Bridging Dynamics of Telechelic Polymers between Solid Surfaces

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Bridging Dynamics of Telechelic Polymers between Solid Surfaces Hossein Rezvantalab and Ronald G. Larson* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *

ABSTRACT: We employ Brownian dynamics simulations combined with forward flux sampling and theoretical firstpassage time analyses to describe the rates of transitions between loops and bridges of telechelic polymers between solid surfaces representing e.g. latex colloids in the limit that the telechelic stickers bind strongly enough to the surfaces to make free chains very rare. It is shown that the bridge formation rate can be expressed by combining times for two processes, namely, the escape of one end sticker from the narrow-but-deep association well near the colloidal surface and the longer-range motion of the chain end to the other surface inhibited by stretching free energy. We find that when using multibead chains to represent the telechelic polymers, the longer-range motion requires use of a multidimensional first passage time analysis that we borrow from the work of Likhtman and co-workers, which was originally developed to describe polymer end fluctuations in a one-dimensional reptation tube. From these ingredients, we develop analytical expressions for the loop-to-bridge and bridge-to-loop transition rates as functions of the number of beads per polymer, the ratio of gap to the equilibrium chain length, and the end sticker association energy to the colloids/surfaces. We also suggest that a 20-to-1 mapping of Kuhn steps to springs may allow the analysis to be applied to real chains, rendering the analysis applicable to a broad range of industrial and biological processes.

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INTRODUCTION Several essential transport problems in biology, pharmaceuticals, personal care products, and commercial coatings involve fluids or materials with multiple levels of structure.1−3 Efficient prediction and design of the properties of such materials require the development and integration of computational methods and a theoretical framework that can span the range of length and time scales over which the transport of constituent species occurs. An important example is that of colloid/ polymer mixtures, which contain micrometer-sized particles or aggregates that interact with long polymers and/or surfactants and are widely used in latex paints, polymer composites, colloid-filled melts, and biological fluids.4,5 The polymers bridge the colloidal particles into transient networks that break and reform as their end groups dissociate from and associate onto the surfaces. The binding of polymer to the colloid surface is regulated by atomistic forces (e.g., hydrophobic interactions), and the flow properties are controlled by the strength of these interactions in addition to the fluid mechanics of the suspended particles. A thorough understanding of the dynamics at all these levels requires a multiscale approach that can capture the spatial and temporal response of the system in sufficient detail across all relevant length and time scales. While there has been considerable interest in quantifying the rheological properties of polymer networks6 and polymer/ particle interactions, e.g., through Monte Carlo simulations,7,8 a theoretical framework for polymer/particle networks is largely lacking. We aim to build a simplified, yet informative, model for the polymer/colloid mixture and investigate the kinetics of © XXXX American Chemical Society

formation/destruction of the networks that control the rheological properties of the material, focusing on polymers and colloids used in latex paints and water-borne coatings.9 Considering colloidal particles whose radii are much larger than the radii of gyration of the polymers and assuming that the particle rearrangement occurs on a significantly longer time scale than the polymer association/dissociation, the polymer dynamics can be reduced to the formation and breakage of bridges between stationary solid surfaces. Here, the polymer is taken to be telechelic, i.e., to have functionalized end groups that favor adsorption to the solid surfaces. This is the case for common rheology modifiers such as hydrophobically modified ethoxylated urethane (HEUR),10,11 which has poly(ethylene oxide) as the hydrophilic backbone and aliphatic alcohol, alkyl phenyl, or fluorocarbon groups as the hydrophobic end groups. The association free energy between the end group and the surface is controlled by the size of the end group, for example, the number of carbon atoms in the alkane hydrophobe.6 The kinetics of network formation is also affected by the stretching free energy of the polymer across the gap between the representative colloidal surfaces, which particularly depends on the polymer coil size and molecular weight. The ultimate goal is to predict the rates of formation/destruction of polymer bridges between colloidal surfaces as a function of these parameters. Received: July 17, 2017 Revised: November 21, 2017

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DOI: 10.1021/acs.macromol.7b01517 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules In the following, we first discuss the coarse-grained methods used for modeling polymer molecules and their interactions with solid surfaces. We then explain different approaches for modeling polymer bridging as well as a theoretical scheme, resulting in simple analytical expressions for bridge formation/ destruction times in terms of the polymer properties and interparticle gaps. Simulation results along with theoretical predictions are then presented, followed by concluding remarks.

where H is the stiff Fraenkel spring constant taken to be large enough to maintain the length of each link close to a Kuhn segment length bK. In all representations used here, the polymer is modeled as a phantom chain with no excluded volume interactions between the atoms and with hydrodynamic interactions neglected. (The effect of excluded volume is considered, however, in the Supporting Information.) Furthermore, all friction is assumed to be concentrated at the beads, which act as drag centers. Polymer−Surface Interaction. The polymer chains are placed in a domain bounded by two flat solid surfaces each perpendicular to the z-axis with a fixed gap d between them and periodic boundaries along x and y coordinates. The end beads of the polymer chains interact with the solid surfaces, which represent the surfaces of colloids much larger than the polymers, via the following Lennard-Jones potential

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MODEL DEVELOPMENT Polymer Molecule Representation. Efficient simulation of a polymer molecule is achieved through coarse-grained bead−spring models in which the polymer is represented as a series of beads with given friction experiencing Brownian motion and connected by entropic springs.12,13 The simplest stochastic model is the Rouse model,14 where the springs are harmonic with spring constant 3kBT/b2, where kB and T are respectively the Boltzmann constant and temperature and b is the root-mean-square end-to-end distance of the spring.15 The stretching potential for each spring in a Rouse chain can be written as 1 3kBT 2 URouse(r ) = r 2 b2

⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ , z < zc ⎝ ⎠ ⎝ z⎠ ⎦ Uads(z) = ⎨ ⎣ z ⎪ ⎪ 0, z ≥ zc ⎩

where z is the distance perpendicular to the surface between the end bead and the flat surface, σ and ε are respectively the width and depth of the interaction potential, and zc = 2.5σ is the distance beyond which the sticker adsorption energy is cut off. To prevent the interior beads of polymer chains from penetrating the solid surfaces, we apply only the repulsive part of the aforementioned potential to them by setting the cutoff for the interactions of interior beads with the surface to the minimum point of the potential, zmin = 21/6σ. Additionally, since we are taking the chains to be noninteracting, we are effectively assuming that the polymer solution is dilute so that polymer adsorption is in the “mushroom regime” where the adsorption of one polymer to the surface does not influence the dynamics of other chains.21 Model Parameters. Brownian dynamics (BD) simulations are performed using the LAMMPS commercial package in the NVT ensemble by solving the Langevin equation of motion15

(1)

where r is the separation distance between any two adjacent beads connected by a harmonic spring corresponding to NK,s = NK/N Kuhn steps each having a Kuhn segment length bK, with NK being the total number of Kuhn steps per chain.15 Note that N here is the number of springs representing the chain, and NK,s is the number of Kuhn steps per spring. Although the Rouse model is simple enough to provide many properties of the polymer analytically, it suffers from the unphysical assumption of infinite extensibility due to the harmonic spring law. The finitely extensible nonlinear elastic (FENE) model15,16 introduces an alternative spring law that fixes this defect by allowing the force to become arbitrarily large as the molecular extension approaches the fully extended length of the molecule. The stretching potential for the FENE model is expressed as 3 UFENE(r ) = − NK,skBT ln[1 − (r /L)2 ] 2

mi

1 H(r − bK )2 2

d2ri dt

2

= −ζi

dri + FCi + FiR dt

(5)

where mi, ri, and ζi are respectively the mass, friction coefficient, and position vector of bead i, t is the time, FCi = −∂U/∂ri is the total conservative force applied to bead i due to the summation of polymer stretching and surface interaction potentials discussed above, and FiR = 6ζikBT ξ(t ) is the random Brownian force where ξ(t) is a random number uniformly distributed between −1 and 1 from a stationary Gaussian process that satisfies the fluctuation−dissipation theorem. We consider polymer chains with NK = 100 Kuhn steps while varying the number of springs N from unity, representing a simple dumbbell with all friction lumped into two beads connected by a single spring, up to N = 100, corresponding to a multibead chain having one Kuhn segment per spring. The bead friction coefficient is set to ζi = 100[m]/[t] in LAMMPS units, which ensures that the simulations are in the overdamped limit where inertial effects are negligible. A time step of Δt = 0.01[t] is found to provide reasonable computational efficiency while allowing convergence in solving the equations of motion. We use the thermal energy [kBT] as the unit of energy, the bead mass mi = [m] as the unit of mass, and the Kuhn segment

(2)

where L = NK,sbK is the fully extended length of each spring and NK,s bK = b. Note that deviations from the elastic spring model are only large for fractional extensions r/L > 0.3. A finer-grained representation of the polymer molecule is the freely jointed chain (FJC) model15,17 which is resolved at the level of links each corresponding to a single Kuhn step, and the flexibility that is distributed over several backbone bonds in the real chain is lumped into a single free joint. While a freely jointed chain is typically represented by Kramers bead−rod model,18 an alternative approach which is in principle equally accurate yet computationally more tractable is to use stiff Fraenkel springs.19,20 The Fraenkel spring generates a force proportional to the deviation between the actual and natural (equilibrium) length of the spring, so that the stretching potential is written as UFJC(r ) =

(4)

(3) B


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Figure 1. (a) Snapshot of the domain for direct BD simulation of polymer bridging between solid surfaces. The telechelic polymer molecules are represented as bead−spring chains with end stickers shown in yellow. (b) Schematic of the adsorption potentials near the surfaces to which the polymer end groups attach. Note that the gap d is taken to be the distance between the minima in adsorption potentials on opposite surfaces and is somewhat smaller than the distance between the infinitely repulsive surfaces.

ever ε is greater than the chain stretching free energy, which in units of kBT is (in the linear or Hookean limit) roughly the chain stretch ratio, which is the end-to-end vector relative to its root-mean-square equilibrium value. For chains as short as 100 Kuhn steps, even when ε is as low as 6kBT, the non-Hookean limit is exceeded before free-chain intermediates occur. For longer chains, with say 1000 Kuhn steps, and relatively small ε = 6kBT, our theory would be limited to transitions in which the chain stretch remains in the Hookean regime but would extend to the non-Hookean regime if ε is higher. These cases cover most of the transitions that are of interest, at least in quiescent solutions. If the suspension is under strong flow, additional, nonlinear, factors are likely to affect transitions between loops and bridges, and these are not considered here.

length [bK] as the unit of length, with the time scale derived using these choices as [t ] = [bK ] [m]/[kBT ] . Within the LAMMPS code, all of these quantities are set to unity. The dimensionless results can be converted to physical quantities for the particular system of interest by assigning values to these fundamental units and using them to calculate the numerical values of all other dimensional quantities. For instance, we set the unit length [bK] to the Kuhn length of a poly(ethylene oxide) (PEO) chain, namely 1.1 nm. As described in the Supporting Information, using a PEO molecular weight of 13 700 as a reference, we set the unit time by matching the estimated relaxation time of this polymer, namely 48.5 ns, to the relaxation time of a Rouse chain resolved at the dumbbell level given by ζiNKbK2/(12kBT), yielding [t] = 58.2 ps for the above choice of NK and ζi. The unit energy is set to the thermal fluctuation energy at 25 °C, namely, [kBT] = 4.11 × 10−21 J. Furthermore, the Lennard-Jones potential width parameter is taken as σ = 1[bK], and the depth is set to ε = 8[kBT], unless otherwise stated. In physical units, this corresponds to the adsorption free energy of a HEUR hydrophobic “sticker” with roughly eight carbon atoms.6 Later, we will consider the effect of the value of ε in some detail. A telechelic polymer chain can have four different adsorption states during the course of simulation: a “loop” conformation if both ends are adsorbed onto the same surface, a “bridge” conformation if the end beads absorb to different particle surfaces, a “dangling” chain conformation if only one sticker is adsorbed to a surface, and a “free” chain state if both stickers are detached from the surfaces. Note that a bead is considered “adsorbed” onto the surface if it lies within the cutoff of the adsorption potential given by eq 4. The latter two conformations are expected to occur less frequently if the association energy is noticeably larger than thermal energy kBT. We found that a sticker interaction energy of ε ≥ 6kBT yields a fraction of dangling/free chains less than 5%, so that the properties of the polymer/latex network are then mainly influenced by the rates of formation/destruction of loop and bridge conformations. Thus, our analysis of the transitions between loops and bridges will be limited to cases in which only one sticker at a time is released from the surfaces of the particles; i.e., the transition from bridge to loop or loop to bridge does not involve a free chain as an intermediate or transitional state. This places restrictions on both the range of sticker energies (ε ≥ 6kBT) and the degree of stretch of the polymer molecules required for the transition, which is the gap between particles. For large gaps, bridges are rare, and transitions to such bridges may involve free chains as intermediates, unless ε is especially high. However, for high enough ε, say ε ≥ 10kBT, even transitions involving rather highly stretched chains will occur without the formation of free-chain intermediates. In general, transitions without free-chain intermediates will occur when-

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SIMULATION METHODOLOGY Direct BD Simulation. The most straightforward method for analyzing the bridging dynamics is through direct BD simulation of a dilute solution of polymer molecules bounded between latex surfaces, as shown in Figure 1a. The polymers are represented by the bead−spring or FJC model with their sticky end groups having a large (ε ≥ 6kBT) adsorption energy to the solid surfaces. The “true” gap, d, between particles, taken as the spacing between the minima in the potential Uads of the two surfaces, is somewhat smaller, by 2zmin, than the gap between the hard (infinite potential) boundaries between the two surfaces, as depicted in Figure 1b. The simulation starts with an equal number of loops distributed randomly on each surface, and no bridges or dangling/free chains, and subsequently tracks the evolution of these species with time until reaching a steady state characterized by a constant number of each species (within the statistical noise). Provided that the fraction of dangling/free chains is negligible, the process can be modeled as a reversible reaction between loop and bridge species: loop ⇌ bridge, with forward reaction rate kf defining the rate of polymer bridging between the surfaces from loop states and the reverse reaction rate kr describing the rate of destruction of bridges into loops. Solving the reversible reaction problem with proper initial conditions yields the following fraction of loops vs time c loop(t ) =

k f exp[−(k f + k r)t ] + k r kf + kr

(6)

while the fraction of bridges increases from zero according to cbridge(t ) = 1 − c loop(t ) =

kf {1 − exp[− (k f + k r)t ]} kf + kr (7)

where the fraction of each polymer species is the number of loops or bridges Nloop or Nbridge (note that we are reusing the symbol “N”, which elsewhere is the number of springs in a polymer), divided by the total number of chains taken as Nloop C


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has to cross all interfaces λi in order to reach the final state λn. The FFS methodology involves two stages:22 (1) a calculation of the crossing rate of the first interface and (2) successive conditional probability estimations for transitions from one interface to the next. In the first stage, a simulation of sufficiently long time t1 is performed in the initial state to generate a collection of M1 configurations corresponding to crossings of the first interface λ1 by trajectories coming from λ0. The initial crossing rate is then calculated as

+ Nbridge, thereby neglecting other conformations. Therefore, the two reaction rate constants kf and kr can be estimated by fitting exponential functions given by eq 6 or 7 to the simulated fraction as a function of time. Note that identical values for kf and kr are expected if we start with all polymers bridging the two surfaces, and we measure the reaction rates in the opposite direction. Although our analysis here is focused on the evaluating the reaction rates, we also investigated the thermodynamics of bridge−loop equilibrium by assessing the steady-state fraction of bridging species at multiple gaps. As shown in the Supporting Information, we found good agreement of our simulations with the available equilibrium theory for telechelic associative polymers between flat surfaces. While direct BD simulations have the advantage of resolving details of the polymer/latex interaction and network formation, they can be computationally expensive for chains composed of large numbers of beads: the whole conformation space has to be sampled by averaging over many initial loop states, and the motions of all beads within a chain have to be resolved. This can be computationally tractable for polymer dumbbells, but as we describe below, the dumbbell model cannot accurately capture the loop-to-bridge transition rates of real polymers near solid surfaces due to the effect of the interior local modes on chain relaxation parallel and normal to the surface. Even for dumbbells, it is computationally expensive to evaluate the transition rates for large gaps between surfaces, approaching the fully extended length of the polymer. The simulation time for reaching steady state becomes exponentially longer with increasing gap, while most of the effort is wasted on the intermediate dangling/free chain conformations, most of which do not lead to bridging chains. Forward-Flux Sampling Scheme. To improve the computational efficiency, an alternative is to employ a rareevent sampling scheme. A widely used example is forward-flux sampling (FFS),22,23 which has been used to capture the firstpassage time of unlikely events and which is applicable to our polymer bridging problem. FFS segments phase space using n + 1 interfaces defined such that the starting point of each simulation trajectory is on the zeroth interface and the final point is on the nth interface, as shown in Figure 2. In our case, the reaction coordinate is the z location of the free terminal bead in the chain, with the other terminal bead frozen to the minimum in the attractive potential on the left surface in Figure 2. The value of the reaction coordinate z increases from one interface to the next, and the interfaces are placed such that a trajectory, defined by the z location of the free terminal bead,

k1 =

M1 t1

(8)

The second stage involves running several consecutive simulations starting from interfaces λ1 to λn−1 in a ratchet-like way, where each simulation uses multiple crossed configurations at any interface as starting configurations for the next trial runs. In the variant of FFS used here, each simulation at the interface λi starts from a random selection of stored configurations on that interface and ends when the trajectory for that starting configuration either reaches the next interface λi+1 or returns to the zeroth interface. The probability to progress from one interface to the next can be estimated as the fraction of successful runs reaching λi+1 to the total number of trials launched from λi. Consequently, the mean first-passage time for reaching the final state λn can be expressed in terms of the initial crossing rate and successive success probabilities n−1 ⎧ ⎫ ⎪ ⎬ k P ( ) τ(λn) = 1/⎨ λ | λ ∏ 1 + 1 i i ⎪ ⎭ ⎩ i=1 ⎪

⎪

(9)

Note that the statistical error and computational efficiency of the FFS algorithm are highly dependent on the number and positioning of the interfaces as well as the number of trial runs initiated from each interface.24 It is recommended to select interface distances such that P(λi+1|λi) > 0.3,25,26 which is the criterion we follow to ensure having sufficient successful trajectories without the need to increase the number of trials launched from each interface. This was achieved by using approximately two interfaces per unit length [bK] and Mi = 1000 trials per interface. Here, we consider polymer chains in loop conformations (as the initial state) placed next to a surface with one end bead permanently fixed to the surface at the potential minimum corresponding to λ0 = 0 and the other end bead interacting with the surface according to the Lennard-Jones potential prescribed by eq 4 with the attractive cutoff defined above for the end bead and the repulsive one for the interior beads. All beads representing the polymer except for the permanently stuck bead undergo Brownian dynamics under the conservative forces resulting from chain stretching and the surface interactions with the surface on the left. A bridge is formed when the free end bead reaches the last interface λn placed at the desired distance d from the fixed end bead, where d is the distance between the energy minima of the polymer/latex interaction potentials on the two surfaces in direct simulations. In our FFS simulations, the attractive and repulsive potentials from the surface on the right are neglected, which should be a safe assumption for gaps larger than the polymer’s average extension, since in that case internal beads rarely cross the location of the surface on the right side of the gap. The inverse of the mean first-passage time for a transition from a loop to an extended (bridge) conformation calculated using eq 9 from FFS simulations

Figure 2. Sketch of FFS simulation scheme for a bead−spring polymer chain with one end fixed at the minimum point of the association potential and the remaining beads following Brownian motion with repulsion but no attraction from the wall on the left. The position of the free end bead normal to the surface is denoted by z, and the λ’s indicate the positions of interfaces used for calculating crossing probabilities. D


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two conformations. We first consider free polymer chains without the end stickers and evaluate the first-passage time for extending a chain from zero extension along a coordinate z to a given distance, d. For the simplest case of a bead−spring Rouse chain, the Milner−McLeish theory27,28 coarse-grains a chain made of N moving beads inside the tube into a two-bead model, with one end at the origin and the other carrying half of the friction of the whole chain ζeff = (N/2)ζi attached to the first bead by a harmonic spring of constant 3kBT/(Nb2). (Note that the accuracy of this approximation deteriorates when there are too few moving beads per chain since ζeff = ζi for the limiting case of a dumbbell with N = 1.) Kramers29 gives the exact solution for the mean first-passage time of this two-bead polymer model as

gives an estimate of the bridge formation rate, kf. On the other hand, we rely only on direct BD simulations to measure the bridge destruction rate, kr, which is computationally tractable in direct simulations due to the relatively fast retraction of the chain toward the loop conformation, but the above FFS scheme (with both surface potentials included and the initial and final states switched) could be used as well. Note that averaging over multiple FFS simulations independently launched from different starting conformations is necessary to achieve reasonable accuracy and low enough statistical error, but the computational cost is significantly lower than that of full-chain direct BD simulations (without biasing) due to the advanced sampling strategy of FFS. We will show agreement between FFS predictions and the direct method for extensions up to 20% the fully extended length of the polymer, thus justifying the use of FFS at somewhat larger extensions where the direct BD simulations are no longer feasible.

τ (z ) =

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THEORETICAL ANALYSIS Population Balance Equations. Considering a twospecies system with only polymer loops and bridges between two colloids i and j placed at a distance d and neglecting intermediate polymer conformations, a possible approach to characterize the polymer/colloid networks is to evolve the following population balance equations8 in time to track the numbers of polymer conformational “species”: dNii = −Niik f (d) + Nijk r(d)/2 dt dNjj = −Njjk f (d) + Nijk r(d)/2 dt dNij = (Nii + Njj)k f (d) − Nijk r(d) dt

≈

ζeff U ′(z)

∫z

⎛ U (x ) ⎞ dx exp⎜ ⎟ ⎝ kBT ⎠ min z

x

⎛

⎞

∫−∞ exp⎜⎝− Uk(xT′) ⎟⎠ dx′ B

⎛ U (z ) ⎞ 2πkBT exp⎜ ⎟ U ″(zmin) ⎝ kBT ⎠

(12)

where the approximation is valid if U(z) ≫ kBT, meaning that the energy barrier for extending to length z is significantly larger than the thermal fluctuation energy. Since we are considering only the z component of the chain fluctuation, and are dealing with Hookean springs whose displacements in each of the three Cartesian directions are independent of each other, the free energy for the z component of fluctuation is given by eq 1, but with r replaced by z, giving a potential U(z) = 3kBTz2/ (2Nb2). After coarse-graining this chain to a dumbbell, the Milner−McLeish prediction for the first-passage time is then simplified to τMM,Rouse(z) =

(10)

where Nii and Njj are respectively the number of loops on particles i and j, Nij is the number of bridges between the i and j particles, and kf and kr are respectively the loop-to-bridge and bridge-to-loop transition rates at spacing d, as defined earlier. Approximate analytical expressions can be found for the reaction rates based on the energetic barriers associated with crossing from one configuration to the other8

⎛ 3z 2 ⎞ Nb π 5/2 exp⎜ τR,c ⎟ z ⎝ 2Nb2 ⎠ 4 6

(13)

where τR,c = 4ζiN2b2/(3π2kBT) is the longest rotational relaxation time of a one-end-fixed Rouse chain in the limit of large N (i.e., a “continuous” Rouse chain), which is 4 times larger than that of the same chain with two free ends. However, we find that the presence of the representative colloidal surface shown in Figure 2 in our polymer-bridging problem reduces the first-passage time along z by roughly a factor of 3 due to the restricted motion in the z direction owing to the bounding surface (see the Supporting Information for details). Taking this into account and upon normalizing the distance from the free chain end to the fixed end z by the root-mean-square endto-end distance of the polymer as s = z /( N b), the result can be written as

⎤ ⎡ 1 k f (d) = Ω exp⎢ − (ΔUads + Uspring(d))⎥ ⎦ ⎣ kBT ⎤ ⎡ 1 k r = Ω exp⎢ − (ΔUads)⎥ ⎦ ⎣ kBT

ζeff kBT

(11)

where ΔUads is the adsorption free energy of the sticky end beads to the latex surface, Uspring is the stretching free energy of the polymer at extension d based on the prescribed spring law, and here the width of the association potential well is assumed to be negligible compared to the gap. However, the rate constants also depend on the thermal fluctuation frequency, or attempt rate Ω, which needs to be obtained either empirically or by matching to simulation results.6,8 First-Passage Time Due to Chain Stretching. As an alternative to the approximate expressions given by eq 11 for the transition rates, we employ available theories for firstpassage problems of Brownian particles and polymer chains and attempt to determine the fluctuation frequency Ω and the associated rate and time constants for transitions between the

τMM,Rouse(s) =

⎛3 ⎞ 1 π 5/2 τR,c exp⎜ s 2⎟ ⎝2 ⎠ s 12 6

(14)

Upon including finite extensibility using the z-component of the FENE stretching potential given by eq 2 with the equivalent spring constant for the whole chain, we find the corresponding expression for the first-passage time τMM,FENE(s) =

E

π 5/2 s2 ⎞ 1⎛ τR,c ⎜1 − ⎟ s⎝ NK ⎠ 12 6 ⎡ 3 ⎛ s 2 ⎞⎤ × exp⎢ − NK ln⎜1 − ⎟⎥ ⎢⎣ 2 NK ⎠⎥⎦ ⎝

(15)


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Moreover, eq 17 for the first-passage time is in reasonable agreement with the Milner−McLeish prediction given by eq 14 for N = 1. It should be emphasized that this result (eq 17) is obtained from the elastic Rouse model while more realistic representations such as the FENE model further complicate the analysis as the necessary transformation to eigenmodes, derivatives of the potential, and Hessian calculation used in deriving eqs 16 and 17 become much more involved. End Sticker Dissociation Time. To superpose the effect of end sticker adsorption onto the above multidimensional chain stretching contribution, we first consider a free Brownian particle with friction coefficient ζi moving in the potential well, Uads(z), defined by eq 4. The escape time can be found from Kramers formula29 based on the structure of the well

with NK the number of Kuhn steps per chain. Note that the ratio τMM,FENE/τMM,Rouse approaches unity for small extensions s2/NK ≪ 1. Inverting the first-passage time calculated from eq 14 or 15 gives the rate of transition from loop to bridge excluding the effect of sticker adsorption energy. It should be emphasized that while the Rouse chain potential U(z) = 3kBTz2/(2Nb2) used in deriving eqs 13 and 14 can accurately represent the behavior for the z component of a threedimensional chain stretched in an arbitrary direction r, this does not hold for the FENE model due to nonlinearity of the spring law. Consequently, the FENE potential cannot be broken down into a sum of contributions from stretch in each direction. Therefore, replacing r in the expression for the FENE potential, eq 2, with the spatial coordinate z becomes inaccurate if there are fluctuating components of the stretch in directions other than z, so that eq 15 derived based on this simplification can be used only as an approximation for FENE chains forming bridges between parallel surfaces. The above two-bead approximation is only reasonable for a polymer dumbbell representation, while it overpredicts the firstpassage time when the chain is represented by more than two beads. This has been shown by Cao, Likhtman, and coworkers,30 who noted that due to the effect of interior modes on end-bead dynamics, the first-passage problem of the end bead of a multibead chain has to be treated as a multidimensional Kramers problem, which presents rich and unexpected behavior, as shown by Likhtman and Marques.31 The theory of Cao et al., originally derived for the problem of fluctuations of the end bead of a Rouse chain along a one-dimensional entanglement tube, is also handy for our problem of loop-tobridge transitions of telechelic polymers. For a Rouse chain composed of N harmonic springs each having a root-meansquare segment length b, combining an exact asymptotic theory with projection onto the minimal action path30 gives the following functional form for the first-passage time at normalized extension s

τesc(Uads) = 2πζi

″ (zmin)|Uads ″ (zc)| Uads

(18)

where z = zmin corresponds to the minimum of the surface adsorption potential and zc is the cutoff beyond which there is no energy barrier associated with particle escape. The barrier height Uads(zc) − Uads(zmin) should be significantly larger than thermal energy kBT for this relation to apply. Overall Bridge Formation/Destruction Time. For the telechelic polymer chain, the escape of the end sticker from the attractive well associated with the latex surface is also coupled to the stretching potential, Uspring(z), for 0 ≤ z ≤ zc. Noting that the width of the association potential well is assumed to be negligible compared to the chain extensions of interest and that the stretching contribution to the actual energy landscape at short extensions is typically much smaller than that of the adsorption barrier, we can decompose the analysis into a twostep process: the escape of the free end bead from the association potential well Uads and chain stretching to some distance d based on Uspring, with d ≫ rc. We note here that the requirement that loop-to-bridge transitions occur without the formation of free-chain intermediates implies that the time for escape of the sticker, τesc, must be much larger than the time for a free chain end to find a surface to bind to. If this were not the case, then the second sticker might come loose from the surface before the first freed sticker has found its way back to a surface, either the opposite surface, where it would form a bridge, or the original surface, where it would rebind to form a loop again. The time for the freed end bead to explore its configurations is roughly τR; thus, we require that τesc ≫ τR. We thus imagine a process in which the freed sticker explores for a time τR after which it reattaches to the original surface if it has not in the meantime reached and attached to the opposite surface. Thus, relative to the multidimensional first passage time of a chain with one anchored and one free bead, the repeated reattachment of the free bead to the original surface should slow down the “attempt rate” for the first passage time by a factor proportional to τesc/τR. Combining the early mode desorption of time scale τesc with the chain stretching mode having the relevant characteristic Rouse time τR (given above just after eq 16), we express the overall first-passage time as

⎛ C (N ) C (N ) ⎞ ⎛3 ⎞ τmulti,Rouse(s , N ) = ⎜ 1 + 2 3 ⎟τR exp⎜ s 2⎟ ⎝2 ⎠ ⎝ Ns s ⎠ (16)

where τR is the Rouse rotational relaxation time for a chain with N springs with one end bead fixed (but no influence of the wall), given by τR = ζib2/(12kBT) sin−2[π/(4N + 2)]. Although the analysis of Cao et al.30 yields analytical expressions for C1(N) and C2(N), comparison with stochastic simulations reveals that this provides the right scaling but incorrect prefactor, which is attributed to the non-Markovian nature of the process after the trajectories are projected onto the minimal action path and the degrees of freedom are reduced. To repair the theory, Cao et al. obtained the functions C1(N) and C2(N) by fitting expression (16) to results from FFS simulation data over a broad range of N = 1−128, giving30 τmulti,Rouse(s , N ) =

exp[(Uads(zc) − Uads(zmin))/kBT ]

⎞ ⎛3 ⎞ 1 ⎛ 3.57 1 + τ exp⎜ s 2⎟ ⎜ −1.41 3⎟ R ⎝2 ⎠ 3 ⎝ Ns + 0.83)s ⎠ (N (17)

⎛ τ (ε) ⎞ τf (s , N , ε) ≈ ⎜1 + esc ⎟τmulti,Rouse(s , N ) τR ⎠ ⎝

where we introduce the additional factor 1/3 to account for the presence of the bounding surface along the z direction. This semianalytical result is found to be accurate for all chain lengths at dimensionless extensions s > 1.5, below which the stretching energy barrier is comparable to kBT and the distribution of firstpassage times is then no longer a single exponential in s2.

(19)

where τmulti,Rouse is substituted from eq 17 with the normalized distance evaluated at z = d giving s = d /( N b), and τesc(ε) is F


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Figure 3. First-passage time for loop-to-bridge transition for Rouse and FENE dumbbells without end sticker as a function of the gap between the surfaces normalized by the root-mean-square end-to-end distance of the polymer: (a) ratio of τf to chain relaxation time on a semilog scale; (b) the ratio renormalized by exp(3s2/2)/s. The symbols are from FFS simulations, and the lines correspond to the theoretical expressions given by eqs 14 and 15.

obtained by replacing Uads(z) given by eq 4 into the Kramers formula (18) with σ = 1[bK], zmin = 21/6σ, and zc = 2.5σ as τesc(ε) =

2πζi exp[0.984ε/kBT ] 2.489ε /kBT

kr =

⎪

⎪

⎪

⎪

(22)

where s = d /( N b). The sticker adsorption potential is left in the general form Uads, and τesc = 1/kr is used to obtain the second expression. Substituting kf and kr into the population balance equations (10) then gives the evolution of the number of species with time at any given separation d between the latex surfaces which, in turn, controls the stress tensor and rheological properties of the suspension. These expressions could be used in a simulation of the dynamics of a network of colloidal particles joined by bridging chains. It should be emphasized that in evaluating the bridge destruction rate in both the theoretical analysis and the FFS scheme, we are neglecting the contribution from polymer chain stretching in lowering the energy barrier for release of the end sticker that remains attached to the surface. Our estimates suggest that for the entropic free energy at large chain stretch to be competitive with the free energy barrier from the end sticker potential with ε = 8[kBT], a chain with NK = 100 Kuhn steps would need to be stretched to well over 30% of its fully extended length, where no more than 2% of the loops will form bridges (see Figure S4 in the Supporting Information).

■

(21)

RESULTS AND DISCUSSION Polymer Dumbbell Level. We start from the simplest representation of the polymer molecule, namely, the elastic dumbbell where all degrees of freedom are lumped into two beads connected by a harmonic spring. Excluding the sticker adsorption energies, we first evaluate the first-passage time purely due to chain extension. This is shown in Figure 3a on a semilog scale for a dumbbell with b = 10[bK] over a range of extensions corresponding to 0.7 ≤ s ≤ 3.6. Excellent agreement is found between the theoretical prediction of the Milner− McLeish formula27,30 and stochastic simulations based on the FFS scheme over a broad range of nearly 8 orders of magnitude in time. Note that here and in what follows the transition time

independent of the number of beads representing the chain and the extension for d ≫ rc. This has a similar form to the inverse of the rate given by eq 11, with the thermal fluctuation frequency expressed as Ω=

2πζi

⎡ U (z ) − Uads(zmin) ⎤ exp⎢ − ads c ⎥ kBT ⎦ ⎣

⎧⎛ 1 ⎞⎛ 3.57 k f = 3 exp( −3s 2 /2)/⎨⎜ + τR ⎟⎜ ⎠⎝ Ns ⎩⎝ k r ⎫ ⎞ 1 + ⎟⎬ (N −1.41 + 0.83)s 3 ⎠⎭

(20)

The loop-to-bridge formation rate at a normalized extension s for the N-spring Rouse chain with end stickers having a latex adsorption energy ε is then determined as kf = 1/τf. While eq 19 is somewhat ad hoc, we show in what follows that it describes well the first passage time of both dumbbells and 10bead chains in the limit of τesc(ε)/τR ≫ 1 and of course also applies when one bead is permanently fixed to the surface and τesc(ε)/τR = 0, for the mobile end bead. The intermediate case of τesc(ε)/τR ∼ 1 is not relevant because, as we noted above, we require that τesc ≫ τR to avoid free chains appearing during the loop-to-bridge transition. To evaluate the bridge destruction rate, we assume that the process is again separable into desorption of the moving end bead from the potential well near z = d, followed by retraction of the chain from the extended state to a loop conformation. Noting that the latter occurs much faster than the dissociation stage for the experimentally relevant range of extensions due to reduction in conformational free energy, the first-passage time for bridge-to-loop transition can be approximated as

τr(s , N , ε) ≈ τesc(ε)

″ (zmin)|Uads ″ (zc)| Uads

″ (zmin)|Uads ″ (zc)| /(2πζi) = 2.489ε /kBT /(2πζi) Uads

according to the Kramers formula. The bridge destruction rate is thus obtained as kr = 1/τr, yielding the following set of equations for the transition rates G


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Figure 4. Effect of end sticker adsorption energy on bridging dynamics of the polymer: the bridge formation time τf for Rouse dumbbells with end stickers having association well depths of ε = 6−12[kBT] as a function of the normalized gap between the surfaces: (a) ratio of τf to the sum of chain relaxation time and sticker escape time from FFS simulations in comparison with the Milner−McLeish theory given by eq 14; (b) the ratio renormalized by exp(3s2/2)/s.

Figure 5. First-passage time for loop-to-bridge transition for polymers represented as 10-spring Rouse chains without end stickers as a function of the normalized gap between the surfaces: (a) ratio of τf to Rouse chain relaxation time on semilog scale; (b) the ratio renormalized by exp(3s2/2)/s. The symbols are from FFS simulations and the solid line corresponds to the theoretical expression given by eq 17.

τf is scaled with the Rouse time, τR. Since the first-passage time grows exponentially with extension as exp(3s2/2)/s, one can present the data in reduced coordinates τf(s)/[τR exp(3s2/2)/s] to magnify the deviation of simulation results from the theory. (We affectionately call such a plot a “Likhtman plot” in memory of Alexei Likhtman’s style of plotting.) As shown in Figure 3b, the FFS simulation result for the Rouse (or Hookean) dumbbell fluctuates around the constant value predicted by the theoretical two-bead model, but the deviation is within 25%, which we consider to be acceptable, noting the statistical error associated with the number/positioning of the interfaces in FFS and the relatively few (up to 10) independent runs used for sampling the space. Therefore, the FFS scheme is reliable for the elastic dumbbell. The analysis is repeated after adding finite extensibility to the polymer dumbbell model. The first-passage time for a FENE dumbbell of similar root-mean-square end-to-end distance as the Rouse (or Hookean) dumbbell and with a fully extended length L = 100 is also shown in Figure 3a. The FFS predictions closely match the theoretical values from eq 15. While we note from eqs 14 and 15 that the first-passage time for the FENE dumbbell has a dependency on s different than that of the Rouse dumbbell, we nevertheless normalize τf in a similar fashion to bring the data within 1 decade along the vertical axis, allowing a better comparison between the theory and simulation. We can see in Figure 3b that the agreement is reasonable for intermediate extensions s < 3, while the relative

error increases for larger extensions presumably due to the nonlinear coupling of transverse fluctuations with the spring force in the z direction, as mentioned in the previous section. Note also that neither FFS nor the theoretical formula is reliable at s < 1.5 since in this range Uz(s) becomes comparable to kBT. Furthermore, comparing the two spring laws reveals that τf for the FENE dumbbell deviates only modestly from that for the Rouse dumbbell for extensions up to 2.5 times the rootmean-square end-to-end distance (corresponding to d < 0.25L), while the transition time for the FENE dumbbell becomes drastically larger than that for the Rouse dumbbell at larger extensions. Next, we consider the effect of adding the end sticker association energy to the polymer on the loop-to-bridge transition time for the case of a Rouse dumbbell with b = 10[bK]. FFS simulations are therefore performed on chains whose end beads are adsorbed to the latex surface with ε = 6− 12[kBT]. As the association well depth increases, the bridge formation time increases due to the contribution from the sticker escape time. The approximation in eq 19 for N = 1 suggests that this effect can be normalized out if we divide τf by (1 + τesc(ε)/τR) at any given extension. This is done in Figure 4a for FFS simulations with different sticker adsorption energies ε and is compared there with the Milner−McLeish prediction given by eq 14 for the elastic dumbbell with no stickers. We observe that all data collapse onto a universal curve across a broad range of time scales, thus confirming the validity H


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Figure 6. Bridge formation time for 10-spring Rouse chains having end stickers with adsorption free energy ε = 8[kBT] as a function of the normalized gap between the surfaces: (a) results from direct BD simulations in the narrow range of gaps computationally accessible, compared to results from the FFS scheme; (b) FFS simulation results over a broader range of gaps compared with the bridging theory given by eq 19, with the inset showing the same data on the axis renormalized by exp(3s2/2)/s.

consequently the characteristic time, τf, at any gap. The loop-tobridge formation time measured from the two approaches are shown in Figure 6a, revealing excellent agreement. This confirms the validity of the FFS scheme for multibead chains. Note the increased first-passage time in Figure 6a relative to that in Figure 5a at the same normalized distance of around s = 2 due to sticker dissociation from the surface. The FFS simulations are then extended to larger gaps and compared with the theoretical estimate given by eq 19 in Figure 6b. The result suggests that our bridging theory is capable of capturing bridging times as large as 108 times the Rouse relaxation time of the end-tethered polymer. Renormalizing the data by exp(3s2/2)/s in the inset reveals that the agreement is extremely good for s ≥ 2, while some deviation is detected for extensions in the range 1.5 < s < 2 where the theory of Cao, Likhtman, and co-workers30 still holds for the stretching contribution to the first-passage time. Therefore, the error in Figure 6 is evidently associated with the two-stage approximation for separating the sticker dissociation and chainstretching contributions. When the chain is represented by more beads, the coupling between the two effects gets stronger as the moving end bead will bear memory of the particular conformation it had before escaping the adsorption barrier. The coupling is gradually lost at larger extensions so that the approximation holds for s ≥ 2. We believe that such range is indeed sufficient for characterizing the polymer/colloid mixture as the particles barely get closer than s = 2 due to the repulsion created by a brush of polymer loops on each surface. Freely Jointed Chain Model. More refined models of the above polymers can be obtained by increasing the number of springs per chain, N, while adjusting the spring length to maintain a similar root-mean-square end-to-end distance N b = 10 for all chains. An even finer-grained representation of the polymer is achieved using the freely jointed chain (FJC) model based on stiff Fraenkel springs introduced in the Model section, where each spring corresponds to a single Kuhn step. The analogue of the above Rouse chain with NK = 100 that has the same root-mean-square end-to-end distance is thus a freely jointed chain with N = 100 Fraenkel springs each having an equilibrium length bK and a sufficiently large spring constant taken as H = 1000[kBT]. Having verified the validity of the FFS results for both dumbbells (N = 1) and multibead (N = 10) chains, we now rely on the analytical expression given by eq 17 for the first-passage time of Rouse chains with other values of N

of the approximation given in eq 19 for all extensions. Upon renormalizing the data by exp(3s2/2)/s in a “Likhtman plot” in Figure 4b to magnify the deviations, we observe fluctuations in the FFS data similar to that in Figure 3b for the simple elastic dumbbell with no stickers. This confirms that the two-step approximation for separating the contribution of the colloidal surface interaction from chain stretching is reasonably good within the FFS statistical error, at least for the elastic dumbbell. Multibead Rouse Chain Model. The two-bead (or dumbbell) representation of the polymer molecule is, however, incapable of accurately resolving the kinetics for the current problem. The model shows large deviations of the first-passage time from that of a more realistic multibead chain due to neglect of the interior modes and ineffective sampling of the conformation space. As shown by Cao et al.,30 the first-passage time gets faster when the same chain is represented by more beads, assuming that the bead drag coefficient is normalized by the number of beads to keep the drag coefficient (and diffusion coefficient) of the whole chain independent of the number of beads. When constrained between solid surfaces, the interior beads can also influence the association/dissociation of end stickers by populating near the surface. A step toward a more realistic representation of the polymer molecule is thus the multibead Rouse model introduced in the Model section. We first consider a 10-spring chain with NK = 100 Kuhn steps and similar root-mean-square end-to-end distance to that of the Rouse dumbbell considered above and evaluate the first-passage time purely due to chain extension (without the attractive interaction of the sticker with the surface). The FFS data for normalized extensions up to s = 3.3 are shown on a semilog scale by symbols in Figure 5a, while the solid line corresponds to the theory of Cao et al.30 given by eq 17. The data are also plotted with the vertical coordinate normalized by exp(3s2/2)/s in Figure 5b to magnify the error. Very good agreement is observed between the theory and FFS simulation within the range of applicability of both approaches, namely s > 1.5, with the average relative error being less than 16%. Upon adding end stickers with adsorption free energy ε = 8[kBT], we first perform both direct BD simulations on chains adsorbed to latex surfaces in loop conformations as well as FFS simulations over an intermediate range of extension 0.7 < s < 1.6 where both methods are computationally tractable. In direct BD simulations, exponential fitting of the bridge fraction as a function of time according to eq 7 yields the bridging rate and I


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Figure 7. First-passage time for loop-to-bridge transition as a function of the normalized gap for polymer chains with no sticker represented by different numbers of springs according to the Rouse model in comparison with a freely jointed chain (FJC) model of similar equilibrium length but with each stiff spring corresponding to one Kuhn step: (a) ratio of τf to chain relaxation time τR; (b) the ratio renormalized by exp(3s2/2)/s. The symbols are from FFS simulations, and the solid lines correspond to the theoretical expression given by eq 17 for different numbers of springs.

from the loop state. Development of analytical expressions for more realistic chains incorporating such effects would depend on the particular form of the excluded volume potential and can become much more involved due to complications associated with potential derivatives and the eigenmode transformation. One could rely on mesoscale simulations, e.g., the BD methodology used here, coupled to a proper sampling scheme for polymer lengths/gaps that are computationally accessible. For instance, we found that the end-to-end distance distribution and radius of gyration of a poly(ethylene oxide) chain with 300 repeating units (PEO300) modeled with the dry Martini force field32,33 could be replicated at the coarsergrained level of a Rouse chain with NK = 100 and only N = 10 springs if the beads in the latter exclude each other via a 6−12 Lennard-Jones potential with σ = 1.3[bK] and ε = 1.3[kBT]. As shown in the Supporting Information, such a polymer chain exhibits similar scaling of first-passage time with normalized gap, s, as an equivalent Rouse chain without the excluded volume potential, with minor reduction in τf. This minor effect can be rationalized to the result of the minor effect of excluded volume on the frequency of extended chain conformations needed to bridge distances larger that the chain’s radius of gyration. Other complexities such as torsional potentials can also be incorporated in polymer chain representation,12,34 but such analysis is beyond the scope of the present work and the computational cost also becomes exceedingly higher. We believe, however, that the effect of torsional potentials is likely subsumed into its effect on the Kuhn step length, which we have already accounted for as described above. A remaining issue is the description of the sticker and its effect on the first passage time within the finer-grained FJC model. In the bead−spring model, the sticker is assigned one bead, and we have given it the same drag coefficient as the other beads of the bead−spring chain. Since the drag coefficient assigned to each bead changes inversely with the number of beads used, this means that the assigned “escape time” τesc(ε) given by eq 18 depends on the resolution of the chain, becoming larger when the chain resolution is lower, even for bead−spring chains. While one might suppose that the “escape time” should be independent of chain resolution, the resolution of the chain will affect the frequency with which the sticker bead is able to reattach to the original surface after it has escaped. We tested eq 19 using eqs 17 and 18 for both a dumbbell (N = 1) and for a 10-spring chain (N = 10), and as

without the end stickers and compare the result with that of an FFS simulation for the FJC polymer. Note that the bead drag coefficient for each chain should be normalized by its number of beads, so that the drag coefficient of the whole chain remains independent of the degree of coarse-graining. The first-passage times, normalized both by τR and by τR exp(3s2/2)/s, are shown in Figure 7 as a function of normalized gap (note that the characteristic time τR for the FJC model is determined from the end-to-end vector autocorrelation function, which agrees with the longest relaxation time of a Rouse chain with same number of springs).17 While the transition time gets faster for larger N, the freely jointed representation approaches the N = 5 Rouse chain prediction at large extensions where both FFS and theory are reliable. This suggests that 100 constrained modes in the FJC model with NK = 100 is approximately equivalent to five effective Rouse modes. Since the stiff springs in the FJC model correspond to almost rigid rods that cannot change length, they mimic Rouse chains with approximately 20-fold fewer flexible springs. We verified this 20-to-1 mapping for polymers of different lengths in the range NK = 20−150 and found reasonable agreement unless the gap gets close to the fully extended length of the polymer (see Supporting Information for details); therefore, we expect that it could be applied to longer polymer chains as well. The mapping is extremely important as it shows that the firstpassage problem for the more realistic FJC representation of the polymer can be replaced by a coarser-grained bead−spring Rouse model with N ≈ NK/20 springs. Without such a mapping, results from the multimode first passage time theory of Cao et al. would be practically useless, since in this theory the transition rate depends on the number of springs N into which any real chain is discretized. Thus, along with the theory a rule for choosing the value of N, the number of springs, must be given, if predictions for real chains are to be made. Consequently, here we propose that the bridge formation time for any real polymer chain with NK Kuhn steps can be estimated from eq 19 where τmulti,Rouse(s,N) is calculated by setting N ≈ NK/20 in eq 17. Presumably, a similar rule could be applied to the problem of end fluctuations in an entanglement tube, for which the theory of Cao et al. was originally derived. Furthermore, while we have neglected excluded volume interactions between the beads representing the polymer chain, such interactions generally favor more extended conformations and can promote bridge formation and faster rate of transition J


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modifiers.35 Therefore, the bridge formation/destruction rates given by eq 22 are expected to provide reliable predictions of bridging dynamics in polymer/colloid systems.

shown in Figures 4 and 6, these equations appear to work for both of these cases. If extrapolated all the way to the freely jointed chain model, the sticker would be resolved into Kuhn steps and assigned appropriate drag coefficients. The sticker is typically an alkane chain with 10−20 carbons, which in a model resolved at the level of a freely jointed chain would contain around 1−2 Kuhn steps, since for an alkane chain, a Kuhn step represents around 10 backbone bonds. What remains to be done is to explore how well a description at the level of a FJC of an entire chain (including the sticker) is represented by eqs 18 and 19. While such simulations would be computationally expensive, they might be carried out for relatively short freely jointed chains, allowing the theory to be tested in this limit and to validate more thoroughly the success of our proposed coarse-graining strategy. We hope to do this in future work. Bridge Destruction Rate. Finally, we evaluate the validity of the theoretical approximation given by eq 21 for the bridgeto-loop transition time. Direct BD simulations are performed on a dilute solution of polymers with NK = 100 represented by N = 10 elastic springs and end sticker energy characterized by ε = 8[kBT] at different gaps between the solid surfaces. Exponential fits to the temporal evolution of bridge fraction as in eq 7 yield the bridge destruction rate and consequently the characteristic time, τr, in addition to the bridge formation times, τf, shown in Figure 6a. The result for the bridge destruction rate is plotted in Figure 8 at multiple normalized

■

CONCLUSIONS A generalized framework was constructed for quantification of bridging dynamics of telechelic polymers between solid surfaces representing, e.g., latex colloids. Taking the polymer molecules to be much smaller than the colloids, we approximated the latter by flat surfaces, while the polymer molecule was modeled as a bead−spring or freely jointed chain with different levels of coarse-graining. Because of the strong association of the polymer end groups to the colloids, the two dominant conformations for modest gaps between surfaces are a loop with both ends adsorbed to one surface and a bridge where the end groups stick to opposite surfaces. Dangling chains, with one end attached to a single surface, form the transitional state between the loop and bridged states. The analysis derived here provides rates of polymer bridging between the two surfaces starting from a loop state, kf, and the rate of destruction of bridges into loops, kr. We used direct Brownian dynamics (BD) simulations, forward flux sampling (FFS) BD simulations, and theoretical analyses using a recent multidimensional first passage time theory of Likhtman and co-workers to obtain bridge formation and bridge destruction rates for gaps between surfaces greater than the root-mean-square end-to-end distance of the polymer but less than around 30% of the maximum extension of the chain. We found that the bridge formation rate can be expressed in terms of a composite expression combining the times for two processes, namely, the escape of one end sticker from the narrow-but-deep association well and the longer-range transit of this sticker to the opposite surface by chain stretching. The bridge destruction rate, on the other hand, is mainly controlled by the depth and shape of the association potential and can be approximated by the escape rate of a Brownian particle. Analytical expressions were derived for the characteristic transition times τf and τr (inverse rates) that depend on the gap normalized by the equilibrium chain length, the number of beads used in the coarse-grained representation of the polymer, and the end sticker association energy to the surfaces. We further demonstrated that results for a more realistic freely jointed chain (FJC) model with one anchored end and one free end with NK Kuhn steps can be mapped approximately onto a coarse-grained Rouse chain with N ≈ NK/20 harmonic springs. For the FJC model, the effect of a fine-grained sticker with small drag coefficient on the transition rate needs also to be tested to determine if our coarse-grained model with composite expression for the loop-to-bridge transition time is accurate in this limit. If so, then our analysis, which is based on simple harmonic springs, could be applied to real polymers of arbitrary length bridging colloids/surfaces.

Figure 8. Bridge destruction time for polymers represented as 10spring Rouse chains having end stickers with adsorption free energy ε = 8 as a function of the normalized gap between the surfaces. The symbols are from direct BD simulations, and the solid line is the theoretical approximation given by eq 21.

gaps, along with the theoretical prediction of eq 21, which assumes that the bridge destruction rate is controlled by the dissociation of end stickers and not by the polymer extension across the gap. The agreement is reasonably good even at small extensions, with the maximum error being

Bridging Dynamics of Telechelic Polymers between Solid Surfaces

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