Tuning Adsorption Duration To Control the Diffusion of a Nanoparticle

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Letter

Tuning Adsorption Duration to Control the Diffusion of a Nanoparticle in Adsorbing Polymers Xuezheng Cao, Holger Merlitz, and Chen-Xu Wu J. Phys. Chem. Lett., Just Accepted Manuscript • Publication Date (Web): 23 May 2017 Downloaded from http://pubs.acs.org on May 27, 2017

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The Journal of Physical Chemistry Letters

Tuning adsorption duration to control the diffusion of a nanoparticle in adsorbing polymers Xue-Zheng Cao,∗,†,‡ Holger Merlitz,∗,¶,§ and Chen-Xu Wu∗,¶ Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China, Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599, United States, Department of Physics, Xiamen University, Xiamen 361005, P.R. China, and Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany E-mail: [email protected]; [email protected]; [email protected]

Abstract Controlling the nanoparticle (NP) diffusion in polymers is a prerequisite to obtain polymer nanocomposites (PNCs) with desired dynamical and rheological properties, and to achieve targeted delivery of nanomedicine in biological systems. Here we determinate the suppression mechanism of direct NP-polymer attraction to hamper the NP mobility in adsorbing polymers, and then quantify the dependence of the effective viscosity ηeff felt by the NP on the adsorption duration τads of polymers on the NP, using scaling theory analysis and molecular dynamics (MD) simulations. We propose and confirm that participation of adsorbed chains in the NP motion break up at time intervals beyond τads due to the rearrangement of polymer segments at the NP surface, ∗

To whom correspondence should be addressed Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China ‡ Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599, United States ¶ Department of Physics, Xiamen University, Xiamen 361005, P.R. China § Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany †

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which accounts for the onset of Fickian NP diffusion at a timescale of t ≈ τads . We develop a power law, ηeff ∼ (τads )ν where ν is the scaling exponent of the dependence of polymer coil size on the chain length, that leads to a theoretical basis for the design of PNCs and nanomedicine with desired applications, through tuning the polymer adsorption duration.

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Designing polymer nanocomposites (PNCs), such as PNCs of polystyrene (PS)/silicananoparticles (NPs), PS/gold-NPs, PS/PS-NPs, carbon black or zinc oxide/vulcanized rubber, and polyethylene glycol-grafted (PEGylated) NPs/Polymethyl methacrylate (PMMA) etc., with specified properties has been the focus of much recent attentions in the fields of nanomaterial science and engineering. 1–3 After incorporating NPs within a polymeric host, the relaxation dynamics of polymer chains, and thereby, the corresponding dynamical and rheological properties, change with the diffusivity of embedded NPs. 4–6 It is imperative to control the diffusion of NPs in polymers in order to achieving polymer-based nanomaterials with desired dynamics-related characteristics, e.g., targeted glass transition and viscoelasticity. Another research driver of studying the NP dynamics is to explore the motional behaviors of nanomedicines in crowded cells. NPs intended for biomedical applications must navigate what is a complex biological environment consisting of bio-macromolecules, cellular and tissue structures. 7–9 A thorough understanding of the diffusional behaviors of tracer NPs in diverse polymers is crucial for the design of some PNCs and nanomedicine with targeted applications. However, the NP motion in polymers has very sensitive dependence on the attributes of NP and polymers. 10–17 NP-polymer attraction exists widely in PNCs and in biological systems. Recently, Mun et al. 18 and Griffin et al. 19 detected in separated experiments, by probing the diffusion of silica NPs in various attractive polymers, that a specific NP-polymer attraction resulting in polymer adsorption onto the NP surface can act as a dominated effect to hamper the NP mobility compared to its diffusion in athermal polymers (without NP-polymer attraction). In the present letter, based on developing scaling theory and a verification using molecular dynamics (MD) simulations, it is revealed that the adsorption duration of polymers on the NP is at the core to suppress the NP diffusivity in the presence of NP-polymer attraction, therefore the NP diffusion is targetedly controllable by tuning the adsorption duration. The mobility of a NP in athermal polymers is normally coupled to the relaxation modes of polymers. As a result, there exists a critical timescale τcr up to which the NP motion

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Figure 1: (color online). Sketchs of a NP diffusing in polymer melts: (a) Without polymer NP adsorption on the NP surface, ǫNP−P < ǫads NP−P and τcr ≈ τrel ; (b) With stable adsorption of ads short chains, ǫNP−P > ǫNP−P and τcr < τads ; (c) With transient adsorption of long chains, ǫNP−P > ǫads NP−P and τcr ≈ τads . The NP is shown in black spherical particle. The nonadsorbed, adsorbed, and transiently adsorbed polymer chains are shown in blue, black and gray respectively. The red arrow gives a random move of the NP. crosses over from sub-diffusion to Fickian diffusion. Scaling theory studies 20,21 provided by Cai et al. indicate that the τcr value is decided by comparing the NP diameter σNP with the characteristic length scales of polymer matrix including the correlation length ξ, the entanglment tube diameter a of entangled polymers, and the end-to-end distance Rete of unentangled polymers. In the particle size regime of ξ < σNP > τads . In this situation, though the terminal relaxation of polymer ∼ N 3.4 and τrel τrel

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ter chains are not reached at timescales of τads < t τads . Therefore, it is assumed that, if τrel >> τads ,

the onset of Fickian NP diffusion is at the timescale as large as the adsorption duration, i.e. τcr ≈ τads . Here the unrealistic case that with extremely large ǫNP−P , in which the ter corresponding τads is comparable with τrel or the reptation time of entangled polymers, is

not considered. In the case that with τcr ≈ τads , the NP motion at the crossover from subdiffusion to Fickian diffusion is not coupled to the relaxation mode of a polymer subsection with coil size as large as σNP , but coupled to the mode of a polymer subsection with relaxation time sub τrel = τads . The transient polymer adsorption decided onset of Fickian NP diffusion at eff t ≈ τads is equivalent to the τcr of an effective NP with size σNP = (Neff )ν b, diffusing in an

athermal polymer matrix, where b is the kuhn monomer size, ν is the scaling exponent of the dependence of polymer coil size on the chain length as derived in Flory theory, and Neff is the sub chain length of a polymer subsection with τrel = τads . Based on the Rouse relaxation theory ter sub which works for the case of Neff < a when τcr ǫads NP−P . Here, σNP = 3σM = 3σ0 and ρ = 0.68. σNP = 3σ0 , respectively, with σ0 being the length unit used in the simulations. In the simulations, the connectivity between monomers is enforced by a finite extensible nonlinear elastic (FENE) potential. 24 The system temperature was fixed at T = 1T0 , where T0 is the temperature unit. The monomer-monomer interaction was modeled as truncated and shifted Lennard-Jones (LJ) potential, with interaction strength ǫMM = 1.0 (in unit of kT0 ) and cutoff distance rc = 1.12σM to avoid an attraction when different monomers are in surface contact, 25 where k is the Boltzmann constant. It is easily verified that without any cutoff the LJ potential has got a minimum at rc = 1.12σM with the depth ǫMM = 1.0. 26 Therefore, an athermal potential is simulated for the monomer-monomer interaction using the truncated and shifted LJ potential. In addition, a full range LJ potential (with cutoff at 8


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rc = 2.5σNP−M ) was implemented for the NP-monomer interaction 25

UNP−M (r) = 4ǫNP−M

"

σNP−M r

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σNP−M − r

6 #

,

(3)

with σNP−M = (σNP + σM )/2. Thus there is an excess enthalpic energy gain ǫNP−M for each monomer close to the NP. A variation of the parameter ǫNP−M then modifies the strength of the enthalpic polymer-NP interaction. The monomer and NP mass is mM = m0 and mNP = (σNP /σM )3 m0 , respectively, with m0 being the mass unit. The characteristic time q

τ0 = σ0 m0 /kT0 is defined as the time unit. The equation of motion for the displacement of a particle (monomer or NP) is given by the Langevin equation. 27,28 All simulations started from a phase in which polymer chains were distributed homogeneously in a cubic box with fixed size d = 30σ0 , and the boundary conditions in all directions are periodic. The number density of monomers ρ = NMonomer /d3 was used to define the concentration of polymer matrices, where NMonomer is the total number of monomers included in the system. In ter running simulations, each system was relaxed by a simulation timestep of the order of 10τrel , ter followed by a longer timestep (of the order of 103 τrel ) of data acquisition, during which a

trajectory of thousands of conformations was stored for the subsequent data analysis. The ter τrel value was determined from calculating the stress relaxation modulus G(t) of polymer

matrices. It has been tested in the simulations that the used relaxation timestep is long enough to have the studied systems arrived in equilibrium. The simulations were carried out using the open source LAMMPS molecular dynamics package. A set of simulations were firstly performed at N = 64, ρ = 0.68 and varying ǫNP−P . For the considered polymer concentration and chain length, it has been shown in previous simulation work that ξ ≈ σM and Rete ≈ 11σM , 22 so ξ < σNP , of the NP, was calculated and shown − in the upper panel of FIG.2, where → r (t) is the NP position vector at time t. The NP diffusion slows down and the crossover of the NP motion from subdiffusion to Fickian diffusion is

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increasingly delayed as increasing ǫNP−P . The Fickian NP diffusion was fitted by the equation, gDiff (t) = 6Dt, where D is the diffusion coefficient. The insert gives the ratio, g(t)/gdiff (t), of the NP MSDs obtained from the simulations to the corresponding fitting functions. A larger deviation of g(t) from gdiff (t) at short t indicates a longer duration of subdiffusion at stronger ǫNP−P . The effective viscosity ηeff "felt" by the NP in the Fickian diffusion regime, was computed by using the Stokes-Einstein relation, ηeff = kT /6πDσNP . The dependence of ηeff on ǫNP−M is provided in the lower panel of FIG.2. As stated above in the scaling analysis, there exists a critical NP-polymer attraction at ǫads NP−P ≈ 1.6 below which ηeff remains unchanged with the change of ǫNP−P . ηeff is equivalent to its value for the case of a NP diffusing in athermal polymer matrix if ǫNP−P < ǫads NP−P . Simultaneously, a power law is ads observed as, ηeff ∼ ǫ1.2 NP−P , for the dependence of ηeff on ǫNP−P when ǫNP−P > ǫNP−P .

Moreover, the NP diffusion in polymer matrixs have been simulated at fixed chain length N = 256 and varying ǫNP−P . Note that the matrix chain length having been considered in this situation is long enough to have polymer chains to be well entangled. 29 As shown in the lower panel of FIG.2, the ηeff value is not changed upon having increased the chain length from N = 64 to N = 256, for all the considered cases with ǫNP−P > ǫads NP−P . The simulation results confirm the conclusion of scaling analysis that ηeff is the effective viscosity felt by a NP with its mobility coupled to the relaxation mode of polymer subsections with chain length decided by the adsorption duration which changes solely with the change of ǫNP−P . Therefore, the ηeff value has no dependence on the matrix chain length when it is ter long enough to have τrel >> τads .

The conformations adopted by adsorbed polymer chains at the NP surface can be distinguished into three structures: 30 train-like that a sequence of an adsorbed polymer chain with all the contained monomers, loop-like that a sequence of an adsorbed polymer chain with only the both ends, and tail-like structure that an adsorbed polymer chain with only one monomer, in contact with the NP surface, respectively. Going outward from the NP surface one first has the train layer which is densest, then follows a zone dorminated by loops

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Figure 3: (color online). Adsorption duration of polymer chains on the NP surface as a function of the NP-monomer attraction. The black dashed line gives the τads value for the case of the NP in athermal polymers. The red sold line is the best power law fit to the simulation data when ǫNP−P > ǫads NP−P . The insert gives the adsorption autocorrelation function, of monomers distributed inside the adsorption layer, at varying NP-monomer attractions. Here, σNP = 3σM = 3σ0 , ρ = 0.68 and N = 64. and finally a periphery where mostly tails occur. Within the adsorption correlation range of 31–34 ξads ∼ kT /(ǫNP−P − ǫads the monomer density profile falls with the distance rNP−M NP−P ),

from the NP surface. Then it remains to be as large as the bulk polymer concentration when rNP−M > ξads . Note that polymer chains can stay in stable adsorption onto the NP surface only when ξads is comparable with or smaller than the correlation length of matrix chains. 25,35 For the considered case being in the concentrated regime, there is ξ∼c−1 . 22,36 Therefore, ξads ≈ ξ ≈ σM if ǫNP−P > ǫads NP−P , which indicates that the thickness of adsorption layer of monomers onto the NP surface is of the order of σM . A calculation of the density profile of monomers at the NP surface have confirmed that the monomers within rNP−M ≈ (σNP/2 + σm ) are in direct contact with the NP. 25 The autocorrelation of monomers distributed inside the adsorption layer surrounding the NP, p(t) =

1 ads < NMonomer >

,

(4)

id(t) id(0)

ads was calculated to describe the dynamical relaxation of polymer adsorption, where NMonomer

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is the number of monomers inside the adsorption layer, id(0) and id(t) being the monomer ID (which has been distributed to each monomer at the begining of every simulation running) at the time t = 0 and t, respectively. The simulation results of p(t) at different ǫNP−P can be found in the insert of FIG.3. Here, the time interval ∆t that starting from a certain time t to the moment t+∆t at which the last one of the orginally adsorbed polymer chains (at t) begin to desorb from the NP surface, is defined as the adsorption duration of polymer chains on the NP. So there is τads = ∆t when the ∆t satisfies p(∆t) =

1 ,

where < Nchain > gives the

averaged number of adsorbed polymer chains. For each adsorbed polymer chain, there is at least one monomer in direct contact with the NP. The dependence of τads on ǫNP−P is shown in FIG.3, with a crossover appeared at the ǫNP−P value of being almost identical with the ǫads NP−P observed in the dependence of ηeff on ǫNP−P . Prior to the crossover, the polymer chains at the NP surface are actually not adsorbed. If ǫNP−P < ǫads NP−P , as predicted in the scaling analysis, the NP-polymer attraction has very limited interference on the rearrangement of polymer segments at the NP surface. Therefore, the τads value, remains unchanged compared to the case of the NP in athermal polymer matrice. However, after the crossover, the polymer adsorption on the NP stays longer as increasing ǫN P −P . The simulation result shows that 1.9 τads ∼ ǫNP−P when ǫNP−P > ǫads NP−P . 1.9 Based on the observed simulation results of ηeff ∼ ǫ1.2 NP−P and τads ∼ ǫNP−P at ǫNP−P >

ǫads NP−P , the dependence of the effective viscosity felt by a NP diffusing in adsorbing polymer 0.63 chains on the polymer adsorption duration is gained to be: ηeff ∼ τads . As shown in FIG.4,

the direct simulation result of the relation between ηeff and τads is in good agreement with the ν ter scaling theory prediction that, ηeff ∼τads with ν = 0.6. Note that Neff < a when τcr

Tuning Adsorption Duration To Control the Diffusion of a Nanoparticle

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