Brownian Dynamics Simulations of Colloidal Suspensions Containing

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Brownian Dynamics Simulations of Colloidal Suspensions Containing Polymers as Precursors of Composite Electrodes for Lithium Batteries Manuella Cerbelaud,*,† Bernard Lestriez,† Dominique Guyomard,† Arnaud Videcoq,‡ and Riccardo Ferrando§ †

Institut des Matériaux Jean Rouxel (IMN), Université de Nantes, CNRS, 2 rue de la Houssinière, BP32229, 44322 Nantes cedex 3, France ‡ SPCTS, UMR 7315, ENSCI, CNRS; Centre Européen de la Céramique, 12 rue Atlantis, 87068 Limoges cedex, France § Dipartimento di Fisica and CNR-IMEM, Via Dodecaneso 33, 16146 Genova, Italy ABSTRACT: Dilute aqueous suspensions of silicon nanoparticles and sodium carboxymethylcellulose salt (CMC) are studied experimentally and numerically by Brownian dynamics simulations. The study focuses on the adsorption of CMC on silicon and on the aggregation state as a function of the suspension composition. To perform simulations, a coarsegrained model has first been developed for the CMC molecules. Then, this model has been applied to study numerically the behavior of suspensions of silicon and CMC. Simulation parameters have been fixed on the basis of experimental characterizations. Results of Brownian dynamics simulations performed with our model are found in qualitative good agreement with experiments and allow a good description of the main features of the experimental behavior.

1. INTRODUCTION Polymers are very common in colloidal systems. They are extensively used in colloidal suspensions to alter the stability. They are also often employed for adjusting rheological properties or controlling the microstructure of suspensions. Their use is particularly important in the shaping process of electrodes for lithium batteries, which are often obtained by tape-casting.1 Lestriez et al. have indeed shown that performances of lithium batteries are related to the microstructure of electrodes, which is determined by the structure of the suspensions from which they are fabricated.2,3 Thus, understanding the role of polymers in these suspensions is very important to control the structure of electrodes and, as a consequence, the performance of batteries. Nowadays, in the domain of lithium batteries, most of the studies concern electrodes made of silicon. Indeed, silicon is very promising as anode material because of the high theoretical quantity of electricity stored per unit of mass (3590 Ah kg−1) compared to that of graphite (372 Ah kg−1) presently used in commercial Li-ion batteries. However, during the battery cycling, silicon exhibits large volume expansion and shrinkage, which end in poor cycle life. To tackle this problem, the use of adequate polymer during the process has been proposed.4 One of the most efficient polymers reported in the literature is the carboxymethylcellulose (CMC).5,6 CMC is a derivative of cellulose with carboxymethyl substitutions. The level of substitution is given by its degree of substitution (DS). In an aqueous medium, carboxymethyl groups can dissociate to © 2012 American Chemical Society

form carboxylate anionic groups, which are responsible for the solubility of CMC and for its conformation.7 The number of dissociated groups is related to the pH. Numerous studies on CMC are reported in the literature. Particular attention has been paid to the mechanisms of CMC adsorption on solids. Indeed, CMC is able to bond to a solid via electrostatic interactions8,9 or hydrophobic interactions10,11 or hydrogen bonds.12 Essentially, two aqueous formulations of electrodes for lithium batteries made of silicon, carbon, and CMC can be found in the literature: the first is in deionized water at natural pH13 and the second in a buffer at pH = 3.14 Different bonding mechanisms in the dried electrodes have been proposed to explain their good electrochemical performance.15 However, in both formulations, it is important to study the distribution of the constituents in the mother suspensions to understand the bonding probability between them. In order to analyze the role of each parameter (pH, salt concentration, etc.) in these suspensions for electrodes, numerical simulations can be very useful. For example, Zhu et al. have performed Brownian dynamics simulations on suspensions devoted to the cathode of lithium batteries.16 In their model, the polymer is only implicitly taken into account through the solvent properties. For silicon anodes, since a great Received: May 25, 2012 Revised: June 26, 2012 Published: June 27, 2012 10713

dx.doi.org/10.1021/la302135v | Langmuir 2012, 28, 10713−10724

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interest lies in the polymer role, such an approach does not seem appropriate. Therefore, we propose here to develop an explicit model for the polymer, based on experimental observations. In this study, a coarse-grained model is developed for Brownian dynamics simulations of CMC. Molecules of CMC are modeled as chains of several spheres (“beads”), each one having a diameter equal to its Kuhn’s length.17,18 Here, beads are large enough to keep considering the solvent as a continuous medium. The parameters of the model are directly derived from experimental characterizations. This simple model is applied to study dilute suspensions of silicon and CMC. Particular attention is paid to the role of pH and of the length of CMC chains. We will show that this model, although being simple, is able to shed light on the experimental results. The paper is structured as follows. In section 2, experimental characterizations of the dilute CMC/silicon suspensions are presented. Then, the interaction model and the simulation method are described in section 3, while the choice of parameters employed in the modeling of CMC is detailed in section 4. Finally, section 5 presents the results of simulations performed with silicon and CMC. The simulation results are compared with the experimental ones throughout the text.

Figure 1. Experimental and theoretical ionic conductivity (σ) of the CMC solutions as a function of the weight percentage of CMC (wt% of CMC = mCMC/msolvent × 100) at natural pH (in water) and at pH = 3 (in HCl). conductivity are Na+, OH−, H+, and Cl−, the latter because, at pH = 3, HCl is used to fix the pH. The ionic conductivities used here are as follows: λNa+ = 50.08 × 10−4 m2 S mol−1, λOH− = 190 × 10−4 m2 S mol−1, λH+ = 350 × 10−4 m2 S mol−1, and λCl− = 76.31 × 10−4 m2 S mol−1.19 Figure 1 shows that the calculated conductivity is a bit smaller than the measured one. This small difference may be explained by other sources of conductivity present in the solutions such as the CMC molecules. Nevertheless, the quite good agreement between calculations and experimental measurements allows us to assume that, in water, CMC is fully dissociated, one CMC90 molecule bearing 289 negative charges, and one CMC250 molecule, 803 negative charges. The number of charges corresponds to the number of COO− groups present on the molecule. In HCl solutions, when CMC is added, an excess of HCl is needed to maintain the pH at 3 (see Table 1). This excess is used in the

2. EXPERIMENTAL CHARACTERIZATIONS 2.1. Materials. The silicon powder used in this study is purchased from Alfa Aesar (diameter ds ≃ 50 nm, purity 99.999%, mass density ρ = 2330 kg m−3). Particles are covered by a thin layer of silicon oxide of around 2 to 4 nm. This thin layer makes the silicon hydrophilic and therefore permits the preparation of suspensions in aqueous media. The carboxymethylcellulose sodium salts (CMC) are purchased from Aldrich. In this study, CMC has a degree of substitution DS = 0.7, and two molecular weights are used: Mw = 90 000 (CMC90) and Mw = 250 000 (CMC250). All the suspensions are dilute, prepared with a silicon volume percentage of 1.5% in deionized water or in HCl solution. In the following, when water is used as solvent, the pH is referred to as natural pH. When suspensions are prepared with HCl, the pH is always adjusted to 3. In all the suspensions, the quantity of polymer is kept so small that suspensions remain dilute. Therefore, chains are considered as separated entities and behave independently.17 2.2. Zeta Potential of the Silicon Nanoparticles. The zeta potential of the silicon nanoparticles is determined with a zetasizer nanoseries (Malvern Instrument) with an autotitrator MPT-2. The pH was adjusted with 0.25 M HCl. Smoluchowski’s theory is used to calculate the zeta potential. The natural pH of the silicon suspension in water was found at about 5. At this pH, silicon particles have a zeta potential (ψ) of −40 mV and the suspension is stable. At pH = 3, silicon particles have a zeta potential of +5 mV and the suspension is unstable. When no CMC is added, the ionic conductivity of the aqueous silicon suspension is measured at the very low value of 15 μS cm−1, indicating that no salt is brought by the silicon. 2.3. Ionic Conductivity of CMC Solutions. When the CMC is in aqueous solutions, the ionic conductivity increases linearly with its concentration (see Figure 1). This can be attributed to the sodium ions, which are liberated by the CMC. From the chemical formula of the CMC, its degree of substitution, and its molar mass, it is possible to calculate the number of carboxymethyl groups (−COO−/Na+) borne by one CMC molecule. Assuming a complete dissociation of these groups, the number of Na+ ions liberated and therefore the Na+ concentration can be determined and used to calculate a theoretical conductivity of the CMC solutions as σ=

∑ ziCiλi i

Table 1. Ionic Strength (I), Used in Simulations, as a Function of the Weight Percentage of CMCa wt% of CMC

pH

VHCl (μL)

0.00 0.03 0.24 0.00 0.03 0.24

natural natural natural pH = 3 pH = 3 pH = 3

0 0 0 25 37 139

I (mol L−1) 1.00 5.71 4.59 1.00 2.13 2.74

× × × × × ×

10−5 10−4 10−3 10−3 10−2 10−2

a

VHCl is the volume of 1 M HCl added to maintain the pH at 3 in 25 mL of CMC solutions. protonation of the COO− groups of the CMC. According to the amounts of CMC and HCl used, we found that each molecule of CMC90 adsorbs 170 protons, and each one of CMC250, 471 protons. Considering that CMC loses all its sodium ions in solutions and then adsorbs protons, it comes out that, in HCl at pH = 3, CMC90 and CMC250 bear 119 and 332 negative charges, respectively. The hypothesis that is used to estimate the conductivity allows to describe its evolution as a function of the CMC amount at pH = 3 as well as at natural pH (see Figure 1). Knowing the nature of ions present in suspension and their concentration, the ionic strength of the solutions (I) has been determined as a function of the amount of CMC added. The corresponding values that are used in the following are reported in Table 1. 2.4. Analysis of the CMC Adsorption on Silicon. Adsorption of CMC on silicon particles (or bonding capacity between CMC and silicon) has been determined experimentally by a method based on

(1)

with zi the valence of ion i, Ci its concentration, and λi its ionic conductivity. The various ions considered contributing to the ionic 10714


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viscosimetry. The viscosity of a dilute polymer solutions increases linearly with the polymer concentration. It is thus possible to deduce the polymer concentration from a viscosity measurement. Solutions with different amounts of CMC on one side and 1.5 vol % silicon suspensions with different amounts of CMC on the other side have been prepared and homogenized on rotating rolls during 20 h. Suspensions are centrifuged at 12 000 rpm during 30 min. One batch of suspensions with only CMC is also prepared without centrifugation. The supernatant is removed and held in sealed tubes for one night. The viscosity of this supernatant is then determined with a stresscontrolled rheometer Physica MCR 101 from Anton Paar, using a plate−plate geometry (gap 0.7 mm). The temperature is fixed at 25 °C by a Peltier system. For each composition, 2 consecutive cycles are carried out (1 cycle = an increase of the shear rate from 1 to 1000 s−1 and a decrease of the shear rate from 1000 to 1 s−1). The viscosity is determined as the mean value obtained at the shear rate γ̇ = 158.5 s−1 measured during the decrease of the first cycle and the increase and decrease of the second cycle. The evolution of the viscosity as a function of the CMC concentration is analyzed from the solutions prepared with only CMC, and the coefficients of the master line fitting this evolution are deduced. The same coefficients are found for the solutions that have been centrifuged or not. This parametrization has allowed us to determine the amount of CMC remaining in the supernatant for the suspensions prepared with CMC and silicon. The total amount of CMC adsorbed on silicon is obtained by the difference in the amount of CMC introduced in the initial suspensions. Results are shown in Figure 2. It is observed that adsorption of CMC on

adsorption in mass is larger with a smaller number of chains adsorbed, because the CMC250 polymer chains are much longer. 2.5. Analysis of the Suspension Stability. In order to determine the effect of CMC on the stability of silicon suspensions, sedimentation tests have been carried out. These tests, easy to realize, give a qualitative description of the aggregation in the system. Suspensions are prepared with a silicon amount of 1.5 vol % and with different amounts of CMC. After preparation, the different suspensions are introduced into closed tubes where they are allowed to settle. After 20 h, the height of the cakes is measured and the state of the supernatant observed. The more turbid the supernatant is and the smaller the cake is, the more stable the suspension is. On the contrary, the clearer the supernatant is and the higher the cake is, the more aggregated the suspension is. Suspensions were not allowed to settle for a longer time, because a parasitic reaction occurred in the tubes (probably due to the oxidation of silicon) creating gases and rendering observation of the sediment difficult afterward. Pictures of the sedimentation tubes are shown in Figure 3 and sediment heights are reported in Table 2. First, at pH = 3, aggregation

Figure 3. Images of the sedimentation tubes after 20 h. Suspensions were prepared with different amounts of CMC90 and CMC250: (a) and (c) at pH = 3; (b) and (d) at natural pH.

Figure 2. Evolution of the weight percentage of CMC adsorbed as a function of the weight percentage of CMC introduced. For clarity, all error bars are not shown.

Table 2. Sediment Height (h) Observed in the Sedimentation Tubes after 20 ha

silicon occurs at pH = 3 as well as at natural pH. For this latter case, since the charges of CMC and silicon particles are both negative, CMC adsorption is certainly not only due to electrostatic interactions, but it may originate from hydrophobic interactions, if, for example, the silica layer on silicon is not homogeneous, or from hydrogen bonding between the hydroxyl groups on silicon and CMC. For CMC90, adsorption at pH = 3 is clearly greater than at natural pH. This may be explained by the combination of 3 phenomena: (i) polymer and silicon are oppositely charged at pH = 3, which favors adsorption; (ii) the electrostatic repulsion between polymers is diminished, because ionic strength is higher at pH = 3 (see Table 1); (iii) both CMC and silicon particles are hydroxylated at pH = 3, favoring hydrogen bonding. Concerning CMC250, the polymer is quasi totally adsorbed for all the concentrations, at both pH conditions. At natural pH, the adsorption in the mass of CMC250 is stronger than the adsorption of CMC90. For example, for 0.24 wt % of CMC at natural pH, the mass of CMC250 adsorbed is 1.75 times larger than the mass of CMC90 adsorbed. However, it is worth noting that the number of CMC250 molecules adsorbed is, on the contrary, 1.59 times lower than the number of CMC90 molecules adsorbed on silicon. So, the CMC250

pH

CMC

wt % of CMC

h (cm)

natural pH

CMC90

0.00 0.03 0.24 0.00 0.03 0.06 0.24 0.00 0.12 0.24 0.61 0.00 0.12 0.24 0.61

1.5 1.5 2.6 2.0 3.4 3.5 4.0 2.2 2.7 2.8 3.2 3.4 3.4 4.0 5.0

CMC250

pH = 3

CMC90

CMC250

a

Suspensions were prepared with different amounts of CMC90 and CMC250 at both natural pH and pH = 3. 10715


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⎡ 2 ⎛ r 2 − 4a 2 ⎞⎤ 2as2 A ⎢ 2as s ⎥ ⎜ ij ⎟ UsvdW ( r ) = − + + ln − s ij ⎜ ⎟⎥ 2 2 6 ⎢⎣ rij2 − 4as2 rij ⎝ rij ⎠⎦

is stronger and faster than at natural pH. Indeed, for all the concentrations analyzed, supernatants are clearer and sediments higher. Second, at both pH conditions, the sediment height tends to increase with the amount of CMC, which suggests that CMC favors aggregation, for example, by creating bridges between silicon particles,5 since in all cases, CMC can adsorb on silicon, as shown in Figure 2.

(7)

with A the Hamaker constant. As mentioned in section 2.1, silicon particles present a thin layer of silica at their surface. The Hamaker constants of silicon and silica in water are 9.75 × 10−20 J and 0.46 × 10−20 J, respectively.27 Therefore, the thin layer of silica may have an influence on the van der Waals term. According to the summation developed for core shell particles,28 we have evaluated the interaction potential between two silicon particles with a silica layer of 2 nm. The result (see Figure 4) shows that the presence of the thin silica layer

3. SIMULATION METHODOLOGY 3.1. Interaction Models. 3.1.1. Interaction between CMC Beads. CMC is modeled through a coarse-grained method,17 in which CMC molecules are represented by chains constituted of Nc “beads”, each bead having a diameter equal to the Kuhn’s length of CMC (dc = 11.8 nm).18 The connectivity between adjacent beads is ensured by an elastic potential Ucelas − c(rij) =

kc (rij − 2ac)2 2

(2)

with rij the distance between the centers of beads, ac the bead radius (here ac = 5.9 nm) and kc an elastic constant that will be determined in the following. Nonadjacent beads interact through the electrostatic potential20

Ucelec − c(rij) =

Zc2e 2 e−κrij 4πε rij

(3)

with Zc the bead charge number, e the elementary charge, ε = ε0εr the dielectric constant of the solvent (εr = 81 for water), and κ the inverse Debye length defined as κ=

2e 2z 2NAI103 εkBT

Figure 4. Van der Waals interaction UvdW S−S (7) as a function of the dimensionless center-to-center separation distance rij/(2as), for silicon (A = 9.75 × 10−20 J), silica (A = 0.46 × 10−20 J), silicon with a 2-nmlayer of silica (result of summation) and for an intermediate Hamaker constant (A = 2 × 10−20 J).

(4)

with z the electrolyte charge number (here z = 1), NA Avogadro’s number, kB Boltzmann’s constant, T the temperature (T = 293 K), and I the ionic strength (in mol L−1). To avoid interpenetration between nonadjacent beads, a linear repulsive potential is added. 3.1.2. Interaction between Silicon Particles. The silicon particles are considered as spheres of radius as = 25 nm. Interactions between two silicon particles are modeled by a DLVO potential.21,22 This interaction is the sum of an vdW electrostatic potential Uelec S−S and a van der Waals potential US−S . For the electrostatic contribution, we used the expression23,24 Zs2e 2 ⎛ e κas ⎞ e−κrij ⎜ ⎟ 4πε ⎝ 1 + κas ⎠ rij

changes the interaction between the silicon particles and cannot be neglected. However, the summation method is cumbersome, which is why we decided to take an effective value of A = 2 × 10−20 J for the Hamaker constant. This approximation is quite good for separation distances larger than rij/(2as) = 1.02 and more approximative below (see Figure 4). Nevertheless, because of the divergence of its van der Waals part, the DLVO potential is no longer considered at small separation distances, and thus, we believe that this approximation is sufficient. Indeed, the DLVO potential is cut at −14kBT, and at contact, a repulsive linear potential is added to avoid the interpenetration between particles, as explained and justified in a previous work.29,30 3.1.3. Interaction between CMC Beads and Silicon Particles. To take the electrostatic interaction between a CMC bead and a silicon particle into account, the following expression is used:20

2

Uselec − s (rij) =

(5)

with rij the distance between the centers of silicon particles and Zs their charge number, which is deduced from the zeta potential (ψ)25 ⎡ ⎛ 1 zeψ ⎞ ⎛ 1 zeψ ⎞⎤ kT 4 Zs = 4πε B 2 κas2⎢2 sinh⎜ tanh⎜ ⎟+ ⎟⎥ ⎢⎣ κas ze ⎝ 2 kBT ⎠ ⎝ 4 kBT ⎠⎥⎦

Ucelec − s (rij) =

ZcZse 2 ⎛ 1 ⎞ e−κ(rij − as) ⎜ ⎟ 4πε ⎝ 1 + κas ⎠ rij

(8)

with rij the center-to-center separation distance between the bead and the particle. As mentioned in section 2.4, electrostatic interactions cannot explain by themselves the CMC adsorption on silicon at natural pH. Therefore, as a first approximation, the hydrogen bond and/or the hydrophobic effect are taken into account through a

(6)

With the silicon zeta potentials given in 2.2 and the ionic strengths in Table 1, eqs 4 and 6 give Zs = −104 and Zs = 25 at natural pH and at pH = 3, respectively. For the van der Waals term, we used the expression26 10716


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fraction of silicon particles (ϕ = 1.5%, as in experiments) and by their number. Initial configurations are generated as follows. Silicon particles are randomly placed in the box, avoiding overlapping. The velocity of each silicon particle is randomly chosen such that its norm is equal to [(3kBT)/ms]1/2. The first bead of each polymer is randomly placed at a non-overlapping position. Then, CMC chains are generated by the excluded volume method presented in ref 34 with λ = 2. And again, overlapping between CMC beads or between a CMC bead and a silicon particle is not allowed.

Lennard-Jones potential (technique already used to model hydrophobic interactions31) UcLJ− s(rij)

12 ⎡⎛ ⎛ a + a ⎞6 ⎤ ac + as ⎞ ⎢ s⎟ ⎥ ⎜ ⎟ = 4εa⎢⎜ − ⎜⎜ c rij ⎟⎠ rij ⎟⎠ ⎥ ⎝ ⎝ ⎣ ⎦

(9)

with εa being the depth of the potential well. 3.2. Brownian Dynamics Simulations. The system is simulated by Brownian dynamics.32 The solvent is considered as a continuum medium and its effect upon the solute is represented by a combination of frictional and random forces. The motion of particle i is thus described by the Langevin equation mi

dvi(t ) = −ζivi(t ) + dt

4. DETERMINATION OF THE PARAMETERS FOR THE CMC MODELING In this section, we focus on the simulation of CMC molecules. Specific simulations are carried out to choose the CMC parameters that will be used in the next section (especially the number of beads (Nc) and the elastic constant (kc) that rule the behavior of CMC molecules). These parameters are chosen in order to correctly reproduce the gyration radius and the diffusion of CMC molecules. These properties are first deduced from the semiempirical Mark−Houwink theory. The intrinsic viscosity ([ν]) can be linked to the molar mass of the polymer (M) as35

∑ Fij{rij(t )} + Γi(t ) (10)

j

with mi and vi its mass and its velocity, respectively, ζi = 6πηai the friction coefficient, which depends on the particle radius (ai) and on the solvent viscosity (here η = 10−3 Pa s for water). Fij{rij(t)} are the pair interaction forces between the particles, which depend on the center-to-center distance between the particles (rij(t)) and which derive from the potentials introduced in section 3.1 (Fij = −∇Uij). Γi is the random force obeying the following statistical properties: ⟨Γi(t )⟩ = 0

[ν] = KM α

where K and α are constants that depend on the nature of the solvent and of the polymer, and which are determined experimentally and generally tabulated for the classical polymers. For instance, the following values of K and α can be found for CMC molecules, with DS = 1.06, in water with NaCl:36 K = 7.2 × 10−3 mL g−1, α = 0.95 for NaCl 0.005 M and K = 8.1 × 10−3 mL g−1, α = 0.92 for NaCl 0.01M. According to ref 35, the intrinsic viscosity is also linked to the hydrodynamic radius (Rh)

⟨Γi(t )Γj(t ′)⟩ = Cδ(t − t ′)δij

and

(11)

with C = 2kBTζi to satisfy the thermal equilibrium condition. The Langevin equation is then numerically integrated due to the algorithm developed by Mannella and Palleschi33 in order to get the time evolution of the particle positions. This CMC/silicon system is characterized by two distinct time scales. The velocity relaxation time of a CMC bead (mc/ζc ≃ 10−13 s) is indeed much smaller than the one of a silicon particle (ms/ζs ≃ 10−10 s). That is why we use the integration strategy already employed in ref 29 and summarized here. We chose a time step, δt = 10−11 s, much smaller than the velocity relaxation time of silicon particles. Therefore, we integrate the full Langevin equation for these particles, so that the evolution of their velocities and positions is given by vi(t + δt ) = vi(t ) +

2kBTζs ms

⎡ 1⎢ + −ζsvi(t ) + ms ⎢⎣

[ν] =

(δt )1/2 Yi

ri(t + δt ) = ri(t ) + vi(t ) δt

2kBT 1 (δt )1/2 Yi + ζc ζc

(16)

kBT 6πηR h

(17)

and for real polymers, the ratio between the hydrodynamic radius and the gyration radius is17 Rh = 0.640 Rg

(12) (13)

(18)

Eqs 15, 16, 17, and 18 have been used to estimate Rg and Df for CMC molecules (DS = 1.06) in water with NaCl. Results are shown in Table 3.

where Yi corresponds to uncorrelated Gaussian-distributed random numbers with an average of zero and a standard deviation of 1. For CMC beads, the time step is much larger than their velocity relaxation time, so that the inertia term in the Langevin equation can be neglected and their positions evolve as ri(t + δt ) = ri(t ) +

10πR h3NA 3M

while the latter is related to the diffusion coefficient of the polymer (Df) Df =

⎤ ∑ Fij{rij(t )}⎥⎥ δt j ⎦

(15)

Table 3. Estimation of the Gyration Radius and the Diffusion Coefficient of CMC Polymers with DS = 1.06, for Two Molar Masses and Two Ionic Strengths

∑ Fij{rij(t )} δt

I (mol L−1)

j

5 5 1 1

(14)

Our simulations are performed in a cubic box with periodic boundary conditions. The box size is defined by the volume 10717

× × × ×

−3

10 10−3 10−2 10−2

CMC kind

Rg (nm)

90 250 90 250

27.0 52.7 25.2 48.4

Df (m2 s−1) 1.2 6.4 1.3 6.9

× × × ×

10−11 10−12 10−11 10−12


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The simulation parameters (Nc and kc) are deduced from the comparison between these results and simulation results. In the simulations, the gyration radius is computed by37 R g2 = R g,2 x + R g,2 y + R g,2 z

they are more numerous, they are also less charged. At constant number of beads, the gyration radius decreases when the ionic strength increases, as expected from Table 3. This important characteristic is well-reproduced by the present simple model. Finally, in order to fit with the Mark−Houwink’s predicitions (Table 3) (with an error ≤5%), Nc1 = 13 and Nc2 = 36 beads were chosen for CMC90 and CMC250, respectively. It is worth noting that the molar mass ratio (250/90 = 2.78) is fairly respected by this number ratio Nc2/Nc1 = 2.77. 4.2. Elastic Constant in between Adjacent Beads. By carrying out simulations as in section 4.1, the effect of the elastic constant (kc) has been investigated, varying its value from 10−3 to 10 N m−1, with I = 5 × 10−3 mol L−1. The gyration radii obtained are reported in Table 5. Plots of the bead−bead radial distribution functions and simulation snapshots are shown in Figure 5 for CMC250. The same trends are observed for CMC90.

(19)

with R g,2 x =

R g,2 y =

R g,2 z =

1 Nc

∑ (xi − xg)2

1 Nc

∑ (yi − yg )2

1 Nc

∑ (zi − zg)2

i

i

i

(20)

(21)

(22)

where (xg, yg, zg) are the coordinates of the center of mass. The CMC charge numbers with DS = 1.06 are calculated as in section 2.3, i.e., considering that the CMC is fully dissociated. Then, it is found that molecules of CMC90 and CMC250 bear 387 and 1074 negative charges, respectively. 4.1. Number of Beads in Each Polymer. Brownian dynamics simulations were performed with one isolated polymer, without periodic boundary conditions. Different numbers of beads were tested for both CMC90 and CMC250. The charge number of each bead (Zc) is obtained by dividing the total number of charges of the CMC molecule by the number of beads (Nc). As a first approximation, kc was fixed to 1 N m−1. For each value of Nc tested, the gyration radius is computed after 5 × 10−4 s and its value is averaged over 100 independent simulations. Results are reported in Table 4. Some tests were also performed with values of the

Table 5. Gyration Radii (Rg) Obtained for CMC90 (Nc1 = 13) and CMC250 (Nc2 = 36) with I = 5 × 10−3 mol L−1, Zc = −29.8, and Different Values of kc kc (N m−1)

Rg (nm) Nc1 = 13

Rg (nm) Nc2 = 36

10 5 1 0.1 0.01 0.001

24.5 28.2 28.4 28.4 28.5 30.0

47.3 51.5 51.6 51.7 51.8 53.2

Table 4. Gyration Radii (Rg) Computed from Simulations, For Different Numbers of Beads Nc and Different Ionic Strengths (I), with kc = 1 N m−1 and λ = 2a CMC kind 90

250

Nc

Zc

Rg (nm) I = 5 × 10−3 mol L−1

Rg (nm) I = 1 × 10−2 mol L−1

10 11 12 13 14 26 27 28 29 30 31 33 34 36 39

−38.7 −35.2 −32.3 −29.8 −27.6 −41.3 −39.8 −38.4 −37.0 −35.8 −34.6 −32.5 −31.6 −29.8 −27.5

24.8 (−8%) 25.6 (−5%) 27.4 (+1%) 28.4 (+5%) 29.4 (+9%) 47.6 (−9.7%) 46.0 (−12.7%) 46.6 (−11.6%) 48.9 (−7.2%) 48.2 (−8.5%) 48.4 (−8.2%) 49.5 (−6.0%) 50.9(−3.4%) 51.6 (−2.1%) 53.8 (+2.1%)

22.0 (−13%) 22.6 (−10%) 24.6 (−2%) 25.3 (0%) 26.3 (+4%) 42.3 (−12.6%) 40.9 (−5.5%) 41.6 (−14.0%) 43.9 (−9.3%) 43.3 (−10.5%) 43.5 (−10.1%) 44.7 (−7.6%) 46.0 (−5.0%) 46.9 (−3.1%) 49.4 (−2.1%)

Figure 5. Bead−bead radial distribution functions (Gr) and simulation snapshots for CMC250 and for different values of kc.

a

The errors, as compared to the radii estimated by Mark-Houwink’s theory, are reported in parentheses.

When kc decreases, the gyration radius increases and the main peak of the radial distribution function, at the first neighbor distance, becomes larger. This means that there is more and more superposition and detachment between the polymer beads. Values of kc lower than 0.01 N m−1 are too small to maintain the beads attached together. On the opposite, for kc = 10 N m−1, the gyration radius is very small. This can be attributed to the restructuring of the beads assembly. Indeed, a second peak is found on the radial distribution function, at rij/

excluded volume parameter (λ) different from 2, and no significant difference was observed. The gyration radius varies almost linearly with the number of beads. Small discrepancies may be explained by the competition between the elongation due to supplementary beads and the shortening due to the decrease of the electrostatic repulsion between beads. Indeed, if 10718


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(2ac) ≃ 1.45, and the simulation snapshot shows a bead organization as in a “necklace with two rods of pearls”. For 0.01 ≤ kc ≤ 5 N m−1, the value of the gyration radius does not vary much. In the following, we chose kc = 0.1 N m−1 in order to maintain the beads attached and avoid any restructuring of the bead assembly. 4.3. Polymer Diffusion. Simulations have been performed with a single polymer (at I = 5 × 10−3 mol L−1) to determine its diffusion coefficient from the asymptotic behavior of the bead mean-square displacement: ⟨[ri(t) − ri(0)]2⟩ = 6Df t. The value of the diffusion coefficient is averaged over 50 simulations, each lasting for 9 × 10−4 s. The values found are Df = 2.9 × 10−12 and 1.9 × 10−12 m2 s−1 for CMC90 and CMC250, respectively. They are lower than the expected values (see Table 3). Therefore, in order to correctly describe the relative diffusion between silicon particles and polymers, the solvent viscosity as felt by the polymer beads has been decreased to an effective value (ηe). This trick enhances the polymer diffusion and is actually justified, as the drag force of the solvent on the CMC molecules is strongly overestimated with the polymer chains of our model, which are much thicker than in reality. Different simulations have been performed varying the value of ηe. It is found that Df decreases as 1/ηe, which is consistent with eq 17. In the following, in order to have polymer diffusion coefficients close to the ones expected and given in Table 3, we chose ηe = 0.215 × 10−3 and 0.2 × 10−3 Pa s for CMC90 and CMC250, respectively. We note that essentially the same effective viscosity accounts for both CMC90 and CMC250. The parameters chosen for the CMC simulation are finally reported in Table 6. The gyration radii obtained for these

Indeed, as the polymer chains used in this model are much thicker than the real ones, it is not possible to include in the simulations the correct amount of CMC. 5.1. Adhesion of a CMC Chain on a Silicon Particle. Brownian dynamics simulations have been performed with one silicon particle and one CMC molecule initially placed in such a way that one bead is adsorbed. Different values of the energy, εa, used in eq 9, have been tried. Simulations snapshots are shown in Figure 6.

Table 6. Parameters Chosen for the CMC Simulation along with the Gyration Radii Found for the Two Kinds of Polymer and Two Values of the Ionic Strengtha I (mol L−1) 5 5 1 1

× × × ×

10−3 10−3 10−2 10−2

CMC kind

Nc

kc (N m−1)

90 250 90 250

13 36 13 36

0.1 0.1 0.1 0.1

ηe (Pa s) 0.215 0.200 0.215 0.200

× × × ×

10−3 10−3 10−3 10−3

Rg (nm)

error

28.2 55.8 25.1 49.1

4.3% 5.9% ∼0% 1.4%

a

The error between these values and the expected gyration radii reported in Table 3 is also mentioned. Figure 6. Snapshots of simulations with 1 CMC (in green) and 1 silicon particle (in red) at t = 5 × 10−5 s as a function of εa = 7, 9, or 11 kBT.

parameters are also given in Table 6. When compared to the expected values reported in Table 3, errors lower than 6% are found. Therefore, we estimate that these parameters provide a satisfactory description of the polymer behavior.

At natural pH, the beads and the silicon particles are both negatively charged and adsorption is rendered possible by the Lennard-Jones potential between silicon and beads. For the lowest ionic strength (which corresponds to 0.03 wt % of CMC), the CMC molecules can adsorb on the silicon particles when εa is larger than 7kBT. For εa ≤ 7kBT, attraction between polymer and silicon is not high enough to counter the electrostatic repulsion between the polymer and the silicon particles and so to maintain the adhesion of CMC on silicon. It appears that, the higher εa is, the more numerous the contacts between beads and silicon are. For I = 4.59 × 10−3 mol L−1 (0.24 wt % of CMC), the CMC molecules can adsorb on the silicon even for εa = 7kBT. This can be explained by the screening of the electrostatic repulsions between the constituents since the ionic strength of the suspension is higher. At pH = 3, the polymer adsorbs on the silicon, maximizing the

5. CMC/SILICON SYSTEM: SIMULATION RESULTS For all the following, the simulated system is the one described in section 2, i.e., the DS of the polymers is equal to 0.7, and they bear, at natural pH, 289 and 803 negative charges for CMC90 and CMC250, respectively, and, at pH = 3, 119 and 332 negative charges for CMC90 and CMC250, respectively. Simulations have been carried out in ionic strength conditions that correspond experimentally to two different amounts of CMC (0.03 wt % and 0.24 wt %; see Table 1). Please notice that, in contrast to the experiments, we are studying adsorption of a small number of CMC molecules on silicon particles. The effect of ionic strength is taken into account in the simulations through the electrostatic interactions, and not through the number of CMC chains included. 10719


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For the cases at pH = 3, the obtained configurations correspond to the case of small stiffness of chains in ref 20. 5.2. Aggregation of Two Silicon Particles with One CMC Chain. Brownian dynamics simulations have been performed with two silicon particles and one CMC molecule. Simulation snapshots are shown in Figure 8.

number of contacts between beads and silicon. At this pH, affinity is good between polymer and silicon, because electrostatic interactions are more screened and also because beads and silicon particles are oppositely charged. In conclusion, since CMC adsorption is observed in the experiments also at small amount of CMC, a value of εa ≥ 9kBT is retained for the model. In Figure 6, for the value εa = 9kBT, few beads of CMC250 are adsorbed on the particle. In ref 20, a Monte Carlo study is done by Stoll et al. about the organization of a polyelectrolyte around an oppositely charged particle, which corresponds to the present situation at pH = 3. For different ionic strengths, Stoll et al. have shown that solenoid conformations are achieved by increasing the chain stiffness (angular potential). To observe if such an organization appears in this study, simulations with CMC250 have been performed for a longer period of time. Results after 5 × 10−3 s are shown in Figure 7.

Figure 7. Snapshots of simulations with 1 CMC250 and 1 silicon particle at t = 5 × 10−3 s and with εa = 9kBT: (a) at natural pH for I = 5.71 × 10−4 mol L−1 (0.03 wt %); (b) at natural pH for I = 4.59 × 10−3 mol L−1 (0.24 wt %); (c) at pH = 3 for I = 2.13 × 10−2 mol L−1 (0.03 wt %); (d) at pH = 3 for I = 2.74 × 10−2 mol L−1 (0.24 wt %).

Figure 8. Snapshots of simulations with 1 CMC and 2 silicon particles at t = 5 × 10−4 s as a function of εa = 7, 9, or 11 kBT.

At pH = 3, it is observed that, for all the cases, the silicon particles are aggregated by the polymer, which is consistent with the sedimentation tests (see section 2.5). Nevertheless, at this pH, silicon particles alone can aggregate together because of their small surface charges and because of the high ionic strength. Aggregation is promoted by the polymer as follows: first, the polymer encounters one silicon particle where it adsorbs; then, it encounters the second silicon particle and it also adsorbs on it. The two particles are thus linked together by the polymer. Then, the polymer rearranges itself near the contact point between the particles to maximize the number of contacts between beads and particles (see Figure 8). In these cases, beads are packed around the contact and nonadjacent ones are found in contact. At natural pH, for the cases corresponding to 0.03 wt %, with εa = 7kBT, particles are not linked together, which is consistent with the fact that adsorption is not observed for εa = 7kBT and silicon particles feel repulsive interactions. For εa > 7kBT, the silicon particles are linked by the polymer (“bridging aggregation”). For CMC90, the polymer is rolled around the

At pH = 3, the polymer adsorbs totally on the silicon particle (all beads are in contact with the particle) and no particular order is observed. Nonadjacent beads can be in contact: the ionic strength is high enough to reduce repulsions between the beads. For CMC250 at natural pH with I = 5.71 × 10−4 mol L−1 (0.03 wt % of CMC), it is observed that the CMC is not totally adsorbed on the silicon particle and the extremities of the chain expand in the bulk. When the ionic strength is increased up to the level corresponding to a CMC content of 0.24 wt % (I = 4.59 × 10−3 mol L−1), the polymer can be totally adsorbed on the silicon. Compared to the case corresponding to 0.03 wt % of CMC250 at natural pH, the higher ionic strength allows a decrease in the repulsions between the constituents, but repulsions between beads are still higher than at pH = 3. Indeed, contrary to the cases at pH = 3, the polymer is not “packed” on the silicon, i.e., nonadjacent beads are never found in contact in the this case: CMC250 at natural pH for I = 4.59 × 10−3 mol L−1 (0.24 wt % of CMC). In all the simulations, no solenoid conformation of the CMC is observed. 10720


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contact. For CMC250, only a part of the polymer is rolled and the rest of the chain spreads on both silicon particles. The fact that the long CMC250 chain does not reorganize entirely near the contact can be explained by the strong repulsion between the beads. For a higher ionic strength corresponding to a larger amount of CMC (0.24 wt %) at natural pH, the silicon particles are also linked together by the polymer which is placed in between them. For CMC250, the polymer is more confined near the contact than for the case corresponding to 0.03 wt % but less than at pH = 3. The ionic strength is not high enough to screen the electrostatic repulsions between beads. At natural pH, when no polymer is added, the silicon suspension is stable, which is due to the high charge of the particles. When polymers are added, the ionic strength increases and we can wonder if this would be sufficient to aggregate the silicon particles. To check this, simulations have been performed with two silicon particles without any polymer but with the ionic strength corresponding to 0.03 wt % and 0.24 wt %. No aggregation between particles has been found. Therefore, in the simulations, the increase of ionic strength in this case is not able to provoke aggregation and, when εa ≥ 9kBT, aggregation between silicon particles is promoted by the polymer links. 5.3. CMC Adsorption Capacity on a Silicon Particle. In experiment, it is found that CMC250 chains adsorb more on silicon (in terms of mass adsorbed) than CMC90 chains, especially at natural pH and for the higher mass content of CMC (0.24 wt %). Simulations have been done to verify whether the same behavior is observed. Brownian dynamics simulations are carried out with 1 silicon particle and 22 polymers of CMC90 or 8 polymers of CMC250 in order to introduce the same mass of polymer in both cases. Simulations are done at natural pH and with an ionic strength I = 4.59 × 10−3 mol L−1, which corresponds to a weight percent of 0.24 wt % of CMC. During the simulations, the number of adsorbed polymers tends to fluctuate. Therefore, this number obtained at t = 5 × 10−5 s (time at which it does not vary a lot) is averaged over 3 simulations. Some simulation snapshots are shown in Figure 9 and the mass ratio (rw) between the adsorbed mass of CMC250 and the adsorbed mass of CMC90 as a function of εa is reported in Table 7. In simulations as in experiments, the mass content of CMC250 adsorbed is found higher than the one of CMC90. The value of rw found in simulations with εa = 9kBT is in good agreement with the experimental one: rw(exp) = 1.75. Thus, in the following, εa will be fixed at 9kBT. According to simulations, the fact that a greater mass of CMC250 is adsorbed is due to the larger expansion in the bulk of the longer CMC250 chains than the one of the CMC90 chains (see Figure 9). 5.4. Simulation of a Larger CMC/Silicon System. Other simulations have been performed with 4 silicon particles and 60 CMC90 polymers or 22 CMC250 polymers. Simulation results are shown in Figure 10. At natural pH, the CMC is more dispersed in the bulk compared with the case at pH = 3, where the CMC molecules cover the silicon particles. There is less adsorption of CMC at natural pH than at pH = 3 (see Figure 11), which is in agreement with the experiments. In Figure 11b, except for the lower ionic conductivity, there is no difference in adsorption at the different ionic strengths. This may be due to the fact that the saturation of the CMC adsorption is not obtained in these systems. All the CMC250 introduced in the simulation is indeed adsorbed contrary to what happens with CMC90 at natural pH. Over the simulated time, except for the

Figure 9. Snapshots of simulations with 1 silicon particle and 22 CMC90 molecules or 8 CMC250 molecules at t = 5 × 10−5 s (εa = 9kBT): (a) with all the CMC90 polymers; (c) with only the CMC90 adsorbed polymers; (b) with all the CMC250 polymers; (d) with only the CMC250 adsorbed. For visibility, the box was repeated in all the directions. For (c) and (d), the individual chains of CMC are marked with different colors.

Table 7. Numbers of CMC90 Polymers Adsorbed (N90adso) and CMC250 Polymers Adsorbed (N250adso) as a Function of εaa εa

N90adso

N250adso

rw

9 kBT 11 kBT 14 kBT

10.33 10.66 13.00

6.33 6.00 6.66

1.70 1.56 1.42

a

The mass ratio (rw) corresponds to the ratio of the adsorbed mass of CMC250 over the adsorbed mass of CMC90 (the experimental ratio was found at 1.75).

case of natural pH with I = 5.71 × 10−4 mol L−1 (0.03 wt % of CMC), it is observed that aggregation between silicon has occurred. This is in agreement with the sedimentation tests which show an aggregation for these cases. For the case corresponding to 0.03 wt % of CMC90 at natural pH, during the simulations it appears that some bridges can be made by the polymer between the particles. However, adsorption of CMC on silicon is poor and the CMC adsorbs and desorbs continually, which makes the bridges weak, and globally, no aggregates of silicon are really formed in our simulations. The simulated time is quite short; therefore, aggregation could happen over a longer time, since we have seen in section 5.2 that two silicon particles can be linked by a CMC90 in these conditions. The bridging phenomenon is much more pronounced with the CMC250. Indeed, as these chains are longer, the probability of making bridges is higher and the bridges remain longer during the simulation. In this case, silicon can be aggregated by the bridges between polymers. This observation is consistent with the sedimentation tests, which show that aggregation is more pronounced at natural pH with 0.03 wt % of CMC250 than with the same amount of CMC90. 10721


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Figure 11. Mean number of polymers adsorbed as a function of time: (a) CMC90; (b) CMC250.

strength (parameter related to the CMC amount in the experiments). Even if the model remains simple, simulations are found in qualitative good agreement with experiments and allow some explanations for the experimental observations. First, it was found experimentally that the ionic conductivity and aggregation increase as a function of the quantity of CMC. Simulations at natural pH have allowed us to discriminate between the effects of CMC bridging and those of ionic strength on the aggregation. In simulations, when the CMC chains are longer, the probability of making bridges between silicon particles is higher; therefore, aggregation of silicon is promoted more, especially when few molecules of CMC are present, which is consistent with sedimentation tests. In simulations as in experiments, adsorption of CMC on silicon is stronger at pH = 3 than at natural pH. This is explained by the combination of the effects of charges and ionic strength. Finally, in the same experimental conditions, the mass of adsorbed CMC250 is higher than the mass of adsorbed CMC90. In simulations, this trend is observed and can be explained by the expansion of the long CMC250 chains in bulk. According to the results, the simple model used here seems sufficient to understand the qualitative behavior of suspensions and can help to understand the results of experiments. However, to improve the quantitative results, a better understanding of the interactions between the constituents and the use of more sophisticated models (but also more

Figure 10. Snapshots of simulations with 4 silicon particles and 60 CMC90 or 22 CMC250 molecules at pH = 3 and at natural pH, at t = 5 × 10−4 s, for 0.03 wt % and 0.24 wt % of CMC.

6. CONCLUSION A coarse-grained model has been developed for carboxymethylcellulose molecules. This model consists of dividing the polymer into beads of large size, equal to its Kuhn’s length. Parameters of the model have been fixed in order to respect the experimental gyration radius and the diffusion coefficient of one molecule. This model has been used in Brownian dynamics simulations to study experimental dilute suspensions composed of silicon particles and CMC. Particular attention was paid to the effects of the pH, of the CMC chain length, and of the ionic 10722


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(9) Hoogendam, C.; de Keizer, A.; Stuart, M. C.; Bijsterbosch, B.; Batelaan, J.; van der Horst, P. Adsorption Mechanisms of Carboxymethyl Cellulose on Mineral Surfaces. Langmuir 1998, 14, 3825−3839. (10) Khraisheh, M.; Holland, C.; Creany, C.; Harris, P.; Parrolis, L. Effect of molecular weight and concentration on the adsorption of CMC onto talc at different ionic strengths. Int. J. Miner. Process. 2005, 75, 197−206. (11) Cuba-Chiem, L.; Huynh, L.; Ralston, J.; Beattie, D. In situ particle film ATR FTIR spectroscopy of carboxymethylcellulose adsorption on talc: binding mechanism, pH effects and adsorption kinetics. Langmuir 2008, 24, 8036−8044. (12) Wang, J.; Somasundaran, P. Adsorption and conformation of carboxymethyl cellulose at solid-liquid interfaces using spectroscopic AFM and allied techniques. J. Colloid Interface Sci. 2005, 291, 75−83. (13) Beattie, S.; Larcher, D.; Morcrette, M.; Simon, B.; Tarascon, J.M. Si electrodes for Li-ion batteries−A new way to look at an old problem. J. Electrochem. Soc. 2008, 155, A158−A163. (14) Mazouzi, D.; Lestriez, B.; Roué, L.; Guyomard, D. Silicon composite electrode with high capacity and long cycle life. Electrochem. Solid-State Lett. 2009, 12, A215−A218. (15) Bridel, J.-S.; Azaïs, T.; Morcrette, M.; Tarascon, J.-M.; Larcher, D. In situ observation and long-term reactivity of Si/C/CMC composites electrodes for Li-ion batteries. J. Electrochem. Soc. 2011, 158, A750−A759. (16) Zhu, M.; Park, J.; Sastry, A. Particle interaction and aggregation in cathode material of Li-ion batteries: a numerical study. J. Electrochem. Soc. 2011, 158, A1155−A1159. (17) Teraoka, I. Polymer Solutions: An Introduction to Physical Properties; Wiley-Interscience: New York, 2002. (18) Hoogendam, C.; de Keizer, A.; Stuart, M. C.; Bijsterbosch, B.; Smit, J.; van Dijk, J.; van der Horst, P.; Batelaan, J. Persistence length of carboxymethyl cellulose as evaluated from size exclusion chromatography and potentiometric titrations. Macromolecules 1998, 31, 6297−6309. (19) Lide, D. Handbook of Chemistry and Physics, 89th ed.; CRC Press: Boca Raton, 2008−2009. (20) Stoll, S.; Chodanowski, P. Polyelectrolyte Adsorption on an Oppositely Charged Spherical Particle. Chain Rigidity Effects. Macromolecules 2002, 35, 9556−9562. (21) Derjaguin, B.; Landau, L. Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Acta Physicochim. URSS 1941, 14, 633−662. (22) Verwey, E.; Overbeek, J. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (23) Truzzolillo, D.; Bordi, F.; Sciortino, F.; Sennato, S. Interaction between like-charged polyelectrolyte-colloid complexes in electrolyte solution: a Monte Carlo simulation study in the Debye-Hückel approximation. J. Chem. Phys. 2010, 133, 024901. (24) Rosenfeld, Y. Adiabatic pair potential for charged particulates in plasmas and electrolytes. Phys. Rev. E 1994, 49, 4425−4429. (25) Russel, W.; Saville, D.; Schowalter, W. Colloidal Dispersions; Cambridge University Press: Cambridge, England, 1989. (26) Lyklema, J. Fondamentals of Interface and Colloid Science; Academic Press: London, 1991; Vol. 1. (27) Cappella, B.; Dietler, G. Force-distance curves by atomic force microscopy. Surf. Sci. Rep. 1999, 34, 1−104. (28) P. Viravathana, D. M. Optical Trapping of Titania/Silica CoreShell Colloidal Particles. J. Colloid Interface Sci. 2000, 221, 301−307. (29) Cerbelaud, M.; Videcoq, A.; Abélard, P.; Pagnoux, C.; Rossignol, F.; Ferrando, R. Heteroaggregation between Al2O3 submicrometer particles and SiO2 nanoparticles: Experiments and simulation. Langmuir 2008, 24, 3001−3008. (30) Cerbelaud, M.; Videcoq, A.; Abélard, P.; Pagnoux, C.; Rossignol, F.; Ferrando, R. Self-assembly of oppositely charged particles in dilute ceramic suspensions: predictive role of simulations. Soft Matter 2009, 6, 370−382.

cumbersome) are needed, which will be the subjects for future work. Nevertheless, the model at hand already sheds some fairly interesting light on unexplained results given in the literature on the colloidal suspensions used to prepare composite electrodes and the subsequent battery behavior of the latter. As a matter of fact, the difference in the conformation of the simulated CMC chains adsorbed on the silicon particles as a function of the pH could explain why the Si/CMC-based electrodes prepared at pH 3 show a better cycling efficiency.14 Indeed, the more compact conformation at pH 3 would allow the formation of a better barrier between the Si active mass and the unstable liquid electrolyte. Another unexplained result is the ability of a Si/CMC-based electrode to reversibly sustain the huge variations in volume of the silicon particles upon their alloying/dealloying with lithium (up to 300%).13,15 Indeed, the volume variations are such that, when simply referring to the elongation at break of a CMC macroscopic film (10%), the electrode should rapidly disintegrate upon cycling. However, simulations give a new picture of the details at the molecular scale of how the CMC chains bind together silicon particles. Because the CMC chains arrange themselves near the contact between the particles as shown in Figure 8, it appears here that with this kind of spring-like conformation the CMC chains could more likely maintain bridges between moving silicon particles. Furthermore, the better cyclability of silicon electrodes based on longer CMC15 can be interpreted from these simulations, since numerous bridges between particles can be created with the longer CMC. The model developed here introduces important prospects for the design of new and more efficient binders for lithium battery electrodes, as the effects of the rigidity of the polymer backbone, ionizable group density, chain length, as well as the active mass surface chemistry on the suspension structure can now be predicted to some extent.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Lestriez, B. Functions of polymers in composite electrodes of lithium ion batteries. C. R. Chim. 2010, 13, 1341−1350. (2) Ligneel, E.; Lestriez, B.; Hudhomme, A.; Guyomard, D. On the origin of the pre-plastisizer effect of the composite electrode for lithium battery. Electrochem. Solid-State Lett. 2007, 10, A122−A126. (3) Porcher, W.; Lestriez, B.; Jouanneau, S.; Guyomard, D. Optimizing the surfactant for the aqueous processing of LiFePO4 composite electrodes. J. Power Sources 2010, 195, 2835−2843. (4) Liu, W.-R.; Yang, M.-H.; Wu, H.-C.; Chiao, S.; Wu, N.-L. Enhanced cycle life of Si anode for Li-ion batteries by using modified elastomeric binder. Electrochem. Solid-State Lett. 2005, 8, A100−A103. (5) Lestriez, B.; Bahri, S.; Sandu, I.; Roué, L.; Guyomard, D. On the binding mechanism of CMC in Si negative electrodes for Li-ion batteries. Electrochem. Commun. 2007, 9, 2801−2806. (6) Li, J.; Lewis, R.; Dahn, J. Sodium carboxmethyl cellulose. A potential binder for Si Negative Electrodes for Li-ion Batteries. Electrochem. Solid-State Lett. 2007, 10, A17−A20. (7) Kästner, U.; Hoffmann, H.; Dönges, R.; Hilbig, J Structure and solution properties of sodium carboxymethylcellulose. Colloids Surf., A 1997, 123−124, 307−328. (8) Fujimoto, J.; Petri, D. Adsorption behavior of carboxymethylcellulose on amino-terminated surfaces. Langmuir 2001, 17, 56−60. 10723


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(31) Arnold, C.; Ulrich, S.; Stoll, S.; Marie, P.; Holl, Y. Monte Carlo simulations of surfactant aggregation on adsorption on soft hydrophobic particles. J. Colloid Interface Sci. 2011, 353, 188−195. (32) Allen, M.; Tildesley, D. Computer simulation of liquids; Oxford University Press: Oxford, 1987. (33) Mannella, R.; Palleschi, V. Fast and precise algorithm for computer simulation of stochastic differential equations. Phys. Rev. A 1989, 40, 3380−3386. (34) Stoll, S.; Buffle, J. Computer Simulation of Bridging Flocculation Processes: the Role of Colloid to Polymer Concentration Ratio on Aggregation kinetics. J. Colloid Interface Sci. 1996, 180, 548−563. (35) Wu, H.-S.; Ke, D.-Y. Molecular characterization of a copolymer of 2-allylphenoxyorganocyclotriphosphazene with styrene, methyl methacrylate and vinylbenzyl chloride. Polymer 1997, 38, 431−436. (36) Brandrup, J.; Immergut, E.; Grulke, E. Polymer Handbook, 4th ed.; New-York, Wiley, 1999. (37) Yang, H.; Zhao, X.; Li, Z.; Yan, F. Molecular dynamics simulations on crystallization of polyethylene/fullerene nanocomposites. J. Chem. Phys. 2009, 130, 074902.

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