Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
Three Regimes of Polymer Surface Dynamics under Crowded Conditions Gregory T. Morrin and Daniel K. Schwartz* Department of Chemical and Biological Engineering, University of Colorado Boulder, Boulder, Colorado 80309, United States S Supporting Information *
ABSTRACT: Single-molecule tracking was used to characterize the mobility of poly(ethylene glycol) chains at a solid−liquid interface over a wide range of surface coverage. Trajectories exhibited intermittent motion consistent with a generalized continuous time random walk (CTRW) model, where strongly confined “waiting times” alternated with rapid flights. The presence of three characteristic regimes emerged as a function of surface coverage, based on an analysis of effective short-time diffusion coefficients, mean-squared displacement, and CTRW distributions. The dilute “site-blocking” regime exhibited increasing short-time diffusion, less confined behavior, and shorter waiting times with higher surface coverage, as anomalously strong adsorption sites were increasingly passivated. At intermediate values of surface coverage, the “crowding” regime was distinguished by the exact opposite trends (slower, more confined mobility), presumably due to increasing intermolecular interactions. The trends reversed yet again in the “brush” regime, where adsorbing molecules interacted weakly with a layer of extended overlapping chains.
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were considered to be flights were due to bulk-mediated diffusion, where a molecule desorbed from the surface, diffused through the liquid phase, encountered the surface again, and either readsorbed or executed another hop through the bulk before finally readsorbing. This hopping mechanism was controlled by the so-called sticking coefficient, where a large sticking coefficient implied a high probability of readsorption during a surface encounter.26 Furthermore, the dynamics were consistent with a generalized continuous-time random walk (CTRW) model, a variation on the random walk where a diffusing particle alternated between periods of mobility and immobility, defined by “waiting time” and “flight length” distributions. Further studies from independent laboratories have since confirmed the characteristic intermittent dynamics for poly(ethylene glycol) (PEG) diffusion under dilute conditions.11,12 Here, we studied the mobility of PEG chains at the interface between an aqueous phase and a hydrophobic trimethylsilane (TMS) surface. To observe the effects of surface crowding, the bulk PEG concentration was systematically varied over many orders of magnitude, resulting in values of surface coverage (Γ) that ranged from extremely dilute isolated chains to well above the surface coverage where polymer chains began to overlap (Γ*). Single molecule tracking methods were used to measure a large number of trajectories (∼104) at each level of surface coverage, facilitating robust statistical analyses. Empirical analyses that examined dynamics at various time and length
INTRODUCTION While the behavior of polymers at surfaces is of longstanding fundamental and practical interest,1−4 most experiments and simulations have focused on polymer surface dynamics under extremely dilute conditions,5−13 where bulk-mediated diffusion10−12 has been observed for a variety of experimental realizations. In contrast, little attention has been paid to interfacial dynamics under crowded conditions, even though such conditions are arguably more relevant to most technological applications. Crowding effects play a crucial role in applications including biosensing,14 nanorheology,15 chromatography,16 multicomponent material synthesis,17 confined polymer crystallization,18 and biointerfaces.19−21 The few experiments conducted at high surface coverage either have neglected the effects of bulk-mediated diffusion,22 which is now established as a dominant mode of interfacial transport,10,12 or have focused narrowly on specific conditions relevant to a particular application.18 For example, increased mobility was observed for crowded DNA chains; however, the behavior was uniquely related to the structural characteristics of the DNA.23 Recent studies investigated the effect of concentration on protein surface residence times24,25 but did not investigate interfacial mobility. In sum, the influence of molecular crowding on interfacial polymer mass transport remains poorly understood. Recent single-molecule tracking experiments performed at extremely dilute surface coverage found that polymers exhibited intermittent dynamics at the liquid−solid interface, where individual particle trajectories exhibited periods of either highly confined, slow two-dimensional diffusion or extremely large displacements, termed “flights”.10 Steps within a trajectory that © XXXX American Chemical Society
Received: November 20, 2017 Revised: January 14, 2018
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DOI: 10.1021/acs.macromol.7b02453 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
Fractional Surface Coverage Calculation. The fractional surface coverage of PEG on the TMS surface was calculated using eq 1:
scales as well as model-dependent analyses were performed on the raw trajectory data. The cumulative findings from these experiments provided a comprehensive picture of polymer surface mobility, identifying nonmonotonic changes in mobility that were readily partitioned into three distinct regimes as a function of surface coverage. For convenience, we will designate these three regimes as “site blocking”, “crowding”, and “brush”, respectively, in order from low to high surface coverage.
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( Γ=
c unlab + c lab c lab
)(N
epf )(πR g
As
2
) (1)
where Γ represents the fractional surface coverage, cunlab and clab represent the bulk concentrations of unlabeled and labeled PEG, respectively, Nepf represents the number of localized objects per frame, Rg represents the radius of gyration of an adsorbed PEG molecule, and As represents the surface area of the field of view. The radius of gyration was estimated based on the three-dimensional conformation of PEG-10k in a good solvent. The rationale for this was based on previous experiments performed by Skaug et al., where the interfacial dynamics of PEG on TMS surfaces exhibited scaling behavior with molecular weight consistent with the 3/5 power-law exponent associated with self-avoiding walks in 3D11 (for more details regarding the parameters used to calculate the fractional surface coverage, see page S3 of the Supporting Information). The overlap surface coverage was defined as Γ = Γ* = 1. Effective Short-Time Diffusion Coefficients. Using the method outlined in Mabry et al.,26 the effective diffusion coefficient, Dshort, was calculated at the shortest experimental time lag of 100 ms. The method corrected for apparent displacements associated with localization error by determining a parameterized model for the population of apparently confined steps within the complementary cumulative squared displacement distribution. The remaining mobile steps were then fit to a separate Gaussian mixture model (the approach is detailed on page S4 of the Supporting Information). Waiting Time Distribution Calculations. Distributions were determined directly from measured trajectories. Specifically, a period of immobility was defined as a time interval with apparent displacements
