Hopping Diffusion of Nanoparticles in Polymer Matrices.

We propose a hopping mechanism for diffusion of large nonsticky nanoparticles subjected to topological constraints in both unentangled and entangled polymer solids (networks and gels) and entangled polymer liquids (melts and solutions). Probe particles with size larger than the mesh size ax of unentangled polymer networks or tube diameter ae of entangled polymer liquids are trapped by the network or entanglement cells. At long time scales, however, these particles can diffuse by overcoming free energy barrier between neighboring confinement cells. The terminal particle diffusion coefficient dominated by this hopping diffusion is appreciable for particles with size moderately larger than the network mesh size ax or tube diameter ae . Much larger particles in polymer solids will be permanently trapped by local network cells, whereas they can still move in polymer liquids by waiting for entanglement cells to rearrange on the relaxation time scales of these liquids. Hopping diffusion in entangled polymer liquids and networks has a weaker dependence on particle size than that in unentangled networks as entanglements can slide along chains under polymer deformation. The proposed novel hopping model enables understanding the motion of large nanoparticles in polymeric nanocomposites and the transport of nano drug carriers in complex biological gels such as mucus.

Introduction

Mobility of nonsticky nanoparticles in complex fluids,[1,2] including polymer solutions and melts,[3−10] biomacromolecular solutions,[11−21] cells,[22−27] extracellular environments,[28,29] and colloidal suspensions,[30] reflects local structure and dynamics of these complex matrices. In previous work,[31,32] we showed that the motion of particles in polymer liquids (solutions and melts) is different on short and long time scales because it is determined by the dynamics of surrounding polymers. At relatively short time scales the motion of particles is subdiffusive as it is coupled to the segmental dynamics of polymers, whereas at relatively long times the particle motion is diffusive. The terminal diffusion coefficient of particles does not depend on the polymer molecular weight if the particle size is smaller than the entanglement length of the polymer liquids. The diffusion coefficient of particles larger than the entanglement length decreases with 3.4 power of the molecular weight of linear polymers, because particles probe the terminal dynamics of entangled polymers. These predictions have been verified by a recent systematic experimental study.[33] The diffusion coefficient of these large particles decreases with increasing matrix viscosity; however, the large particles can still move as the polymer liquids relax, no matter how slow the relaxation is. The question then is as follows: What is the particle mobility in polymers that cannot relax, for example, in permanently cross-linked networks? Can particles move through permanently cross-linked networks if their size is larger than the network mesh size? A prior work[43] addressed this question via a mode-coupling approach and obtained the fact that the hopping distance of large particles in polymer melts increases with particle size, while the hopping energy barrier asymptotically varies proportionally to the particle volume, which is qualitatively different from the results of our paper. On the basis of our prior work,[31,32] we present a systematic scaling description for diffusion of such large particles in both unentangled and entangled polymer solids (networks and gels) and entangled liquids (melts and solutions).

We argue that in permanently cross-linked networks the only way for a particle with size exceeding the network mesh size to leave a confinement cage is by hopping—waiting for the fluctuation of a gate (loop) between two neighboring confinement cages to become large enough to slip around the particle. To describe the particle hopping diffusion, we introduce two important parameters, hopping free energy barrier and hopping step size. The free energy barrier is determined by the deformation of a loop (gate) as it slips around the particle. To describe particle confinement in unentangled networks with strongly overlapping chains and to estimate the hopping step size, we introduce a model of “overlapping elementary networks”.

The idea of hopping diffusion of particles in unentangled permanent networks is extended to describe diffusion of particles in entangled polymer networks, which contain both permanent cross-links and topologically trapped entanglements. Unlike the fixed permanent cross-links, entanglements can slide along polymer chains. Therefore, the confinement cages due to entanglements are “softer” in comparison to the network cages formed by permanent cross-links. Consequently, there is a range of particle sizes for which the particle diffusion is dominated by hopping of particles between entanglement cages.

Unlike entangled polymer networks, there are no permanent cross-links in entangled polymer liquids; polymers in entangled liquids can relax on long time scales. We show that there is a certain (polymer molecular weight dependent) particle size above which the hopping diffusion of particles between entanglement cages becomes very difficult; it is relatively easier for the very large particle to diffuse at long time scales by “waiting” for the entangled polymer liquid to relax and flow around the particle.

The paper is structured as follows. In section 2, we discuss hopping diffusion of large particles in unentangled polymer solids (networks and gels). To estimate the step size and entropic free energy barrier for hopping, we model the “real” unentangled network by many overlapping “elementary” networks. Hopping diffusion of particles in entangled polymer solids is discussed in section 3. Section 4 presents the diffusion of probe particles in entangled polymer liquids. Specifically, we compare the particle motion due to the relaxation of polymer chains and that due to hopping and calculate the range of parameters for which each of two diffusion modes dominates. Concluding remarks are presented in section 5.

Unentangled Polymer Solids (N < N)

A typical polymer network has both chemical cross-links and topological entanglements. The properties of the polymer network are dominated by the type of constraints that has higher density. Therefore, we distinguish two types of networks: unentangled and entangled polymer networks, depending on the relative values of network mesh size and entanglement length scale. The network mesh size is defined as the average distance between two neighboring permanent cross-links along the chain and can be measured by the value of unentangled network elastic modulus. The entanglement length is measured by the magnitude of entanglement plateau modulus.[34−36] The case of high density of permanent cross-links, with the network mesh size smaller than the entanglement length, is classified as unentangled polymer network. In the opposite case of entangled polymer network the network mesh size is larger than the entanglement length and the network properties are dominated by the topological entanglements between network strands.

Consider the motion of a probe particle of size d in a dry, monodisperse unentangled permanently cross-linked network above its glass transition temperature Tg and crystallization transition temperature T. Let us denote the number of Kuhn monomers between two neighboring cross-links by N and the size of a network strand by a ≃ bN1/2, where b is Kuhn length. In a typical network there are many network strands overlapping within the volume pervaded by a network strand (see Figure 1). The overlap parameter P ≃ N1/2 is defined as the number of network strands within the volume a3 ≃ (bN1/2)3 pervaded by one network strand.

Figure 1

Unentangled polymer network modeled by overlapping “elementary” networks. (a) Schematic visualization of a particle of size d in an unentangled polymer network with strands of average size a. The solid circles represent permanent cross-links. There are P ≃ N1/2 network strands within the pervaded volume a3 of a network strand. (b) The unentangled polymer network is modeled by P overlapping yet independent “elementary” polymer networks, with details discussed in Appendix A. One of these “elementary” networks is shown by bright black lines while the remaining P – 1 “elementary” networks are shown by dimmed color lines.

Hopping Diffusion

A large probe particle of size d (d > a) is confined inside the unentangled network, as shown in Figure 1a. However, it is still possible for this particle to escape from the cage formed by network strands confining the particle. This escape–a hopping step–occurs by a large fluctuation of the one of these strands. To understand this process, we model the monodisperse unentangled network by P overlapping yet independent “elementary” networks, as shown in Figure 1b. The details of this representation are discussed in Appendix A. There is on the order of one network strand per volume a3 in each of these “elementary” networks.

The “elementary” networks are not static; instead, they are fluctuating all the time. The fluctuation of an “elementary” network cage can be large enough to allow one of the network strands to slip around the particle. In this case a hopping event occurs and the particle enters a neighboring cage of the network. We use the model of “elementary” networks to elucidate the hopping diffusion of large particles.

Hopping Step Size

Each of P “elementary” networks constrains the particle independently and tends to localize the particle at the “center” of its own cage. We model the constraint from an “elementary” network by a virtual chain with one of its two ends attached to the particle center and the other anchored at the center of the “elementary” network cage, as illustrated by the dashed lines and black dots in Figure 2a. The number of monomers, ncage, per virtual chain is determined by equating its elastic energy and the elastic deformation energy of an “elementary” network when the particle is shifted from its equilibrium position by a distance δr (see Appendix C):which gives

Figure 2

Step size of a large probe particle hopping between two neighboring cages in a monodisperse unentangled polymer network. (a) The unentangled polymer network is modeled by P overlapping “elementary” networks with their network cage centers r1, ..., r, ..., r (dots) randomly distributed around the fluctuation center of the particle (its equilibrium position). The constraint applied to a large particle of size d > a from an “elementary” network is modeled by a virtual chain with ncage monomers. The virtual chain has one of its two ends attached to the particle center, and the other anchored to the center of a cage of ith “elementary” network, as shown by the black dots. (b) During a single hopping event the particle leaves its initial equilibrium position O and arrives at a neighboring equilibrium position O′ with a step size Δr. This hop is achieved by the particle leaving the confinement cage of the “elementary” network i with the center r that is most likely to be the furthest from the initial equilibrium position O of the particle and entering the neighboring cage of the same “elementary” network with the center at r + a, while staying in the same cages of all the rest “elementary” networks.

Instead of fluctuating around the cage “center” in a particular “elementary” network, the particle finds an “optimal” position at which the restoring forces from all the “elementary” networks are balanced. The centers of cages in P “elementary” networks are randomly distributed within the volume on the order of a3 around the equilibrium position of the probe particle, as illustrated in Figure 2b. At this equilibrium position the net force exerted by these P “elementary” networks on the particle is zero. The restoring force f applied to the particle from the jth “elementary” network is linearly proportional to the deviation δr of the particle from the “center” of the “elementary” network cage, f = [kT/(b2ncage)] δr, as the confinement potential is parabolic (see eq 1 and Appendix C). Assuming that all virtual chains have the same number of monomers ncage, we haveThe hopping step size for a large probe particle (d > a) moving through the unentangled network is much smaller than that for an “elementary” network because the particle is constrained by many surrounding overlapping “elementary” networks. During a single hopping step the particle moves by a displacement Δr and arrives at a new equilibrium position. The particle most likely escapes from the cage of an “elementary” network i, whose center is at the maximum distance from the equilibrium position of the particle, as the corresponding free energy barrier is the lowest compared with that of other “elementary” networks.[38] Consequently, after leaving the cage center r of the “elementary” network i, the particle enters the neighboring cage, whose center r + a is separated by a vector a from the old cage, and deviates by a vector δr′ = δr + Δr – a from the new equilibrium position. Note that deviations from the cage centers of all other P – 1 “elementary” networks to the new equilibrium position of the particle are changed by a vector Δr (see Figure 2). Since at this new equilibrium position the net force exerted on the particle by P “elementary” networks is still zero, one obtains the equation for the step size Δr of particle hopEquation 4 can be rewritten using eq 3 as PΔr – a = 0, which gives the magnitude of the step size of particle hop in a dry networkIt is important to emphasize that this displacement Δr is P ≃ N1/2 times smaller than the hopping step size a ≃ bN1/2 in a single “elementary” network.

Hopping Entropic Free Energy Barrier

To hop from one cage to a neighboring one, the large probe particle has to overcome a free energy barrier, which is defined as the difference between the maximum and the initial elastic deformation energy of the network strands during the hopping event. To estimate the energy barrier, one might think that it is necessary to consider the deformation energy of all P (d3/a3) network strands affected by the large probe particle,[43] which is the number of network strands d3/(b3N) within its pervaded volume d3. However, not all of the affected network strands are deformed in the same way during a single hopping event. Indeed, it is enough for one network loop to slip around the particle for this particle to hop between neighboring cages. Stretching the slipping loop also results in further deformation of loops connected to it. However, deformation of all other affected loops can be taken into account using the concept of “virtual chains”, which, effectively, only renormalizes the length of the slipping loop. The size of a loop in “elementary” networks is about a (see Appendix A).

The energy barrier due to the slipping loop corresponds to the “transition” state in which the large particle is leaving the initial cage and is at the onset of entering the neighboring cage (see “transition” state in Figure 3). In this state the network loop is stretched from length a to the order of particle size d (in fact, the peripheral length πd of the particle). Therefore, the entropic free energy barrier contributed from the deformation of the loop slipping around the particle (barrier loop) during a single hopping event is

Figure 3

Illustration of a large probe particle hop from one network cage to a neighboring cage with only one network loop (highlighted by red) slipping around the particle.

Dry Unentangled Polymer Networks

A loop of size larger than d can slip around the particle, resulting in a hopping step. The hopping free energy barrier (eq 6) determines the probability of a loop fluctuating to a size larger than the particle size d. This probability is proportional to ∫∞ dx exp (−x2/a2)/a. The waiting time for this hopping step in a dry unentangled polymer network iswhereis the Rouse relaxation time of a network strand, at which a loop attempts to slip around the particle but is unlikely to succeed utill τnet. The monomer relaxation time τ0 in eq 8 iswith ζ corresponding to the monomeric friction coefficient.

The mean-square displacement for a large particle hopping in a dry unentangled polymer network is proportional to the number of steps t/τnet that the particle makes during a certain period of timeThe hopping process occurs on time scales longer than the relaxation time of a network strand τ, but with a very small probability to succeed during this relatively short time interval. The probability of a hop increases with time interval and becomes significant enough for a successful hopping to occur at the waiting time scale τnet (eq 7). In addition to hopping the particle is fluctuating within the network cells without leaving them at times longer than relaxation time of a network strand τ. We model the total restoring force on the particle due to P “elementary” networks by elastic force from a composite virtual chain, which consists of P virtual chains with ncage monomers connected in parallel, as shown in Figure 2. The number of monomers ncage/P of such a composite virtual chain determines the mean-square fluctuations of the particle on time scales longer than network strand relaxation time:One can use microrheological approach to estimate the network modulus from the amplitude of these thermal fluctuations of the particle: G ≃ kT/(d⟨r2⟩fluctnet) ≃ kTP/a3 (see eq C.3).

We would like to stress that the particle motion at times t > τ is due to the superposition of the two processes: fluctuations around the center of a network cage but without leaving it (eq 11) and hopping between neighboring network cages (eq 10). The contribution to the particle mean-square displacement from hopping ⟨r2 (t)⟩hopnet becomes important at certain time scale τhopnet, at which ⟨r2(t)⟩hopnet is comparable to the mean-square displacement ⟨r2⟩fluctnet due to particle fluctuations within the confinement cage. This gives the crossover time at which hopping diffusion becomes observableThe mean-square displacement of the particle does not significantly increase until time scale τhopnet, as shown in Figure 4. At time scales longer than τhopnet the mean-square displacement of large particles is dominated by the hopping diffusion (see eq 10):The particle diffusion coefficient due to hopping in a dry unentangled network (see eq 10) isFor a relatively large particle, the hopping diffusion is extremely slow as the mean-square displacement of particles decreases exponentially with the square of particle size.

Figure 4

Time dependence of the product of mean-square displacement ⟨Δr2(t)⟩ and the particle size d for large particles subjected to the confinement from cages of size a. The motion of large particles (d > a) is not affected by confinement cells at time scales shorter than the relaxation time τ (see eq 23) of a strand. The particle motion is ballistic at very short time scales (t < τbal; eq 18) and crosses over into subdiffusive at longer time scales (τbal < t < τ). At time scales longer than τ the particles are trapped by confinement cells; they cannot move until time scale τ, at which the particles start to hop between neighboring confinement cells. The hopping diffusion becomes experimentally observable on time scale τhop, at which mean-square displacement of the particle due to hopping becomes comparable to that due to fluctuations of the particle within a confinement cell, a2b/d (eq 11). For dry unentangled polymer networks a ≡ a, τ ≡ τ (eq 8), τ ≡ τnet (eq 7), and τhop ≡ τhopnet (eq 12); for entangled polymer networks and melts, a ≡ a, τ ≡ τ (eq 23), τ ≡ τent (eq 22), and τhop ≡ τhopent (eq 26). Logarithmic scales.

At times shorter than the relaxation time τ (eq 8), the motion of a large probe particle (d > a) is unaffected by network cages and is similar to particle movement in polymer melts. The particle motion on times t < τ is subdiffusive with the mean-square displacement proportional to the square root of time, since particle motion is coupled to the segmental dynamics of network strands, as shown by the solid line with the slope 1/2 in Figure 4.[31]The effective viscosity ηeff(t) “felt” by the particle during time interval t corresponds to a melt with chains containing (t/τ0)1/2 monomers coherently moving at this time.[31]

At times shorter than the crossover time τbal, the particle moves along ballistic trajectories. The mean-square velocity of the particle v is determined by the equipartition theoremSubstituting r = υt in eq 16 we find the particle mean-square displacement for ballistic motion (see Figure 4):Matching eqs 15 and 17 at t = τbal gives the crossover time between the ballistic and subdiffusive regimes:The width of the ballistic regime is determined by the time scale τbal. For large particles with higher density ρ in liquids of low viscosity, it is possible that τbal > τ0, or mb/(ζτ0d) > 1. This dimensionless ratio can be written as mb/(ζτ0d) ≃ (d/b)2 (ηref/η)2, where the reference viscosity ηref = (ρkT/b)1/2 is determined by the particle density ρ and Kuhn length b with a typical value ηref ∼ 10–4 Pa·s. Thus, for the ratio of the particle to Kuhn monomer size larger than the ratio of viscosities d/b > η/ηref the ballistic regime ends at time scales longer than τ0 and “eats” part of the subidiffusion regime. Indeed, the ballistic regime is experimentally observable by using particle tracking of ultrahigh temporal-spatial resolutions.[44] The ballistic regime puts the short-time cutoff at τbal to the application of microrheology, since the particle motion at time scales t < τbal is not coupled to the modes of the probed matrix. The long-time cutoff to the application of microrheology is determined by the hopping process, as the finite zero-shear-rate viscosity predicted by the generalized Stokes–Einstein approach from particle diffusion at time scales t > τhopnet does not correspond to the infinite zero-shear-rate viscosity of a permanent network.

Unentangled Polymer Gels

An unentangled polymer gel can be treated as an “effective” unentangled dry polymer network in which the “effective” monomers are correlation blobs. Therefore, the results of particle hopping in dry polymer networks can be directly applied to polymer gels with hopping step size b replaced by the correlation length ξ (see eq D.2) and other parameters replaced by concentration dependent ones (see eqs D.3 and D.4 in Appendix D). The key elements for hopping diffusion of large particles in unentangled polymer gels are summarized in Table 1 and their detailed discussion is presented in Appendix D.

Table 1

Parameters for Hopping Diffusion of Particles in Unentangled Polymer Solids and Entangled Solids and Liquidsa

	unentangled solids		entangled solids and liquids
	dry networks	gels	dry networks and melts	gels and solutions
Δr	b	ξ	b	ξ
ΔU/kBT	d2/ax2	d2/ax2	d/ae	d/ae
Dhop	(b2/(τxd/ax)) exp(−d2/ax2)	(ξ2/(τxd/ax)) exp(−d2/ax2)	(b2/τe) exp(−d/ae)	(ξ2/τe) exp(−d/ae)

Δr is the hopping step size, ΔU is the entropic free energy barrier, and Dhop is the hopping diffusion coefficient. Length scales: b is the size of a Kuhn monomer, ξ is correlation length, a is the network strand size, and d is the particle size. Time scales: τ0 is the monomeric relaxation time, τξ is relaxation time of correlation volume, τ and τ correspond to the relaxation time of a network and entanglement strand respectively.

Note that we consider above only monodisperse polymer networks. However, real polymer networks are polydisperse as they are typically made of strands of different molecular weight. The motion of a large particle in real polymer networks could be affected by the polydispersity. Interestingly, we find that there is still a window in which the particle motion is not affected by the network polydispersity even for particles larger than the average network loop size (see Appendix B). Within this window, the variation of energy barriers due to polydispersity is smaller than the thermal energy kT. As a result, polydispersity of the network becomes important only for very large particles with size larger than average loop size d > al̅1/2, in which l̅ ≃ ln P is the average number of network strands in a loop (see Appendix A). These particles diffuse extremely slow with exponentially small diffusion coefficient. This interesting behavior will be the subject of future explorations.

Entangled Polymer Solids: Entanglement Cages Are “Softer”

Fluctuations of chains in polymer solids (networks and gels) are suppressed by both permanent cross-links and entanglements. In entangled polymer solids the density of permanent cross-links is lower than the density of entanglements so that the tube diameter a is smaller than the network mesh size a. The hopping diffusion of large particles with size a < d < a2/a in entangled polymer solids can be readily obtained by extending the results of hopping diffusion of large particles in unentangled polymer solids with a < a (see section 2). However, unlike the loop formed in permanently cross-linked networks, the loop formed by entanglements does not have fixed number of monomers. Indeed, the number of monomers contained in a particular loop increases when the loop slides over the particle because some chain segments far from the particle are pulled in as the particle slips through the loop. The number of monomers n in the loop is determined by the condition that the tension in the loop is balanced by the tube tension kT/a:in which the term kTd/(nb2) corresponds to the tension in the loop of n monomers, and the tube diameter (entanglement length) a ≃ bN1/2 with N being the number of monomers per entanglement strand. Therefore, the number of monomers in the loop that slides over the particle isand the free energy barrier for particle hopping between entanglement cages isThis free energy barrier for the probe particle to hop between entanglement cages (eq 21) has a weaker (linear) dependence on particle size d/a in comparison to the quadratic dependence for cross-linked networks (see eq 6). This linear free energy barrier (eq 21) represents the softening of the confining potential due to the increase in the distance between entanglements under network stretching.[37] Notice that the polydispersity of chain lengths does not affect the barrier energy (eq 21) in the case of lightly cross-linked networks (N > N). Our prediction for the particle size dependence of free energy barrier in entangled polymer melts (eq 21) is different from prediction of ref (43).

The waiting time for the particle to hop between two neighboring entanglement cages is (see eq 7)in which the relaxation time of an entanglement strand isThis waiting time (eq 22) increases exponentially with the relative size of the particle d with respect to the size of an entanglement strand a, but with a relatively weaker dependence on particle size d than that for unentangled polymer solids (see eq 7). This weaker dependence is due to the lower energy of nonaffine deformation of entanglement strands (see eq 21) in comparison with stronger affine deformation of unentangled polymer networks and gels (see eq 6). For instance, for particles with size d twice larger than the entanglement mesh a or network mesh size a (d/a = d/a = 2) the ratio of two waiting times is τent/τgel ≃ exp ((d/a) – (d/a)2) ≃ exp (2–22) ≃ 10–1.

Since the particle “feels” network modulus G ≃ kT/(a2b), the mean-square fluctuations of the particle trapped by the entanglement net is:The ballistic and subdiffusive motion at shorter time scales are similar to the case of unentangled networks (eqs 15 and 17) with the crossover time at the upper boundary of subdiffusive regime equal to the relaxation time of entanglement strand τ, see Figure 4.

Mean-square displacement of a large probe particle (d > a) due to hopping is proportional to the number of hops that the particle makes during a certain time period t with the same step size b as in the case of unentangled dry network (eq 5)which is determined by the relative size of the particles with respect to the entanglement mesh size a. The crossover time at which the mean-square particle displacement due to hopping diffusion ⟨r2⟩hopent (eq 25) becomes comparable to the mean square particle fluctuation in an entanglement cage ⟨r2⟩fluctent (eq 24) isDiffusion coefficient of large probe particles in entangled polymer solids is exponentially small

Mobility of relatively large particles (a < a < d < d ≃ a2/a) is affected by both entanglements and permanent cross-links, but dominated by the entanglements. This is because the entropic free energy barrier due to permanent cross-links, kT(d/a)2, is smaller than that from entanglements, kT(d/a), for a < a < d < d. The two barriers are on the same order at d ≃ d ≃ a2/a. The motion of a very large particle (d > d > a) is dominated by permanent cross-links and essentially not affected by entanglements, because the entanglements are under large deformation due to the presence of the very large particle, leading to the slippage of them toward the permanent cross-links, as discussed in detail in Appendix E. Therefore, the particle diffuses as if it is in unentangled polymer networks (see Section 2). The crossover at d ≃ d ≃ a2/a between entanglement and cross-link dominated regimes can be approximately described by the sum of the two free energy barriers

The terminal diffusion coefficient of particles of different sizes in an entangled polymer network is presented by the solid line in Figure 5. Hopping diffusion of a particle in entangled polymer gels is similar to that in entangled polymer networks, with the hopping step size b replaced by correlation length ξ (eq D.2) and corresponding parameters replaced by concentration dependent ones (see Appendices D and F). The key elements for hopping diffusion of large particles in entangled polymer gels are summarized in Table 1.

Figure 5

Particle diffusion coefficient. Dependence of particle diffusion coefficient D(d) on particle size d in entangled polymer networks (solid line) and entangled polymer melts (dashed line). In an entangled polymer networks, for small particles with size d < a, the particle diffusion coefficient is inversely proportional to the minus third power of particle size: D ∼ d–3.[31] Particles of intermediate sizes (a < d < a2/a) experience hopping diffusion between entanglement cages: D ∼ exp(−d/a) (see eq 27). Large particles in permanent networks (a2/a < d < d (eq B.19)) experience hopping diffusion between network cages: D ∼ exp(−d2/a2)/d (see eq 14). Extremely large particles d > d are permanently trapped in the network. In an entangled polymer melt, the particle motion is dominated by hopping diffusion for particles with size a < d < d: D ∼ exp(−d/a) (see eqs 27 and 34). Particles larger than d have to wait for the polymer melt relax to diffuse and they “feel” the bulk viscosity: D ∼ 1/d. D0 ≃ kT/(ηb) corresponds to the diffusion coefficient of a monomer. Y-axis is logarithmic; X-axis is linear.

Entangled Polymer Liquids: Competition between Hopping Diffusion and Chain Reptation

In our previous work,[31] we discussed the motion of particles in entangled polymer liquids. Here we revisit this problem taking into account the contribution of hopping to particle mobility. The motion of large particles (d > a) in entangled polymer liquids is due to both hopping mechanism and chain relaxation by reptation. In order to diffuse, the particles either have to hop between confinement cages or wait for polymer liquids to flow around these particles. The particle motion is not affected by the entanglements at time scales shorter than the relaxation time τ of an entanglement strand (see Figure 6 and ref (31)). At time scales longer than τ, the large particles are trapped by entanglement cages and cannot move further until a certain time scale τliq. The physical meaning of the onset time scale τliq of particle diffusion is determined by the fastest of the two processes that dominates the particle motion at long time scales. The motion of large particles in entangled polymer liquids due to chain reptation process has been discussed in ref (31). In section 3, we have discussed the mechanism of hopping diffusion of large particles in entangled polymer solids. These results can be directly applied to describe the hopping diffusion in entangled polymer liquids (melts and solutions) for large particles with size d larger than the entanglement strand size a ≃ bN1/2. In the following, we compare the mean-square displacement of large particles in entangled polymer melts due to hopping mechanism and the mean-square displacement due to the chain relaxation (reptation) process.

Figure 6

Time dependence of mean-square displacement of large particles (d > a) in entangled polymer liquids. Illustration of the case at which τliq is determined by hopping process with τliq ≃ τhopent if d < d or N > N (eqs 33 and 34) with mean-square displacement described by eq 26, as shown by the solid curve. The motion of large probe particles at time scales shorter than τ is not affected by entanglement.[31] At time scales longer than τ large probe particles are trapped by entanglement mesh, but they do not have to wait for the polymer liquids to relax to move further (dash-dotted line); instead, they can diffuse by hopping between neighboring entanglement cages (solid line with unit slope for t > τhopent). Logarithmic scales.

The motion of large particles (d > a) in entangled polymer liquids due to hopping is the same as that presented in section 3; the mean-square displacement on time scale t > τ is described by eq 26. Another process contributing to the particle motion at time scales longer than τ is chain reptation.[31] Large probe particles can also diffuse by waiting for polymer chains to relax at reptation time scale τrepwhich increases as cube of degree of polymerization N and τ is the relaxation time of entanglement strand (eq 23) containing N Kuhn segments.

Mean-square displacement of large particles due to chain reptation process isin which η ≃ [kT/(ba2)]τ(N/N)3 is bulk viscosity of entangled polymer melts. Assuming no coupling between the two processes (chain reptation and hopping diffusion) the net mean-square displacement of the large probe particle in entangled polymer melts can be written as the sum of contributions from both processesin which τhopent is the crossover time between fluctuation plateau and hopping diffusion (eq 26).

The corresponding terminal particle diffusion coefficient is

The polymer relaxation (reptation) time increases as power law of the degree of polymerization (eq 29), and at the crossover value of the degree of polymerization N the reptation time τrep becomes comparable to τhopent.The description of particle mobility in entangled polymer melts can be extended to polymer solutions by substituting Kuhn monomer size b by the correlation length ξ (eq D.2) and including concentration dependence of all other quantities, N (eq F.3) and a (eq F.1) as shown in Appendix F.

The mean-square displacement of the large probe particle in polymer liquids with degree of polymerization larger than N is dominated by the hopping diffusion (see solid line in Figure 6). The mobility of probe particles in liquids with shorter polymers (N < N) is dominated by chain relaxation process. Note that the reptation time τrep is independent of the particle size, whereas the time scale τhopent increases exponentially with particle size d (see eq 26). Therefore, for a polymer solution with fixed polymer length N and concentration above the entanglement onset we can introduce the crossover particle size dat which the hopping time scale τhopent is comparable to the reptation time τrep. Note that in the second line of eq 34, we are assuming d ∼ a, as the numerical solution of eq 34 gives d on the order of a.

Thus, there is an interval of particle sizes (a < d < d) for which the terminal particle diffusion coefficient is dominated by the contribution from hopping diffusion, whereas for particles with size d larger than d (see eq 34) it is dominated by the contribution from chain reptation process.

The interval of particle sizes (a < d < d) within which the particle motion is dominated by the hopping process is of significant width to be tested by experiments or computer simulations. For instance, the crossover particle size could be of one order of magnitude larger than the tube diameter (d ≃ 10a) in a highly entangled polymer liquid with N/N ≃ 50 entanglements per chain and typical ratio of tube diameter a and correlation length ξ of a/ξ ≃ 5. The motion of very large particles with size larger than d (eq 34) is diffusive with their terminal diffusion coefficient inversely proportional to the bulk viscosity and the particle size. Note that above we describe the dependence of terminal particle diffusion coefficient D(d) on particle size d. By including the concentration and molecular weight dependencies of corresponding parameters, ξ, a, τ, and η, one can obtain the dependence of particle diffusion coefficient D on solution concentration ϕ, and degree of polymerization N (polymer molecular weight M), as discussed in Appendix F.

To summarize: (1) There is a range of particle sizes (a < d < d (eq 34)) in which the particle motion is mainly determined by hopping diffusion; (2) the hopping diffusion coefficient of large particles decreases exponentially with increasing particle size d (eq 27 and Table 1); (3) very large particles with size greater than d have to wait for polymer liquids to relax and flow around them in order to diffuse.[31] The dependence of terminal diffusion coefficient of particles on their sizes in an entangled polymer melt is presented by the dashed line in Figure 5. Note that microrheological approach works only for these very large particles with d > d, while for smaller particles it leads to the underestimation of polymer viscosity due to faster particle diffusion by hopping process.

Conclusion

In this work, we have discussed the mobility of large particles subjected to topological constraints. The topological constraints could be network cages in unentangled polymer solids (networks and gels), entanglement nets in polymer liquids (melts and solutions), or both entanglements and network cages in entangled polymer solids. We introduce a novel hopping mechanism to describe the diffusion of large particles with size d larger than the network mesh size a and/or the entanglement mesh size a. We argue that although the large particles experience the topological constraints from the network (entanglement) cages, they can still diffuse by waiting for the fluctuations of the surrounding confinement cages, which could be large enough to slip around the particle.

In unentangled polymer solids (a < a) the large particles are trapped by network cages at long time scales (t > τ). To move (hop) between cages, these particles have to wait for time τ, at which the fluctuations of network strands become large enough to allow the particles to hop between cages. The hopping step occurs by just one network loop out of many overlapping ones that slips over the particle. The resulting hopping step size Δr of a particle is on the order of a monomer size in dry polymer networks or melts and on the order of correlation length in gels and entangled polymer solutions. Note that this hoping step size is much smaller than the network mesh size and is independent of particle size d, which is qualitatively different from prediction in ref (43). Hopping diffusion coefficient of large particles in unentangled networks exhibits exponential dependence on the square of the ratio between the particle size and the network strand size: Dhopnet ∼ (a/d) exp (−d2/a2).

In addition to permanent cross-links, polymer solids can also contain entanglements. Particles diffusing in weakly cross-linked polymer solids are primarily constrained by entanglements. Unlike the chemical cross-links, the constraining effect due to entanglements softens upon chain elongation and thus the corresponding free energy barrier for hopping diffusion between neighboring entanglement cages is weaker. The corresponding diffusion coefficient of large particles (a < d < d ≃ a2/a) in entangled networks has relatively weaker dependence on particle size, Dhopent ∼ exp (−d/a), in comparison to unentangled networks. We would like to stress that our model predicts linear dependence of the hopping energy barrier on particle size d, which is qualitatively different from that in ref (43), in which the hopping barrier is expected to be asymptotically proportional to the particle volume d3. It is worthwhile to note that the barrier height predicted in ref (43) is similar to our estimate for the energy of embedding a particle into the polymer network (Appendix C).

In contrast to particle motion in entangled permanently cross-linked networks, for which hopping is the only mechanism allowing long-time diffusion, large particles in entangled polymer liquids can also diffuse by allowing these liquids to flow around them at time scales longer than the relaxation time. We show that particles with intermediate size a < d < d (see eq 34) diffuse primarily by hopping between neighboring entanglement cages, while extra large particles (d > d > a) have to wait for the polymer liquids to relax as the entropic energy barrier for hopping between neighboring entanglement cages becomes extremely high.

The hopping process provides the mechanism for diffusion of particles with size several times larger than the mesh size of unentangled polymer networks and tube diameter of entangled polymer solids and liquids. For instance, recent experiments studying the diffusion of gold nanoparticles with size slightly larger than the entanglement length in polystyrene solutions show that the particles experience viscosity smaller than the macroscopic value.[41] It is possible that the diffusion coefficient of these particles with size d > a is due to hopping. We are looking forward to more systematic experimental and computer simulation tests that will provide more information about diffusion of particles with size larger than the network (entanglement) mesh size. Furthermore, a natural extension of the results presented in this paper could be the mobility of particles in reversible polymer liquids[42] and solids,[45] which will be presented in a future publication.