Distribution of the Number of Polymer Chains Grafted on

3 days ago - We demonstrate that polymer number distribution (PND) for polymer grafted nanoparticles (NPs) fabricated via the grafting-to technique ca...
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Distribution of the Number of Polymer Chains Grafted on Nanoparticles Fabricated by Grafting-to and Grafting-from Procedures Hong Liu,†,‡ Huan-Yu Zhao,† Florian Müller-Plathe,‡ Hu-Jun Qian,† Zhao-Yan Sun,§ and Zhong-Yuan Lu*,† †

State Key Laboratory of Supramolecular Structure and Materials, Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China ‡ Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische Universität Darmstadt, 64287 Darmstadt, Germany § State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China S Supporting Information *

ABSTRACT: We demonstrate that polymer number distribution (PND) for polymer grafted nanoparticles (NPs) fabricated via the grafting-to technique can be described, without any fitting parameters, as a function of the conversion of polymer chains. This distribution function is convenient to be applied since the variables in PND are directly linked to experimental measurements and easy to be obtained. As an independent validation, the molecular dynamics simulation in this study is important since the experimental approach may be prone to artifacts that result from the complex parameters. This distribution is further generalized to describe the PNDs for polymer grafted NPs fabricated via the grafting-from technique. Our study implies that the grafting process, no matter grafting-to or grafting-from, does not alter the heterogeneity. Our results also provide evidence that the Poisson model, often invoked to describe the PND in previous experiments, is not accurate. We also show that the binomial form function of PND will not break down even in the cases of relatively large polymer chain length, high binding site density, and high polymer concentration. This function is quite effective since it naturally involves most influencing factors through polymer chain conversion. This study helps to better understand the ligand chain number distribution for polymer-grafted NPs fabricated via both grafting-to and grafting-from techniques.

1. INTRODUCTION Dispersity effects, for example in grafting density, particle diameter, graft length, etc., are often ignored in computational modeling and experiment, despite the large effect of dispersity can have on the interactions1 and dynamics2 of polymer systems. On the other hand, polymer-grafted nanoparticles are widely used in many scientific fields, such as high efficiency organic photovoltaics,3 drug delivery systems,4,5 and water purification membranes.6 In the commonly used grafting-to technique to fabricate polymer grafted nanoparticles (NPs)7,8 or NP-like macromolecules,9 a distribution of nanoparticles with different polymer−nanoparticle ratios usually exists, which is further related to the material composition. It is useful to know not only the average number of polymer chains grafted on the nanoparticle10 but also the distribution of the number of polymer chains grafted on nanoparticle (polymer number distribution (PND) for short in the following). Knowing and controlling the PND of NPs are crucial to understand the selfassembly behavior of NPs and design novel materials with advanced functions. Although the arithmetic mean number of polymers per NP is widely adopted to describe the polymer © XXXX American Chemical Society

chain grafted NP characteristics, it is far beyond satisfactory to describe the true material composition.10 Ad-hoc assumptions about its form had been proposed; for example, there was a viewpoint that PND has a Gaussian form centered at the arithmetic mean, but later it was proved problematic on describing the true material compositions.5,11−13 Pons and coworkers reported that the experimental distributions were consistent with the Poisson distribution in their study of quantum dot−protein components, which could be considered as typical nanoparticle-like system.14 This conclusion was later supported by Mullen et al.10,15,16 and Delport et al.17 in their respective experiments. However, a recent study by Bockstaller and co-workers on the poly(ethylene glycol)-conjugated trypsin system demonstrated that a deviation of their distribution occurred from the Poisson distribution.18 For the first time they used a binomial-like form distribution to describe the PND in their system, which was obtained by solving the mathematical Received: February 9, 2018 Revised: May 1, 2018

A

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

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Analogous reaction procedures are applied for a molar ratio (PEGNHS:trypsin) 16:1. For evaluating polymer distribution of PEGylated trypsin by MALDI-TOF (matrix-assisted laser desorption ionization time-of-flight mass spectrometry) analysis, unreacted PEG-NHS and ions are separated from PEGylated trypsin by dialysis for 7 days against water. Dialysis is performed by use of dialysis membrane with molecular weight cutoff 8000−12 000. Following dialysis, the PEGylated trypsin is freeze-dried and stored at −20 °C for longterm stability. For determining the grafting efficiency, the unreacted PEG-NHS for reactions is evaluated by NMR using internal standard method (details are provided below). 5 g of Cu(NO3)2 is added to the final reaction solution, and the solution is stirred for 12 h to the full precipitation of the PEGylated trypsin. The protein is separated by filtration, and then the unreacted PEG-NHS for NMR test is obtained by dialysis (against water for 7 days) and concentration. MALDI-TOF is performed on a Bruker Autoflex speed TOF/TOF mass spectrometer (mass range 1−500 000 Da) equipped with positive and negative ion modes as well as linear and reflector modes. In a typical sample preparation protocol, 1 mg of trypsin or PEG-modified trypsin samples and 1.5 mg of α-cyano-4-hydroxycinnamic acid (matrix) are each dissolved in 0.1 mL of water/acetonitrile/ trifluoroacetic acid (50/50/0.1). Sample and matrix are deposited on sample holders (stainless steel) by successive drop-casting of 1 μL of the respective matrix and enzyme solution and subsequent solvent evaporation. 1 H nuclear magnetic resonance is performed on a Bruker AVANCEIII500. Prior to the test, the samples are treated under vacuum for 12 h. In a typical sample preparation protocol, 2−2.5 mL of dimethyl sulfoxide-d6 is added into the samples and treated under ultrasonic for 5 min. Then certain quality of acetonitrile (15−20 mg) is added as internal standard substance. The well-mixed solution is submitted to test. The quality of PEG-NHS is calculated by the following equation:

ordinary differential equation from the conjugation reactions equation. Generally, the polymer number distribution in nanoparticle-like systems may be influenced by various experimental conditions. Intuitively, these factors include the geometric distribution of binding sites on the surface of nanoparticle (uniformly or randomly distributed), the routes or techniques of binding process (grafting-to, grafting-from, ligand exchange, and so on), the ligand concentration and its relative quantity to the nanoparticle sites, etc. It is very important to better design and optimize the functional nanocomposite materials by clarifying the effects of these conditions on the distribution of polymer numbers in the nanoparticle-like systems. In this study, we present a binomial function of PND that can faithfully describe the ligand chain number distribution of NPs. In this predictive PND function, the only input variable is the conversion of polymer chains, which can be easily measured in experiments. To validate this distribution, we conduct experiments on the conjugation of poly(ethylene glycol) (PEG) chains to NP-like trypsin molecules by nonspecific reaction of poly(ethylene glycol)−N-hydroxysuccinimide (PEG-NHS) with the amino functional group of lysine residues that are exposed on the protein surface. We further perform molecular dynamics (MD) simulations that can faithfully reproduce grafting processes by incorporating the stochastic reaction model. Both the experimental and simulation results prove the correctness of the binomial form of PND. This distribution function is convenient to be applied since the input variable is directly linked to experimental measurements in which the grafting efficiency is comparatively easier to be obtained, for example, by directly measuring the ligand chain conversion during the grafting process. Chain conversion could be obtained by different measurements, e.g., by quantifying the unreacted ligands in the system through 1H NMR with the internal standard method. We also prove that this binomial function can be used to describe the PND for polymer grafted NPs fabricated via the grafting-from technique. Our results also show that the Poisson-like distribution typically adopted in previous experiments on NPs systems can only be used to describe PND in the cases of low grafting efficiency, and it is obviously not suitable for densely grafted NPs systems. The binomial form of PND will not break down with the increase of polymer chain length, binding site density, and polymer concentration. This PND function, derived purely from the probability perspective, naturally involves most of the factors that influence the grafting processes.

AACN = APEG

mACN 41.05 mPEG 44

×3 ×4

(1)

in which AACN and APEG represent integral area of peak at 2.07 ppm (CH3CN) and 3.51 ppm (−CH2CH2O−, PEG), respectively (as shown in Figure S1 of the Supporting Information). mACN and mPEG represent the mass of the acetonitrile as internal standard substance and the desired mass of PEG. Conversion of polymer (PEG-NHS), η, is calculated according to the following equation: mPEG η=1− × 100% mtotal(PEG) (2) with mtotal(PEG) being the initial total mass of PEG. 2.2. Simulation Method. We use canonical ensemble Brownian dynamics simulation method with implicit solvent model to describe the motion of NPs, polymeric chains, and monomers. The standard Lennard-Jones (LJ) potential with truncation at 3.0σ (σ is the length scale of LJ potential) is used to characterize the nonbonded interactions between any two coarse-grained (CG) beads

2. EXPERIMENTAL AND SIMULATION DETAILS

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ULJ(r ) = 4εLJ⎢⎜ ⎟ − α⎜ ⎟ ⎥ ⎝ ⎠ ⎝r⎠ ⎦ ⎣ r

2.1. Materials Preparation and Physical Characterization. Following previous works (e.g., ref 18), we choose to focus on a specific system, i.e., the conjugation of PEG chains to NP-like trypsin molecules by nonspecific reaction of PEG-NHS with the amino functional group of lysine residues that are exposed on the protein surface.18,19 We choose the trypsin molecule to represent an NP because it is well-defined with the total number of reactive sites as g0 = 8. The PEG-NHS is purchased from JENKEM Technology Ltd. with the average mass 5000 Da and the dispersity (Đ) around 1.02. Prior to modification, trypsin is dialyzed in buffer solution (aqueous solution of sodium tetraborate, 10 mM, and calcium chloride, 5 mM) at pH = 8 for 12 h prior to use. For the PEGylated trypsin Try-PEG5000, the modification is as follows: 41.7 mg of PEG-NHS is added to 50 mL of a buffer solution of trypsin (c = 0.5 mg/mL), and the mixture is vigorously stirred for 2 h at 40 °C (this approach corresponds to an 8fold excess of PEG-NHS with respect to available lysine residues).

(3)

The interaction where α = 0.4 indicates a “Θ solvent” condition. parameters between any types of CG beads are set the same. Bonded interactions between adjacent monomers (beads) in the polymer chain as well as between the first monomer of polymer and the anchoring site on the NP surface are both characterized by a finitely extensible nonlinear elastic (FENE) spring potential as 20,21

⎛ 1 r2 ⎞ UFENE = − kr0 2 ln⎜1 − 2 ⎟ 2 r0 ⎠ ⎝

(4)

2

where k = 30εLJ/σ and r0 = 1.5σ according to ref 22. All simulations are performed on NVIDIA Fermi C2050 Graphic Processing Units with the GALAMOST simulation package.23 With the energy scale εLJ, B

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Typical snapshot of the initial configuration in grafting-to reaction. The free polymers (cyan chains) with active ends (red) distribute randomly among the nanoparticles (yellow spheres) with reactive binding sites (green) on the surface. The grafting-to process occurs between the red active ends of polymers and the green sites of nanoparticles. (b) Polymer number distribution (PND) as given by eq 7 with the specific number of reactive sites g0 = 8 as an example. The inset figure shows a schematic model of a polymer grafted nanoparticle. The reactive sites (red) on the nanoparticle (yellow) can react with active ends of free polymers in the vicinity. A part of the sites already carry grafted polymers (cyan), as shown in the figure. the length scale σ, and the bead mass m, the simulations are conducted with time step dt = 0.0005σ(m/εLJ)1/2 and T* = 1.0kBT/εLJ. The grafting reactions, including the grafting-to conjugation and the grafting-from polymerization, are both described by the generic stochastic reaction model proposed in our previous works.24,25 In this reaction model, we introduce the idea of reaction probability Pr to control the reaction process. In each reaction time interval τ, if an active end meets several reactable beads (free monomers or surface anchoring sites) in the reaction radius, first it randomly chooses one of the reactable beads as a reacting object. Subsequently, another random number is generated. Then by checking if it is smaller than the preset reaction probability Pr, we decide whether the chosen reactable bead will connect with the active end or not. If the bond can be formed between the active end and the reacting object, we record the connection information and update the spring forces between them. During the polymerization, the newly connected monomers then turn to be the growth centers in the next propagation step of the same chain to connect other free monomers, so that the active end is transferred forward. Thus, this model is suitable to reproduce the grafting-from reaction process. For typical grafting-to reaction, the binding reaction occurs between the active chain end and the binding site on the NP determined by a probabilistic criterion. After that, both reactants turn to be saturated. This generic stochastic reaction model had been successfully used to describe the polymerizations in different conditions, such as polymerization-induced phase separation26 and surface-initiated polymerization on the flat substrate,27 on the concave surface,24 and on the convex NP surface.28 This generic reaction model had also been used to describe other types of reactions. For example, it had been used to describe curing reactions in epoxy resin systems.25 In a recent paper, we proposed an updated reaction model that is easier to be handled technically. For example, it has lower perturbation on the MD simulations, so that the adoption of thermostat turns to be unimportant.29 More details on reaction kinetics of these two types of reactions29,30 and simulation protocols for both grafting-from28 and grafting-to reactions31 can be found in our previous papers. 2.3. Simulation Model Construction. In our simulations, the NP is constructed by lumping NP-type beads into a spherical structure via the geodesic subdivision method.32 The details of the NP model construction can be found in the Supporting Information. Initially, we set 125 NPs with the same size and the same number of active binding sites (labeled as g0 in this study) in the 5 × 5 × 5 lattice arrangement with periodic boundary conditions in all three directions of the simulation box. The Quaternion method33 is applied on the beads of the same NP to constrain mutual movement of NP-type beads within

the NP. The box size is set large enough to accommodate all the NPs with polymers, so that the distances among NPs are suitably large to allow grafting the polymers in dilute solution. In practice, we set the box size as (120σ)3 for the systems with NP diameter d = 10σ. After the NPs are constructed, free polymer chains are put randomly in the space among them with predefined concentration. The end monomer of the polymer is labeled as active so that it can be bound to the NP via the stochastic reaction model. The chain length of all polymers is set as N = 10 to make sure the movement of polymer is fast enough so that the efficiency of grafting-to reaction is acceptable. The influence of polymer chain length on PND will be discussed specially in the following section. For the simulations in which polymers are in excess quantity, the concentration of the polymers is set as 0.05 chains/σ3 so that the grafting-to reaction is still taken place in a dilute solution. For the systems with insufficient polymers, the total number of polymers is determined by the feed balance parameter, as defined in the following. Figure 1a shows a typical snapshot of the initial configuration in the grafting-to reaction. In the simulations of grafting to process, the reaction occurs between the end monomer of polymer and the active site on the NP. For all the simulations, the data are obtained with the same reaction probability Pr = 0.1 and the same reaction time interval τ = 50dt. According to our previous study, this set of parameters corresponds to a moderate reaction that is slower than the particle diffusion, so that the influence of transport limitation by chain diffusion to the reaction sites is negligible. In our previous paper we had stressed that the large value of reaction probability corresponds to a fast reaction in our stochastic reaction model.27 In the case of fast reaction, the process is obviously diffusion controlled; an extreme example is that the reaction takes place just like between frozen chains. This is not a true condition in most cases because the chain diffusion becomes a key factor to determine the progress of reaction. Especially for long ligand chains, when the local reaction is accomplished, the reaction process is in pause until the reactants diffuse to the reaction site. Therefore, we choose the suitable reaction probability that corresponds to a moderate reaction. The consideration of moderate reaction is based on the common condition in most experiments, in which the active ends of chains have enough chance to try to react with the NPs. Before the grafting reaction is switched on, a period of 2 × 105 time steps simulation is performed to eliminate the influence of initial configuration. Then a period of reaction process is simulated with at most 5 × 106 time steps to collect the data. To obtain more accurate PNDs, parallel samples are required. Typically four parallel samples are used for obtaining each data point in the reported figures. Therefore, C

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules all the PNDs in the paper are based on 500 polymer-grafted NPs with an efficiency obtained from four parallel samples (each sample includes 125 NPs).

this form of PND. The average grafting density of polymers on spherical NPs σg = g0φη/4πR2 (R is the radius of NP) can also be obtained, which is of great importance in experiments as well. In the following, eq 7 is validated by a combination of experiments and computer simulations. By using MALDI-TOF (as shown in Figures 2a and 2c) on the grafted trypsin

3. RESULTS AND DISCUSSION Without consideration of any detailed kinetics during the grafting-to procedure, the PND can be derived directly from the probability perspective based on the already-grafted polymers. Bockstaller and co-workers made pioneering studies on this issue and proposed a correct distribution function.18,34,35 Their distribution is based on the estimation of binding site occupancy, which requires characterizing NPs in experiments (in most cases not cheap and convenient). Following their ideas, we make a new derivation of PND in which the input quantities are easy to be obtained in experiments as compared to characterizing the reactive sites on NPs. Let g0 be the number of reactive sites on each NP surface and A be the molar ratio between polymer chains and NPs originally in the feed. np is the number of NPs in the system, then the number of polymer chains is Anp, and the total number of reactive sites on all NP surfaces is npg0. The conversion η of polymer chains after the grafting-to reaction is defined as the fraction of polymer chains eventually bound to the NPs. Thus, after the completion of the grafting-to reaction, the number of bound polymer chains is Anpη, which is equal to the number of reacted sites on NP surfaces. Therefore, the probability of each surface reactive site to be saturated by polymer is An pη p= = φη n pg0 (5) Here we define φ = (Anp)/(npg0) = A/g0, i.e., the ratio between the total number of polymer chains in the system and the total number of NP reactive sites, in other words, the feed balance (surplus or shortfall) of polymer chains to NPs. φ > 100% means an excess and φ < 100% a deficiency of polymer chains with respect to the number of reactive sites on NP surfaces. The probability of each NP surface reactive site to be grafted is p; then the probability to find g sites out of g0 sites to be grafted is pg(1 − p)(g0−g). On the other hand, there are Cgg0 ways to select g grafted sites out of a total of g0 reactive sites. Thus, the total probability for a NP to carry g polymer chains is P(g ) = Cgg p g (1 − p)(g0 − g ) 0

Figure 2. (a) and (c) are MALDI-TOF spectra of PEG-conjugated trypsin after dialysis with the molar feed ratios of PEG-NHS to trypsin as (a) 8:1 and (c) 16:1, respectively. (b) and (d) are comparisons of polymer number distributions from different origins (i.e., predicted by eq 7, experiment, and simulation) with the molar ratios of PEG-NHS to trypsin as (b) 8:1 and (d) 16:1, respectively. Red hollow circles are MALDI-TOF data from (a) and (c) after peak fitting and correction with the molecular weight of native trypsin. Black circles are data from corresponding computer simulations. The solid lines are PNDs obtained by using eq 7. We also show for comparison the corresponding Poisson distributions based on the same grafting efficiencies as those in experiments (dashed lines). For better comparison between different distributions, all data in (b) and (d) are scaled by taking the peak values as 1.0 in each distribution.

(6)

Namely P(g ) =

g0! g ! (g 0 − g )!

(φη)g (1 − φη)(g0 − g ) (7)

This binomial form function of PND contains two variables, which are directly linked to experimental measurements. The conversion of polymer chains, η, can be easily obtained by measuring the solution concentration of free polymer chains after grafting-to reactions and then dividing it by their initial concentration in the feed. The concentrations can be obtained, for example by titration or liquid chromatography. The other important variable is the feed balance φ, which is known a priori in an experiment. With the two variables in eq 7 known, a determined PND for grafting-to reactions can be described even for situations where there are fewer polymer chains than surface reactive sites. Figure 1b shows a schematic illustration of

molecules after grafting-to reaction, the PND can be obtained in experiments, as shown by the red hollow circles in Figures 2b and 2d. Two sets of experiments are carried out with the feed ratios of PEG-NHS to trypsin as 8:1 and 16:1 by mole. Corresponding computer simulations with the same conditions are performed for comparison. According to the feature of trypsin, the value of g0 is set as 8 for each NP. For the system D

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules with PEG:trypsin = 8:1, 1000 polymer chains are fed so that the balance parameter φ is 100% (cf. Figure 2a). For the system with PEG:trypsin = 16:1, 2000 polymer chains are fed so that the balance parameter φ is 200% (cf. Figure 2c). The graftingto simulations are conducted for long enough time, and a lot of trajectories are stored with small time intervals, so that we can get the exact polymer conversion η corresponding to that in experiment from these trajectories (practically the experimental conversions of PEG-NHS are η = 0.39 for PEG-NHS:trypsin = 8:1 and η = 0.24 for PEG-NHS:trypsin = 16:1, respectively). As shown in Figures 2b and 2d, for different PEG-NHS/trypsin feed ratios, eq 7 predicts well both the results of experiments and simulations. The computer simulation and experiment data both fall on the curve predicted by eq 7 and thereby reasonably validate eq 7 for describing PND in the grafting-to process. Since in computer simulations g0 and feed balance φ can be accurately determined and easily controlled, as a further validation, we perform a series of computer simulations of the grafting-to reaction with other g0 and φ values, i.e., g0 = 16 and 40, and φ = 33%, 66%, 100%, and 150% (see details in Table S1 of the Supporting Information). In the simulation box with 125 NPs, 660, 1320, 2000, and 3000 polymer chains are fed in for g0 = 16, and 1650, 3300, 5000, and 7500 polymer chains are fed in for g0 = 40. The grafting-to processes are conducted for long enough time (5 × 106 time steps, until the polymer conversion η does not change anymore), so that PNDs for these systems can be accurately obtained. Figure 3 shows

P(g ) =

g0! g ! (g 0 − g )!

e−g0ν(e ν − 1)g (8)

where ν is the instantaneous grafting ratio defined by ν ≡ −ln(1 − ε) with ε ≡ ⟨g⟩/g0 being the grafting efficiency of binding sites on the NP (⟨g⟩ is the average number of grafted polymers per NP). Notably, the grafting efficiency of the binding sites on NP is labeled as ε. Quite recently, Kumar and co-workers36 proved Bockstaller et al.’s function in their simulations of the “grafting-to” process. On the basis of eq 8, if we directly put ε = 1 − e−ν into this equation, it is easy to obtain a binomial form similar to eq 7 as P(g ) =

g0! g ! (g 0 − g ) !

ε g (1 − ε)(g0 − g ) (9)

Equation 9 originates from the estimation of binding site occupancy, which requires characterizing NPs in experiments. Instead, in eq 7 we focus on the unreacted polymers by measuring the solution concentration after the reaction, which would be practically easier to be obtained as compared to characterizing the reactive sites on NPs. On the other hand, eq 7 could be reasonably applied to predict the PND in specific biomacromolecules systems via reverse processes like the hydrolytic cleavage of bound polymers because any binding reaction details, for example, fast or slow, reversible or irreversible, even independent bindings or correlated binding of sites with each other, are consistently taken into account when deriving eq 7, since a determined polymer conversion η can always be measured in experiments. Regarding the derivation of eq 7 in this study, it should be noted that it actually follows analogous arguments as presented in Bockstaller and co-workers’ study.34 Here, effectively, we choose the fraction of reacted/unreacted polymer as a reference to determine the probability of each surface reactive site to be saturated, i.e., the value of p. On the other hand, to validate eq 7, we choose the trypsin-PEG model to obtain the substantially bound ligand chains on trypsin molecules via MALDI-TOF experiment. Actually, the same approach was used to validate eqs 8 and 9 by Bockstaller and co-workers,18 in which they also made the same experiment of PEG conjugating on the trypsin molecules. In the above grafting-to simulations, the chain length of all polymers is set as N = 10. As a further investigation, we compare the PNDs for other polymer chain lengths; the results are shown in Figure 4. For both the early stage (1 × 106 time steps) and the late stage (4 × 106 time steps) of the reaction, we can find that the simulation results fall on the corresponding lines of eq 7 with η obtained from the simulations. It is quite easy to understand that the binding of long chains on NPs will be influenced by the already bound chains on the neighboring sites. Thus, it is not a random or independent event of grafting, but the whole grafting process is correlated more or less. Our finding is that the function to describe PND eq 7 will not break down with the increase of the polymer chain lengths. The local cooperative binding in long-chain system leads to that an already bound long chain on the nanoparticle may shield the neighboring binding site and reduce its possibility of being further grafted, which directly lowers the overall binding efficiency, the polymer conversion (η), and the average probability of each surface reactive site to be bound (p). In this process, the intrinsic kinetics of each polymer chain to be grafted does not change; the foundation and basic assumptions

Figure 3. Comparison of computer simulation results (symbols) with theoretical curves based in eq 7 on PNDs for systems with different polymer−nanoparticle balance φ in “grafting-to” process: (a) with g0 = 16 and (b) with g0 = 40. For better comparison between different distributions, all data in the figure are scaled by taking the peak values as 1.0 in each distribution.

the PNDs obtained by computer simulations and the predictions by eq 7 with η measured in simulations. They agree well with each other for all systems. Figure 3 also shows that even with φ = 100%, the reactive sites on NP surfaces are not fully saturated by polymer chains. It should be noted that Bockstaller and co-workers made pioneering studies and proposed a distribution function as18,34,35 E

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

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Figure 4. Comparison between simulation results (symbols) with theoretical curves based in eq 7 on PNDs for different polymer chain length systems in “grafting-to” process with g0 = 40. A period of 5 × 106 time steps simulations are conducted to collect the data. The two sets of plots correspond to the period of 1 × 106 time steps of reaction (hollow symbols) and 4 × 106 time steps of reaction (solid symbols).

NP, g, its average, ⟨g⟩, and the grafting efficiency ε (i.e., fraction of surface initiators being involved in grafting-from reaction) by ε = ⟨g⟩/g0. Here, a grafted chain is assumed to exist, if at least one monomer has grown from the initiator. Since in graftingfrom reactions there is no polymer conversion (because only free monomers are fed), we transform eq 7 so that it is represented by the efficiency of surface initiators ε (the overall efficiency of surface initiators exactly equals to the average probability of each site to be grafted, so p can be directly replaced by ε in eq 6 according to the physical meaning of eq 6) and obtain eq 9. The PNDs based on the computer simulations are shown in Figure 5a with different values of g0. It is clear that the simulation results agree with eq 9 (black lines) quite well. This result is not reported previously and apparently nontrivial since it clarifies that PND in the cases of graftingfrom reactions can still be described by a distribution function originally proposed for use in the grafting-to technique. This result can be explained if we focus on the initiation process of the grafting-from technique, in which the number of polymers on NP surface is actually decided by the reaction between the first monomer and the surface initiator. In this first-step reaction, the grafting-from can be considered as a simplified version of grafting-to except that the agent to be grafted-on is the first monomer and not an entire polymer chain. As a result, the PND form should be the same in both grafting-to and grafting-from reactions. We further validate the PNDs for “grafting-from” systems with different free monomer concentrations; the results are shown in Figure 6. For the systems with ρm = 0.06−0.72 monomers/σ3, it is clear that eq 9 still describes PNDs very well. Thus, the concentration of free monomers has no influence on the PND of NPs. Besides, since all the systems reported in Figure 6 are performed with the same grafting-from reaction time, we also find that the systems with lower free monomer concentration have relatively lower grafting efficiency, which is consistent with experimental observations. It should be noted that as an independent validation of eqs 7−9, the application of MD simulation to model heterogeneity in polymer-graft reactions is very important. It is because the experimental approach (MALDI-TOF) is prone to artifacts that result from the complex parameters that determine peak height in MALDI-TOF spectra (i.e., peak height does not always correlate with frequency of ionized species). It is therefore an

for deriving the binomial distribution eq 7 do not change. Thus, this function should be still valid in the case of grafting with long chains. Our simulation results, as shown in Figure 4, well support this analysis. The results show that the polymer number distribution can still be satisfactorily described by eq 7. Actually, in the case of grafting with long chains, it commonly corresponds to a quite low conversion of polymers η. Therefore, by using η, we have automatically taken into account the influence of nearby grafted chains. This conclusion can also be partially supported by a comparison of our experiment and that of Bockstaller et al.18 In ref 18 the authors chose the PEG-NHS chains with obviously higher molar masses, e.g., 3.5 and 7.5 kDa. Our study uses the PEG-NHS with 5000 Da. The validity of this binomial form of PND in such a wide range of molar masses of ligand chains well supports our conclusion that the PND is independent of the polymer chains. In sum, one may expect that the PND should be dependent on the grafting density as well as the degrees of polymerization of the ligands. However, our result proves that eqs 7−9 remain applicable even for high molecular weight ligands. This conclusion is important because it shows that ligand interactions do not fundamentally alter the statistics of the grafting process. Another important route to fabricate polymer-grafted NPs is via grafting-from reactions (also called surface-initiated polymerization to grow the chains from the surface). Its PND has received little attention in the literature, as compared with that of grafting-to reaction, despite its obvious importance. We had conducted computer simulations to elucidate the grafting efficiency in this set of systems.28 In the initial simulation configuration, the free monomers are homogeneously distributed in the space among NPs. The free monomer concentration is set to 0.24 monomers/σ3 to ensure a reasonable chain-growth efficiency. Figure S2 shows a schematic snapshot of the initial simulation configuration in the grafting-from reaction. Parameter settings in grafting-from simulation are basically the same as those in grafting-to simulation, except for the reaction probability. Here, the reaction probability is set as Pr = 0.002 so that the process of chain growth is moderate. In this grafting-from scheme, let g0 denote the number of initiators on each NP surface before the grafting-from reaction. After the same period of grafting-from reactions, we calculate the number of grafted chains on each F

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aspect should be of interest to the community since graftingfrom is often considered as the “universal solution” to all problems. This result helps to better predict the PND of grafted chains fabricated via the grafting-from procedure. It is necessary to discuss the relationship between the binomial form distribution and the Poisson distribution. Pons et al. reported in experiments that the distribution of polymer number on NPs was consistent with Poisson distribution.14 This was later supported by the experiments of Mullen et al.10,15,16 and Delport et al.,17 whose respective data could be generally fitted on Poisson distributions. Also, Uddayasankar et al.37 emphasized that the Poisson model was the simplest to describe PNDs. According to ref 18, the Poisson distribution can be described as PPoisson(g) = e−ννg/g!, where ν is the instantaneous grafting ratio as that in eq 8. Now let us go back to Figure 5. For clarifying the validity of Poisson model on this issue, we also put the curves described by Poisson distribution in Figure 5a (red lines). Obviously, there is a large deviation of simulated data from Poisson distribution. For discovering the reason for the deviation, we choose a system with the specific value of g0 (g0 = 40) and calculate PNDs at different stages of grafting-from reaction, as shown in Figure 5b. The grafting efficiency ε changes as the grafting-from reaction proceeds. The peak positions of PNDs increase with increasing grafting efficiency. At the early stage of reaction (e.g., ε = 0.17) the PND only slightly deviates from a Poisson distribution. As the grafting-from reaction proceeds to higher grafting efficiency, the PND markedly deviates from Poisson distribution. However, the binomial form of eq 9 can still well describe all PNDs in the cases of different ε without using any fitting parameters. It strongly suggests that eq 9 is able to describe correct PNDs at any stages of grafting-from reactions. On the other hand, the fact that the deviation from Poisson distribution turns larger with increasing ε indicates that the Poisson-like (or called skewed Poisson) distributions observed by Delport et al.17 and Mullen et al.10,15,16 may be due to the low grafting efficiency in their respective systems (generally with ε < 50%, see detailed analysis in the Supporting Information). From Figures 2b and 2d, we can further compare the Poisson distribution (dashed lines) with the measured data. Clearly, eq 7 describes the PNDs obtained from both experiments and simulations better than the Poisson distribution for grafting-to reactions. A number of factors, e.g., mass transport, solubility, autocatalysis, cooperativity in binding, steric blocking of sites, etc., could account for a deviation of the true PND from Poisson distribution.9 However, eq 7, derived only from the probability perspective, naturally takes into account the influence from all above factors and correctly describes PNDs in different chain grafting conditions. It should be noted that in ref 18 a quantitative relationship between eq 8 (which is identical to eqs 7 and 9) and the Poisson distribution had been established. As indicated by ref 18, eq 8 reduces to the Poisson distribution for large dendrimer or colloidal systems (with large g0) in the limit of very low coupling efficiencies (with low ε or ν). It actually rationalizes the reason that the use of Poisson for describing PND had been persistent in the literature. Basically the Poisson distribution provides a good enough representation of PND in the limit of large g0, while it breaks down for small particle systems (with small g0). Since eq 7 is derived from the probability perspective, it can also be used to describe PNDs in the cases of different NP shapes. For spherical nanoparticles and flat surfaces, their different shapes correspond to different Gaussian curvatures. If

Figure 5. (a) Polymer number distributions obtained by simulations (symbols) and the corresponding predictions by eq 9 (black lines) and Poisson distributions (red lines) for systems with different g0 values in the cases of “grafting-from” reactions. The black and red lines are obtained by using ε and g0 from simulations. (b) Polymer number distributions for different grafting efficiencies ε obtained by simulations (symbols) and the corresponding predictions by eq 9 (black lines) and Poisson distributions (red lines) at different stages of grafting-from reactions with g0 = 40. The same style of black and red lines (solid, dash, dot, and dash-dot) is used to indicate that they correspond to the same set. For better comparison between different distributions, all simulation data in the figure are scaled by taking the peak values as 1.0 in each distribution.

Figure 6. Comparison between simulation results (symbols) and theoretical curves based in eq 9 for PND in systems of different free monomer concentrations in “grafting-from” process with g0 = 40. The PNDs are calculated after the same period (3 × 105 time steps) of the grafting-from reaction.

important task to have an independent validation. The mutual agreement of observations in MALDI-TOF data and simulation results evidently proves that the validation is feasible. Our simulation study clarifies that PND in the cases of grafting-from reactions can still be described by a distribution function originally proposed for use in the cases of grafting-to reactions. Although not surprising, this conclusion is important because it implies that the grafting process, no matter graftingto or grafting-from, does not alter the heterogeneity. This G

DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

observations in MALDI-TOF data and simulation results evidently proves that the validation is feasible. We also show that the Poisson-like distribution observed in previous experiments may be due to the low grafting efficiency in their respective systems. The binomial form function of PND will not break down as the increase of polymer chain length, binding site density, and the polymer concentration. This study helps to better understand the chain number distribution of polymer grafted NPs fabricated via both grafting-to and grafting-from techniques.

a nanoparticle changes its size from several nanometers to a very large value, the corresponding particle curvature decreases monotonically. Actually, the flat surface can be taken as an infinitely large nanoparticle with Gaussian curvature equal to 0. It is apparent that with decreasing particle curvature the crowdedness of grafted chains increases, which will effectively reduce the probability of each surface site to be grafted with polymer chains. But in our theoretical derivation, this effect has been automatically taken into accounti.e., in eq 5 we have built relation between chain grafting efficiency and the conversion of ligand chains. It means that no matter whether the nanoparticle is large or small (or even for a flat surface), by measuring ligand chain conversion, we can immediately obtain the grafting efficiency, with which we can describe ligand chain number distribution using eq 7. This also shows the prediction power of our theoretical eq 7. For nanorods, the situation seems more complicated since the curvature of the “end-caps” may be different from that of “nanorod stem”. Such a difference will result in different ligand chain crowdedness on nanorod surface. But in our derivation of ligand chain number distribution of nanoparticles, the key physical property is the grafting efficiency. For nanorods with the same shape and size, the grafting efficiency on each nanorod would be the same, even though on each nanorod surface the grafting efficiency may be locally different. The “on-average” same grafting efficiency can still be obtained by measuring the ligand chain conversion during the grafting process. It further implies that our theoretical result (eq 7) is still correct for describing ligand chain number distribution of nanorod systems. Last but not least, for inorganic particle, for example, the gold NP, each gold atom exposed at the surface can be regarded as an active site which is possible to bind the ligand chains with the end groups like thiol. Therefore, g0 may be a quite large value, and it is unknown. For this kind of NP with unknown g0, we also have strategy to handle it and obtain the predicted PND. The details can be found in the Supporting Information.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00309. Model and simulation method; supporting discussion (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Z.-Y.L.). ORCID

Hong Liu: 0000-0002-7256-5751 Florian Müller-Plathe: 0000-0002-9111-7786 Hu-Jun Qian: 0000-0001-8149-8776 Zhong-Yuan Lu: 0000-0001-7884-0091 Author Contributions

H.L. and H.-Y.Z. contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful for Ms. Mei-Ting Li for experimental preparations and Dr. Yi-Han Zhou for helping us to do the MALDI-TOF measurements. This work is supported by the National Science Foundation of China (21474042, 21534004, and 21774051). H.L. gratefully acknowledges the support from the Alexander von Humboldt Foundation. Z.Y.L. is thankful for the support of the JLU-STIRT program at Jilin University.

4. CONCLUSIONS We have shown herein that the distribution of the number of polymer chains per NP for polymer-grafted NPs fabricated via the grafting-to technique can be accurately described by a binomial form function that is obtained from a probability perspective. It is then validated by both experiments and computer simulations. This function is easy to use, without requiring any fitting parameters or expensive instrumental analysis and characterizations. To apply this function, one only needs to measure the conversion of polymer chains, which can be easily obtained by, for example, titration or liquid chromatography. Our simulation study also shows that eqs 7−9 remain applicable even for high molecular weight ligands. This conclusion is important because it implies that ligand interactions do not fundamentally alter the statistics of the grafting process. This function is also applicable to describe grafted chain number distribution for grafting-from reactions. It clarifies that PND in the cases of grafting-from reactions can still be described by a distribution function originally proposed for use in the cases of grafting-to reactions. Although not surprising, this conclusion is nontrivial because it implies that the grafting process, no matter grafting-to or grafting-from, does not alter the heterogeneity. On the other hand, as an independent validation, the MD simulation is important since the experimental approach may be prone to artifacts that result from the complex parameters. The mutual agreement of

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REFERENCES

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DOI: 10.1021/acs.macromol.8b00309 Macromolecules XXXX, XXX, XXX−XXX