number of random points (108) are generated and the percentage of accessible points (vacc%) is obtained. Accessible volume is then simply the percentage of accessible points multiplied by the volume of the simulation box. Note that the radius, r, and thickness, h, are the sizes of the confinement with smooth walls when we first relaxed the polymer chains, while the effective confinement dimensions (reff, heff) are obtained from the accessible volume evaluation (e.g., reff = (vacc/(πlz))1/2). From the equilibrated configurations, MD simulations are run until the diffusive regime of the MSD is reached and then polymer configurations are recorded for post analysis. To improve the statistics of our results, configurations at seven different times separated by 3 × 106τ are used, except for the systems with r/σ = 10, 15 for N = 500, σ = 20, 25 for N = 350, and h/σ = 40 for N = 350. For those systems the diffusion time is prohibitively long, and so connectivity-altering Monte Carlo moves are used in combination with MD simulation to obtain uncorrelated configurations. Table 1 lists the range of systems studied for cylindrical and thin film confinement. Table 2 summarizes the results of these calculations. To study the entanglement properties of our systems, there are three standard techniques in the community: the two geometric methods (CReTA25 and Z126−29) and the “classical primitive path algorithm” (PPA30). In isotropic systems it has been argued that CReTA and Z1 give essentially identical results, and in this work we explicitly verify this for a subset of our cylindrical data. In contrast, the PPA finds paths that minimize elastic energy of the chains (as opposed to the contour-length minimization of the geometric methods), and it has been argued that contour-length minimizing algorithms do a better job of reproducing the Doi−Kuzuu distribution of primitive path lengths.27 Further complicating the use of the PPA, chain tension in the energy-minimizing algorithm can shift the position of the entanglement point, possibly changing any signature of orientational order of the primitive paths. For this reason, the original PPA method by definition does not allow one to extract the entanglement length in an anisotropic sample. Thus, combined with the fact that the PPA algorithm is not parameter free, we believe that comparing it with the geometric methods is unwarranted. Because of its computational efficiency, then, we primary focus on the Z1 algorithm but also include some comparisons with CReTA results. B. Simulation Results. The effective cylinder radius or film thickness (reff, heff) is obtained from the accessible volume, and

understand our results on both the equilibrium chain conformational changes and the entanglement network dilution in a unified theoretical framework. Our perspective is that strong confinement modifies global polymer conformations; this modification is transmitted to the PP level, inducing a change in the orientational distribution of PP segments, which in turn affects the number of entanglements per chain. Following this perspective, we extend existing equilibrium and dynamic theories and show that we can quantitatively predict how the confined polymers’ average conformation and number of interchain entanglements change as a function of film thickness or cylinder radius. The remainder of the article is organized as follows. Section II details the simulations we have performed and our results on the conformational changes and entanglement losses experienced by polymer melts under different confinement geometries. Section III discusses our theory for predicting chain conformation based on confinement and quantitatively compares it with our simulations. Section IV first briefly reviews our prior theory connecting chain orientational order with average degree of entanglement in the bulk and then extends it to treat the confined melts of present interest. Quantitative comparisons with theory and simulations are also performed, and predictions are made for the spatially resolved gradient of entanglement density. The paper concludes in section V with a brief summary and discussion.

II. SIMULATIONS OF CONFINED POLYMERS A. Coarse-Grained Model and Methods. Our simulations were performed using the Kremer−Grest model20 with nonbonded interactions governed by the repulsive part of the Lennard-Jones (LJ) potential and units normalized by the potential strength, ϵ, the monomer size, σ, and the time τ = σ(m/ϵ)1/2, where m is the monomer mass. Since we report all units in terms of the monomer or “bead” size, some care must be taken when comparing with other results in the literature, which have sometimes been presented in units of the bond length or the mean-square end-to-end distance per monomer. All simulations are run with the LAMMPS MD simulation package with the velocity-Verlet algorithm.21 To generate confined polymers at the same density as the bulk polymers, we first confine our polymers with smooth, repulsive walls and equilibrate the system in an ensemble where the pressure is held constant in the unconfined directions. The pressure is set to the average pressure calculated from an unconfined NVT simulation at a monomer density of ρ = 0.85σ−3. For entangled polymer chains Monte Carlo connectivity-altering moves are used to assist chain relaxation.22−24 The mean-square displacement (MSD) is calculated to ensure each monomer has moved a distance comparable to Ree, the root-mean-square chain endto-end distance. The pressure is computed in the unconfined directions to obtain the corresponding equilibrium box length in these directions, where we ensure that this length is larger than Ree to prevent polymer chains interacting with themselves across the periodic boundaries. To prevent polymer crystallization for both the thin film and cylindrical cases, we impose geometric confinement via amorphous immobile particles of the same size and interacting with the same repulsive LJ potential as the polymer monomers. To do this, a simulation box is set up with desired sizes in both unconfined and confined directions and filled with LJ monomers at a density 1.3/σ3. These beads are relaxed under constant volume conditions, and a subset is removed to create a 6463


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modestly expanded parallel to the surface, and there is a near collapse of these conformational confinement effects normal to the surface, despite the difference in the number of confining directions. We also note that our results for planar confinement agree very well with previous simulation studies,12,13 as seen in Figure 2A. Figure 3 illustrates the changes in the PP orientational probability distribution, g(u⃗), where u⃗ is a unit vector describing the orientation of the PP step, for representative degrees of confinement. Given the symmetry of films and cylinders, in Figure 3 we plot the distribution of orientations relative to the ẑ axis, g(u⃗) ∝ f(θ) sin θ, where cos θ = u⃗·ẑ, so that f(θ) = 1 represents an isotropic distribution. As discussed below, we find that this distribution closely follows the orientational distribution of chain end-to-end vectors. Under planar confinement, the PP steps tend to lie down in the plane of the film, and so f(θ) has a maximum near θ = π/2. Under strong cylindrical confinement the PP steps align along the axis of cylindrical symmetry, and hence f(θ) has a maximum near θ = 0. Figure 4 establishes how confinement induces a significant reduction in the number of entanglements per chain, Z ≡ N/ Ne, and the effects of confinement on Z are significantly stronger under cylindrical confinement. Under planar confinement, the strongest reduction in the entanglement density was only approximately 20% for films with a thickness as small as δ = 0.25, while cylinders with a diameter corresponding to δ = 0.25 had an approximately 50% reduction in the entanglement density. To ensure that our results were not sensitive to our choice of primitive path algorithm, we have also used the CReTA algorithm25 to compute entanglement reduction in a subset of our systems. It had previously been argued that Z1 and CReTA should yield qualitatively similar results under both isotropic and anisotropic conditions;19,27 here we explicitly demonstrate this for the cylindrically confined systems. Comparing Figures 2 and 4 shows that while the confinement geometry has only a modest effect on chain conformations normal to the surface, the number of confined dimensions strongly affects PP statistics. If one assumes that even in anisotropic systems there is a simple relationship between the number of entanglements per chain and the tube diameter, Z ∼ N/Ne ∼ σ2/dT2, then the observed reduction of Z corresponds to an increase in the tube diameter dT = σ√Ne. We find that δ ≳ 2.0 is required for the entanglement network to become bulk-like. This result agrees qualitatively with polymer nanocomposite experiments that find a change in the entanglement plateau modulus when the separation between nanoparticle surfaces is less than ∼2.5Ree.5 Neutron scattering experiments on cylindrically confined systems reported an increase in dT of ∼15% for a system with δ ≈ 0.42.13 We typically find a larger enhancement; e.g., our model cylindrical system with δ ≈ 0.49 has an effective dT ∼ 28% larger than the bulk. Given the changes in chain dimensions, one possible phenomenological ansatz for the entanglement reduction is an anisotropic extension of packing-length arguments. The number of interpenetrating chains in the bulk scales as Ree3/ (NVmon) ∼ σ√N/p, where p = (ρsσ2)−1 is the invariant packing length.31,32 Packing a fixed number of chains in the pervaded volume implies Ne ∼ (p/σ)2. While it is known that the average number of chains in the pervaded volume of a given chain is not precisely constant as a function of confinement,33 as a first order estimate one can neglect this effect and approximate (RxRyRz)/(NVmon) ∼ constant, suggesting Z/Zbulk = (RxRyRz/

Table 1. Details about Each Simulation System for Cylindrical and Thin Film Confinement (All Length Units Are in σ) Cylindrical Confinement size of the simulation box radius (r) N = 50

N = 350

N = 500

bulk 3 5 10 20 bulk 5 7 10 15 20 25 bulk 5 7 10 15

no. of chains (M)

lx (= ly)

lz (unconfined)

100 18.05 80 10 80 15 100 25 640 45 90 33.34 30 15 40 19 60 25 60 35 90 45 180 55 80 36.1 20 15 30 19 50 25 60 35 Thin Film Confinement

18.05 226.77 71.61 20.36 20.49 33.34 179.28 121.35 85.95 36.96 30.76 38.33 36.1 179.28 130.23 102.33 52.9

effective radius (reff) N/A 2.57 4.58 9.58 19.58 N/A 4.57 6.57 9.57 14.58 19.57 24.58 N/A 4.58 6.57 9.57 14.58

size of the simulation box

N = 350

thickness (h)

no. of chains (M)

8 10 14 20 30 40

30 40 50 70 90 120

lz

lx = ly (unconfined)

effective thickness (heff)

13 15 19 25 35 45

41.42 42.31 39.32 38.80 35.63 35.49

7.20 9.15 13.25 19.23 29.20 39.24

we take the cylindrical axis and the normal to the thin films to lie along ẑ. We parametrize the degree of confinement by δ ≡ heff/Ree,bulk or δ ≡ 2reff/Ree,bulk, the film thickness or cylinder diameter in units of the average bulk end-to-end distance. Our main results from the simulations concern the components of the root-mean-square end-to-end vector Ree, the number of beads between entanglements, Ne, and the primitive path (PP) configuration of the system as generated by the Z1 algorithm.26−29 This algorithm uses geometrical moves to monotonically reduce chain contour lengths to the limit of infinitely thin PP thickness, and we report the average entanglement number, ⟨Z⟩, as the average number of kinks per chain in the resulting PP network. From the full PP configuration we calculate orientational distributions of PP steps in bulk and under confinement, which is a key point of comparison against our theoretical analysis. Representative simulation images of polymers in bulk and cylindrical confinement are shown in Figure 1, which also shows the corresponding PP steps of those chains. We note that this algorithm has previously been used to study anisotropic entanglement networks.19 Figure 2 shows the changes in the components of Ree relative to the confining surfaces as a function of δ. Generically, the chains are significantly compressed normal to a surface and 6464


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Table 2. Reduced Density and Accessible Volume for the Simulated Systems Cylindrical Confinement N = 50

N = 350

N = 500

N = 350

r

equilibrated

vacc

3 5 10 20 5 7 10 15 20 25 5 7 10 15

221.57 69.61 19.61 19.58 182.70 117.06 82.37 35.41 29.38 37.23 173.66 125.30 98.04 50.57

h

equilibrated

vacc

density

scaled

re-evaluated

density

8 10 14 20 30 40

39.77 40.87 38.45 37.98 35.13 35.12

11390.93 15369.66 19684.72 27615.69 36020.76 48384.76

0.9179 0.9109 0.8890 0.8872 0.8945 0.8680

41.42 42.37 39.32 38.80 35.63 35.49

12357.54 16375.76 20479.95 28950.31 37076.87 49436.14

0.84987 0.8549 0.8545 0.8463 0.8496 0.8496

density

scaled

re-evaluated

density

4597.92 0.8670 4574.59 0.8744 5666.82 0.8823 23606.02 0.8896 11988.93 0.8758 15888.21 0.8812 23675.87 0.8870 23666.13 0.8873 35388.98 0.8901 71989.52 0.8751 11396.26 0.8775 16978.94 0.8836 28179.48 0.8872 33739.06 0.8892 Thin Film Confinement

226.77 71.61 20.36 20.49 188.25 121.35 85.95 36.96 30.76 38.33 179.28 130.23 102.33 52.90

4708.42 4717.43 5869.17 24690.78 12342.31 16474.09 24719.42 24677.13 37013.16 72775.50 11978.11 17679.73 29418.03 35320.36

0.8495 0.8479 0.8519 0.8505 0.8507 0.8498 0.8495 0.8510 0.8510 0.8657 0.8476 0.8484 0.8498 0.8494

The first result of this calculation is the mean end-to-end vector normal to the wall as a function of the distance from the terminal random walk step to the wall, Ree,z,film2(z,h) and Ree,r,cyl2(r,h) for thin film and cylindrical confinement, respectively. This function is integrated to obtain a film- or cylinder-averaged change in the conformational extent of the chains normal to the confining surfaces, e.g.

Rx,bulk3)2 < 1. That is, in this simple packing estimate the relative change in the pervaded volume of an oriented chain is directly mapped to the number of chains it is entangled with. Figure 4B shows that this estimate, using conformations from either the simulations or theoretical results (eq 1, discussed below), predicts a much stronger loss of entanglement than we observe. This overprediction is not a simple consequence of neglecting changes in local chain self-density as a function of confinement (i.e., the approximation of (RxRyRz)/(NVmon) ∼ constant). Thus, a deeper understanding of entanglements is required to treat confined systems.

αfilm

2

1 ≡ h

∫0

h

R ee, z ,film 2(z′) dz′ R ee,bulk 2/ 3

(1)

It is these quantities we will make use of in our subsequent predictions in section IV of confinement-induced entanglement reduction. In Figure 2 we compare the theoretical predictions derived below with our simulations and find the theory quantitatively explains the data over the entire range of confinement strength. We now first review Silberberg’s original argument for the conformation in thick polymer films and then extend this argument to thin films and cylindrical geometries. A. Principle of Conformational Transfer: Thick Films. Silberberg’s original calculation was for “thick” films, h ≳ Ree, and the argument for the conformational change of a randomwalk polymer near a single wall can be written as follows.34 For simplicity, we will treat this problem as one-dimensional, with the chain being a random walk in the z direction with a surface at z = 0. As seen in Figure 2, the simulations do observe modest chain swelling parallel to the surface; this one-dimensional simplification corresponds to neglecting this swelling. Operationally, one starts with a space-filling population of random walks with a Gaussian distribution of end-to-end vectors. The presence of the surface breaks the symmetry of the system and allows one to identify the “start” and “terminus” of the chain, z0 and zt, by the end that is closer to or farther from the surface. To respect the surface constraints, the direction of every chain is reversed whenever it violates a boundary constraint; this replaces every bulk conformation with an equally likely

III. PREDICTING CONFINED CHAIN CONFORMATIONS It is clear from the simulation data in Figure 2 that the dominant contribution to changes in the polymer conformation is chain compression normal to the surface. There are many different formal approaches one could take to predict confined chain conformations; to capture these effects, we have generalized Silberberg’s principle of conformational transfer34 to treat our polymer melts in strongly confined geometries. We fully expect that alternative approaches, e.g. adopting an equivalent Green’s function approach,35 would lead to conclusions identical with what we have derived here. The Silberberg model treats the polymer as a random walk, neglecting the excluded-volume interactions that lead to chain extension parallel to a surface, and uses reflecting boundary conditions to compute changes to the chain conformation in the presence of a wall. The model assumes that under confinement the chain is still locally able to adopt bulk-like conformations, so it is not appropriate for confinement scales h < p. Above this length scale, however, the model makes quantitative predictions for the statistics of confined chain conformations in the direction normal to the surface without adjustable parameters. 6465


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conformations that have one of their chain ends at z*, where we assume a Gaussian distribution of chains with end-to-end distance Ree,z2 = ⟨(zt − z0)2⟩ of a given value: Ω(z 0 , zt ) = Ω tot

⎛ −(z − z )2 ⎞ 2 t 0 ⎟ ⎜⎜ exp ⎟ 2 πR ee, z 2 ⎝ 2R ee, z ⎠

(3)

Near the surface (which we indicate with a superscript σ) the assumption is that the number of new conformations due to conformational transfers must equal the number of original conformations. Thus, the number of chains “starting” at z* postswap is Ωσ (z 0 = z*, zt ) = Ωtot

2 πR ee, z 2

⎡ ∞ ⎛ −(z − z*)2 ⎞ t ⎟⎟ dzt ×⎢ exp⎜⎜ 2 ⎢⎣ z * ⎝ 2R ee, z ⎠

∫

−z *

+

∫−∞

⎛ −(z −z ∗)2 ⎞ ⎤ t ⎟⎟ dzt ⎥ exp⎜⎜ 2 R 2 ⎝ ⎠ ⎥⎦ ee, z

(4)

That is, for a chain near the surface with a left-most end at z*, the other end could have either originally been in the region z* < zt < ∞, or it could have been in the region −∞ < zt < −z* and then ended up to the right of z* following the conformational swap. Similarly, the number of chains terminating at z* postswap is Figure 1. Representative configurations of the simulated systems and the corresponding primitive path networks (as obtained via the Z1 algorithm). The top pair of images correspond to a bulk configuration and the bottom pair to a cylindrically confined system, where the orientational ordering of the primitive paths along the cylinder is clear from visual inspection.

Ωσ (z 0 , zt = z*) = Ωtot

∑ zt > z *

Ω(z 0 = z*, zt ) =

∑ z0< z *

⎛ −(z* − z )2 ⎞ 0 ⎟ exp⎜⎜ ⎟ dz 0 2 ⎝ 2R ee, z ⎠ *

z*

×

∫−z

(5)

From here it is straightforward to calculate ensemble averages of interest: one weights the postswap variable of interest with the exponential probability factors of the unperturbed, preswap conformations. For instance, the mean square end-to-end distance for all chains with one end at z* can be written as

conformation that respects the boundary condition. As an introduction to the formulation of the method, we begin by simply counting conformations. In the bulk there are Ωtot z* =

2 πR ee, z 2

Ω(z 0 , zt = z*) (2)

Figure 2. (Left) Root-mean-square component of the end-to-end vector for cylindrical (solid blue line; circles) and thin-film (dashed red line; diamonds) confinement. Lines are the predictions of eq 1, and filled symbols are the simulation results of this work. Upper points indicate components parallel to the surface, lower points indicate components normal to the surface, and the dash-dotted line indicates the bulk value of Ree,bulk/√3. The confinement parameter, δ, is either the effective film thickness or cylinder diameter divided by the bulk root-mean-square end-toend chain distance. (Right) The same plot, but where additionally thin film simulation results from refs 13 and 14 are included as open squares. 6466


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Figure 3. Representative comparisons between theoretical (curves) and simulation (points) results for the orientational distribution of PP segments. (Left) Thin film f(θ) for δ = 1.7 (solid curve; circles) and δ = 0.40 (dashed curve; squares). (Right) Cylindrical confinement f(θ) for δ = 0.84 (solid curve; circles) and δ = 0.34 (dashed curve; squares).

Figure 4. (Left) Normalized number of entanglements per chain for cylindrical (thick solid line; large circles) and thin-film (thick dashed line; large diamonds) confinement. Large points are the simulation results of this work, and the thick curves are the predictions of eq 17. Light blue hexagons are ⟨Z⟩/⟨Z⟩bulk as calculated by the CReTA algorithm for a subset of the cylindrically confined systems.25 (Right) The same plot, but with additional data from the literature. Small diamonds are experimentally estimated data for three different length polymers in thin-film confinement from ref 3, and small squares are thin-film simulations from ref 15. The two thin lower curves show the “anisotropic packing length” argument estimate for entanglement loss in cylinders (thin solid line) and films (thin dashed line) using the theoretically predicted changes in chain end-to-end distances.

2⟨R ee, z 2(z*)⟩ =

B. Conformations in Thin-Film Confinement. To extend this calculation to thin films, where a given chain could interact with both surfaces, we perform a method-of-images-like calculation. In principle, this involves a sum over an infinite number of conformational swaps, but given how rapidly the Gaussians decay the number of terms needed to have a desired degree of quantitative accuracy grows quite slowly. One would need a large number of terms only for very thin films, h ≪ Ree,bulk. For our purposes this criterion is satisfied only either for chains much longer than those we simulate or for film thicknesses smaller than the packing length, h ≲ p. In the latter case the assumptions of the theory no longer hold, since locally the chain would no longer be a random walk. For convenience we denote the bulk mean-squared end-to-end distance as β ≡ (2Ree,bulk2/3)−1, after which we can write the number of chains starting at z* as

2 πR ee, z 2

⎡ ∞ ⎛ −(z − z*) ⎞ t ⎟⎟ dzt (zt − z ∗)2 exp⎜⎜ ×⎢ 2 ⎢⎣ z * ⎝ 2R ee, z ⎠

∫

⎛ − (z + z ∗ ) ⎞ t ⎟⎟ dzt (zt − z ∗)2 exp⎜⎜ 2 2 R ⎝ ⎠ * ee, z ⎛ z* −(z ∗ − z 0) ⎞ ⎟⎟ dz 0 (z* − z 0)2 exp⎜⎜ + 2 0 ⎝ 2R ee, z ⎠ ⎤ ⎛ − (z * + z ) ⎞ z* 0 ⎟ ⎥ (z* − z 0)2 exp⎜⎜ d z + ⎟ 0⎥ 2 0 ⎝ 2R ee, z ⎠ ⎦ +

∫z

∞

∫ ∫

(6)

The first two integrals average over chains starting at z* and the last two correspond to integrals over chains ending at z*; the factor of 2 on the left-hand-side of the equation is to avoid double counting. Evaluating these simple integrals and normalizing by the bulk end-to-end distance gives 2

R ee, z (z*) R ee,bulk 2/3 +

=1−

4 3 z* 2πR ee,bulk

⎛ ⎛ 6(z*)2 ⎜ 3 z* ⎜ − 1 erf ⎜ 2⎜ R ee,bulk ⎝ ⎝ 2R ee,bulk

Ωσ (z 0 = z*, zt ) = Ωtot2 −z *

⎛ −3(z*)2 ⎞ ⎟ exp⎜⎜ 2⎟ ⎝ 2R ee,bulk ⎠

⎞⎞ ⎟⎟ ⎟⎟ ⎠⎠

+

∫−h

+

∫h

+

∫−2h+z

2h − z *

3h

(7) 6467

∫2h+z

∫

exp( −β(zt − z*)2 ) dzt

−h

+

β⎡ h exp( −β(zt − z*)2 ) dzt ⎢ π ⎣ z*

*

exp( −β(zt − z ∗)2 ) dzt exp( −β(zt − z ∗)2 ) dzt

⎤ exp( −β(zt − z*)2 ) dzt + ...⎥ ⎦ *

(8)


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Figure 5. Average mean-squared end-to-end distance normalized by the bulk value as a function of position within the film. The dotted curve is Silberberg’s “thick film” calculation, the solid curve is the first correction, and the dash-dotted curve is the second correction. (Left) corresponds to a film thickness of h = (2/√3)Ree,bulk and (Right) to h = (5/√3)Ree,bulk.

Parsing this expression, the first term corresponds to chains with a terminus originally between z* and the second surface. The next two terms correspond to chains with a terminus originally in the regions −h < zt < −z* or h < zt < 2h − z* and thus needed a single conformational swap to respect the boundaries. Similarly, the next two terms correspond to chains that required two swaps to have the terminus in the appropriate z* < zt < h region, and so on. Completely analogously, the number of chains terminating at z* postswap is Ωσ (z 0 , zt = z ∗ ) = Ωtot2 0

+

∫−z

*

∫2h−z

*

∫−2h ∫2h

1/(βπ ) (erf( β (h(n + 1) − z ∗)) + erf( β (nh + z*))) ⎡ h ×⎢ (z − z*)2 exp(− β(zt − z ∗ )2 ) dzt ⎣ z* t

∫

h

exp(− β(z* − z 0)2 ) dz 0

+

∫z* (zt − z*)2 exp(−β(zt + z*)2 ) dzt

+

∫h

+

∫0

+

∫0

+

∫2h−z

2h − z * z* z∗

2h

2h + z *

+

z*

=

exp(− β(z* − z 0)2 ) dz 0

− 2h + z *

+

∫0

R ee,bulk 2/3

exp(− β(z* − z 0)2 ) dz 0

2h

+

β⎡ ⎢ π⎣

R ee 2(z*)

exp(− β(z* − z 0)2 ) dz 0

⎤ exp( − β(z* − z 0) ) dz 0 + ...⎥ ⎦

(z ∗ − z 0)2 exp(− β(z* − z 0)2 ) dz 0 (z* − z 0)2 exp(− β(z* + z 0)2 ) dz 0 ⎤ (z* − (2h − z 0))2 exp( − β(z* − z 0)2 ) dz 0 + ...⎥ ⎦ * (11)

This corresponds to a straightforward (if tedious to write out) combination of exponentials and error functions. Figure 5 shows a few representative results and a comparison with the unmodified Silberberg calculation and the first few thin film corrections. To make this comparison, we have done the most natural thing and used the “thick film” version of the calculation up to half of the film thickness and then mirrored the result for the other half of the film. As the figure makes clear, for reasonably thick films (e.g., h ≥ (5/√3)Ree,bulk) the original calculation is nearly indistinguishable from the corrected version (except near the middle of the film), but for thinner films the original calculation is both quantitatively and qualitatively in error. C. Conformations in Cylindrical Confinement. The case of cylindrical confinement is no more complicated than the extension to thin films above. Here instead of boundaries at z = 0 and z = h we work in cylindrical coordinates and impose a boundary at r = h/2, where h is the diameter of the desired cylinder. The distribution of chain end-to-end vectors is

2

(9)

As a minor point, because we are truncating the infinite series, note that some population of chains with (preswap) ends far outside of the surfaces are neglected. The total number of chains with either end at z* is Ωσ = Ωtot(erf( β (h(n + 1) − z*)) + erf( β (nh + z*)))

((2h − zt ) − z ∗ )2 exp(− β(zt − z*)2 ) dzt

(10)

where n is the maximum number of swaps/reflections we consider. Letting n → ∞ gives Ωσ = 2Ωtot, as one would expect. Applying precisely the same logic as before, where the postswap distances are weighted with the preswap probabilities, the mean-squared end-to-end distance normalized by the bulk value can be computed. To keep the following equation manageable, we explicitly write down only the first correction to Silberberg, i.e., considering at most one chain end swap across either surface. Manipulating some of the limits on the integrals for convenience, the result is

Ω(r , θ , r0 , θ0) =

3r 2πR ee,bulk 2 ⎛ r 2 + r 2 − 2rr cos(θ − θ ) ⎞ 0 0 0 ⎟ × exp⎜⎜ − ⎟ 2R ee,bulk 2/3 ⎝ ⎠ (12)

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mimicked at the primitive path level. The simplest assumption is that the global changes in end-to-end vectors are transmitted to the orientation of the primitive paths via an affine transformation. That is, we take the deformation tensors to be

One proceeds exactly as before to calculate, for instance, the average mean-squared x component of the end-to-end-vector as a function of position within the cylinder. Explicitly writing just the first few terms, again allowing up to n reflections, one obtains 3R ee, x 2(r′, θ′) R ee,bulk 2 ⎡ ×⎢ ⎣

1

=

∫0

h /2

∫r′ ∫0

nh /2

⃡ Efilm

2π

∫0 Ω(r , θ , r′, θ′) dθ dr

2π

2

(rt cos θt − r′ cos θ′)

⎛1 0 0 ⎞ ⎜ ⎟ = ⎜ 0 1 0 ⎟; ⎜ ⎟ ⎝ 0 0 αfilm ⎠

+

∫h/2 ∫0

2π

g (ui⃗ , E⃡ ) =

((h − rt ) cos θt − r′ cosθ′)2

r′

∫0 ∫0

2π

(r′ cos θ′ − r0 cos θ0)2

× Ω(r′, θ′, r0 , θ0) dθ0 dr0 h

+

∫h−r′ ∫0

2π

(r′cos θ′ − (h − r0) cos θ0)2

⎤ × Ω(r′, θ′, r0 , θ0) dθ0 dr0 + ...⎥ ⎦

1 4π ⟨|E⃡ ·ui⃗ |⟩0

⎛

∫ du⃗′ |E⃡ ·u⃗′|δ⎜⎝ui⃗ −

E ⃡ · u ⃗′ ⎞ ⎟ |E⃡ ·u ⃗′| ⎠

(15)

where ⟨...⟩0 = ∫ ...du⃗i/4π is an average over an isotropic distribution of orientations.36 Figure 3 compares this prediction with the orientational distribution of PP steps obtained in simulations for different degrees of confinement in the two geometries studied. The agreement between simulations and theory is striking, supporting our assumption that global chain conformations are transferred to the primitive path level through an affine transformation. There are slight systematic discrepancies consistent with our neglect of chain extension parallel to the confining surfaces, but this is a strong indication that at the level of entanglements the confined melt is dominated by changes in conformation normal to the surface. Two of the authors have recently proposed a microscopic dynamical theory relating the tube diameter to the PP orientational order.37 The theory starts by coarse graining each chain into a random walk of Z primitive path steps of size Le = σ√Ne. Crucially, Ne is not an adjustable parameterit is self-consistently determined based on a relationship between Le and the transverse localization of the primitive path step, rl ≡ dT/2. To describe chain uncrossability, the interactions between pairs of PP steps are treated as if they were uncrossable needles;38,39 this models chain uncrossability and the orientational dependence of the uncrossability constraint in the simplest possible way. After quenching the longitudinal motion of PP steps, a self-consistent expression for the size of the transverse fluctuations as a function of orientational order is derived:37

× Ω(rt , θt , r′, θ′) dθt drt +

(14)

The PP orientational distribution function can then be written as

× Ω(rt , θt , r′, θ′) dθt drt h−r′

⃡ Ecyl

⎛ αcyl 0 0⎞ ⎜ ⎟ = ⎜0 αcyl 0 ⎟ ⎜ ⎟ ⎝0 0 1⎠

(13)

Averaging eq 13 over the circle r′ < h/2 gives the confinementaveraged Ree,x2 for a cylinder of diameter h. Figure 6 compares

Figure 6. Component of the mean-squared end-to-end distance normal to a surface normalized by its bulk value as a function of confinement. The solid curve is the film-averaged Ree,z2 for a thin film of thickness h, and the dash-dotted curve is the cylinder-averaged Ree,x2 for a cylinder of diameter h.

⎛L ⎞ ρPP Le 3 1 = F⎜ e ⎟ 2 rl 16π 2rl 2 2 ⎝ rl ⎠

∫ du1⃗ du2⃗ g(u1⃗ )g(u2⃗ )

1 − (u1⃗ ·u 2⃗ )2

(16)

where ρPP is the density of PP steps and F(x) is a function whose analytic form can be found in ref 40; this function captures the effect of the path step aspect ratio, Le/rl, on the localization length. This ratio must be specified, but we have shown that as long as Le ≳ rl the theory is only very weakly sensitive to the aspect ratio,37 and the localization length quickly asymptotes to a constant value. For concreteness we adopt the standard Doi−Edwards coarse graining36 of Le = dT = 2rl, and for an isotropic g(ui⃗ ) eq 16 can be rewritten as rl = 2p√2/(πF(2)) → L e = 2r l = 10.2p. This result is commensurate with the rheological estimation of dT ≈ 17.7p for hundreds of flexible chain polymer melts.32 Furthermore, it has recently been argued that rheological measurments of the tube diameter should be a factor of √2 larger than purely topological measures of it (as in our PP analysis);41 this corresponds to an experimental estimate of the “topological”

the result of this cylinder-averaged change in the end-to-end vector with the film-averaged change in Ree,z2 for thin film geometries. As expected, there is a modestly larger effect in cylindrical confinement since the boundaries are restricting a larger amount of space, enforcing conformation swaps in some cases where the thin film geometry would not. Nevertheless, the differences are relatively minor.

IV. CONNECTING CONFORMATIONAL CHANGE TO ENTANGLEMENT STRUCTURE The conformational changes predicted in the previous section and observed in our simulations clearly also induce a change in the orientational distribution of end-to-end vectors. As we are interested in the entanglement properties of the confined melt, it is natural to ask whether the global chain orientations are 6469


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tube diameter of dT,topo ≈ 12.2p, suggesting an even closer agreement between our calculation and experiment. In anisotropic melts, and particularly in the present case of confinement where there is a spatially dependent local orientational order, employing eq 16 and defining just one “tube diameter” corresponds to studying a spatially averaged tube diameter as a function of orientational order. A more sophisticated version of the theory could be written down, with three coupled, self-consistent equations defining rl,x, rl,y, and rl,z. This would address the question of not just how the average entanglement density of a system changes as a function of orientational order, but the anisotropy of a “tube diameter” when the PP steps have a net orientational order. One could perhaps study this in simulations by investigating the distribution of PP step lengths as a function of the local PP orientation. Alternately, following the isoconfigurational averaging work of Bisbee and co-workers, the transverse fluctuations of the chain segments belonging to a given PP step can be investigated in molecular dynamics simulations as a function of the local orientational environment of that step.42 In our current simulations we did not have enough data to extract statistically significant results for such effects, and so we adopt the coarser level of description embodied in eq 16. Thus, to study the changes in entanglement density in the confined systems of present interest, we combine this theory with the results of section III. For a given confinement geometry and scale δ the generalized principle of conformational transfer gives the change in end-to-end vector. An affine transformation gives the distribution of PP orientations via eq 15, which is then used in our dynamic theory to calculate a filmor cylinder-averaged tube diameter. The mean change in entanglement number for the two confinement geometries

As an illustration, Figure 7 shows precisely this calculation of the spatially resolved average degree of entanglement reduction

Figure 7. Degree of entanglement reduction for thin films as a function of the position of chain ends in the film. From bottom to top, curves correspond to δ = 0.5, 0.75, 1.0, 1.5, and 3, and z* is in units of the bulk end-to-end vector magnitude.

(17)

for thin films with δ ranging from 0.5 to 3. For the thickest film (top solid curve), bulk behavior is retained for chains with an end precisely at the surface, followed by a small reduction of entanglement density close to the wall and then a recovery of bulk behavior. For δ = 1.5 the behavior at the wall is comparable to the bulk, but the degree of entanglement now attains a minimum near the film center. For the thinnest films the entanglement density is everywhere less than the bulk value, and the minimum continues to be achieved for chains with ends near the middle of the film. The apparent fact that the results reported in Figure 7 have a value close to the walls that depends only very weakly on film thickness is partly due to the fact that z* refers to the position of a chain end and not the chain centers of mass (the distribution of which we have not computed here).

is reported in Figure 4. The agreement between theory and simulation is nearly quantitative and again demonstrates how powerfully multiple confinement directions affect the entanglement density of the confined system. Whether, as seen in Figure 3, the anisotropy of the PP network always directly reflects the anisotropy of chain end-to-end vectors is unclear, but for the chain sizes and confinement scales studied here we find that this is the case. Since the global chain properties are much easier to calculate than PP-scale subchain conformations, this raises the exciting prospect that entanglement properties of confined systems can be predicted from first principles with relatively modest effort. Although in this work our simulations do not provide sufficient statistics to compare against, our theoretical approach can straightforwardly predict entanglement reduction due to confinement in a spatially resolved way. This question is of great interest for problems such as wall slip near surfaces and inhomogeneous entanglement density in polymer nanocomposites. To address this problem, note that eqs 11 and 13 give predictions for average chain conformations as a function of position within the sample. Instead of averaging this data to get a system-averaged conformation to use in the deformation tensor in eq 14, one could instead compute, e.g., a spatially resolved thin film deformation tensor based associated with the average conformation of chains with an end at z* and use this to construct ⟨Z(z*)⟩/⟨Z⟩bulk.

V. CONCLUSION In summary, we have systematically studied average measures of conformational change and entanglement density dilution in strongly confined polymer melts using both theory and molecular-dynamics simulations. Studying two distinct confinement geometries with neutral interactions with the confining walls, our simulations reveal conformational changes in response to geometric confinement. The simulations further show a systematic reduction in the entanglement densityas studied by multiple primitive-path measuresas the confinement induces progressively more orientational order in the polymer melt. To understand these results, we have generalized theoretical ideas to allow first-principles, quantitative, adjustable-parameter-free predictions for the change in the end-toend vector and average number of entanglements per chain. Very good agreement between theory and simulation is found. In addition to direct analyses of polymer liquids, this suggests that our theoretical perspective connecting local orientational order to local entanglement density may be used to inform other more highly coarse-grained approaches, such as slip-link models of entangled polymer liquids.43 Our results for thin films are roughly consistent with the findings of Baschnagel and co-workers15 except for the thinnest films they studied. On the basis of that chain length and our model, we expect Z would be only slightly greater than 2; this is close to the disentanglement crossover, and so we speculate

⟨Ne⟩bulk ⟨r 2⟩ ⟨Z(δ)⟩ = = l 2 bulk ⟨Z⟩bulk ⟨Ne(δ)⟩ ⟨rl (δ)⟩

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that chain-length effects may be present in their analysis. We also note that in that work entanglements were analyzed using the classical PPA, which as noted above may also affect primitive path orientation distributions in an undesirable way. In contrast, for moderate to strong confinement our thin film simulation and theoretical results are in modest disagreement with prior experiments.3 However, we note that the experiments employed a nonlinear crazing measurement to extract the linear response quantity Z using an approximate isotropic model of the entanglement network. This point accentuates our primary finding: the entanglement network becomes highly anisotropic under confinement, and this anisotropy directly translates to a loss of entanglement. Taking this observation a step further and studying the tensorial nature of the tube diameter in anisotropic systems is an avenue of future study, albeit one that is much easier to study theoretically and in simulations than experimentally. In this work we studied system-averaged quantities, but as discussed in section III, the principle of conformational transfer allows one to predict chain conformation properties as a function of distance from a boundary. Thus, as illustrated in Figure 7, one can apply our methods to predict the tube diameter as a function of position in the sample. This implies the theoretical formulation is relevant not only to strongly confined polymers, but more broadly to the general phenomena of entanglement dilution at boundaries and interfaces, where questions such as slip and lubrication arise.44,45 The change in the entanglement network in polymer nanocomposites is a subtle issue that the present work could also be extended to address, but where the shape, size, spatial distribution, and volume fraction of the nanoparticles are crucial features.4,46−49 We note that the theoretical approach employed here has been recently extended to polymer nanocomposites.50 Hard fillers are predicted to result in tube narrowing for both entangled needles and flexible chains, with the latter in good agreement with a recent simulation.48 Finally, we have established that confined polymer melts exhibit anisotropic changes in polymer conformation and topological entanglement, and our future work includes exploring how this anisotropy impacts the melt diffusion, rheology, and elastic response in these confined systems.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (D.M.S.). *E-mail [email protected] (R.A.R.). Author Contributions

D.M.S. and W.-S.T. contributed equally to this work. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS D.M.S. acknowledges support from DOE via DE-FG0203ER46088 and NSF EFRI13-31583; K.S.S. acknowledges support from DOE-BES Materials Science and Engineering via Oak Ridge National Laboratory; W.S.T. and K.I.W. acknowledge NSF-DMR-Materials World Network (1210379); R.A.R. acknowledges funding from the University of Pennsylvania.

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REFERENCES

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