Stability of Negatively Charged Platelets in Calcium-Rich Anionic Copolymer Solutions

On the Stability of Negatively Charged Platelets

in Calcium-Rich Anionic Copolymer Solutions

Martin Turesson,∗,†,‡ Andre Nonat,† and Christophe Labbez∗,†

ICB, UMR 6303 CNRS, Universite de Bourgogne, F-21078 Dijon Cedex, France

E-mail: [email protected]; [email protected]

Abstract

Controlling the stability of anisotropic particles is key to the development of ad-

vanced materials. Here, we report an investigation, by means of mesoscale MD sim-

ulations, of the stability and structural change of calcium rich dispersions containing

negatively charged nanoplatelets, neutralized by calcium counterions, in the presence

of either comb copolymers composed of anionic backbones with attached neutral side

chains, or anionic-neutral linear block copolymers. In agreement with experimental

observations, small stacks of platelets (tactoids) are formed, which are greatly stabi-

lized in the presence of copolymers. In the absence of polymers, tactoids will grow and

aggregate strongly due to large attractive Ca2+-Ca2+ correlation forces. Unlike comb

copolymers which only adsorb on the external surfaces, block copolymer are found to

intercalate between the platelets. The present results show that the stabilization is the

result of a free energy barrier induced by the excluded volume of hydrophilic chains

while the intercalation is due to bridging forces. More generally, the results shed new

light on the recent finding of the first hybrid cementitious mesocrystal.

∗To whom correspondence should be addressed†ICB, UMR 6303 CNRS, Universite de Bourgogne, F-21078 Dijon Cedex, France‡Current address: Chemical Center, Lund University, P.O. Box 124, S-221 00, Lund, Sweden

1

Introduction

The stability of nanoparticles dispersed in solution and their phases,1,2 have been studied

for over 50 years and found to be a delicate balance of entropic and energetic contributions

to the total free energy. For example, it is possible to form a stable liquid or a gel as well as

a colloidal glass or crystal, with identical nanoparticles whose only difference is the charged

state of their surface.3–6 More recently, the phase stability and the controlled aggregation

of nanoparticles dispersed in polyelectrolyte solutions have attracted intense scrutiny7 and

is challenging to predict because of the computational difficulty in accessing the relevant

length and time scales. Flory theories, density functional theories, and molecular simulations

methods provide essential guidance, although accurate calculations are restricted to simple

systems with two or at most few spherical nanoparticles in the relevant size regime.8,9 Despite

these difficulties, a vast array of self assembled hybrid nano-objects and nanomaterials are

emerging with advanced functions and properties.10–12

Although widespread, one often overlooked example of nanomaterial is hydrated cement

paste used in construction materials, e.g. concrete, as well as in applications such as synthetic

bone or teeth repairing. It is composed of nano-hydrates of various shapes and nature,

which germinate and grow during the hydration of the initial cement grains (clinker) under

the condition of high pH (> 12) and salt concentration (mainly Ca(OH)2). Due to this

complexity, the origin of cement cohesion was only discovered ten years ago! It was shown to

be the result of strong attractive electrostatic ion correlation forces, acting between the main

cement hydrates, i.e, calcium silicate nano-hydrates, commonly abbreviated C-S-H.13–16

In this context, the use of anionic comb copolymers dispersants, consisting of negatively

charged backbones (e.g. polyacrylic acid) with grafted neutral hydrophilic side chains (most

often polyalkene oxides), has allowed the emergence of highly pumpable and self compacting

concrete with high mechanical performance, essential for construction of incredible skyscrap-

ers.17 Recently, similar copolymers were employed to stabilize and control the gelification

of C-S-H suspensions which, when used in concrete, were shown to be excellent hardening

2

accelerators.18–20

From the self assembly of stabilized C-S-H suspensions with linear block copolymers,

Picker et al. just obtained the first cementitious hybrid mesocrystal21 with impressive elas-

tic properties. These results constitute the initial steps toward a new revolution in the field of

advanced nano-materials for construction. In this paper, we shall present the first large scale

Figure 1: Simulation snapshots (for system s10), illustrating the setup used for (left) po-tential of mean force calculations between two parallel platelets arranged face-to-face alongthe z-axis. The shaded areas show a time averaged iso-density surface for the side chainmonomers (white) which are connected to the polymer backbones (red). The divalent ions(Ca2+) and monovalent anions (OH−) are shown in green and blue, respectively. The rightfigure is a snapshot from the corresponding many-platelet simulation (at large t), from whichstatic and dynamic properties were extracted. For clarity, only the platelets are shown.

mesoscopic simulations of an assembly of highly charged platelets dispersed in like-charged

comb copolymer solutions. Based on these extensive molecular dynamics simulations com-

plemented with free energy calculations, see snapshot in Figure 1, we will identify the kinetic

and thermodynamic mechanisms responsible for the observed stabilizing effects. We will fur-

ther show the self assembly of charged platelet tactoids with intercalated copolymers, which

can be viewed as the first bricks toward the self-construction of hybrid C-S-H mesocrystals.

3

Description of the model system

All simulations were performed on the level of the full primitive model, including ions,

platelets and copolymers explicitly, but considering water as a structureless dielectric con-

tinuum, with a dielectric constant ǫr, equal to 78.4. In the primitive model, two charges, qi

and qj , separated by a distance rij interact via Coulombs law: u(rij) =qiqj

4πǫ0ǫrrij, where ǫ0 is

the permittivity of vacuum. Ions (here Ca2+ and OH−) were approximated as freely moving

divalent and monovalent point charges, respectively. On top of the electrostatic interactions

between charges in the system, all species were subjected to a strictly repulsive truncated

and shifted Lennard-Jones potential UTrSLJ ,

UTrSLJ (rij) =

ǫLJ

[

(

σij

rij

)12

− 2(

σij

rij

)6]

+ ǫLJ if rij < σij

0 otherwise.(1)

ǫLJ = 1kT , where T (here 298 K) and k is the temperature and Boltzmann constant,

respectively. Furthermore σij = (σi + σj)/2 and σCa2+ = σOH− = 0.4 nm.

b)a)

c)0.5 nm

0.5 nm0.4 nm

0.4 nm

7 nm

n

xnx x

i

[P cn,x]

[P bn,x]

C1

C3

j k C2

Figure 2: 2-D schematics of the repeating units of a) the comb copolymer of type P cn,x b)

the block copolymer of type P bn,x. The rightmost picture in Figure c, zoom in on the C-S-H

platelet, showing the hexagonal surface pattern and the symmetry axes, C1, C2 and C3,along which pairs, ij, and triplets, ijk, define bonds and angles, respectively.

The same model for comb polyelectrolytes as used in ref.22 (sketched in Figure 2a), was

4

employed. In brief, the hydrophilic side chains are modeled by chains of neutral monomers

attached in regular intervals to a chain of backbone monomers. Each monomer in the

backbone, not being a side chain grafting point, carries a negative unit charge. No bond

angle potential was implemented, yielding a freely jointed chain. Throughout the text, comb

copolymers will be notated P cn,x, with n being the number of repeating blocks, and x being

the length of the side chains, see Figure 2a. More specifically, in this work we have studied

two different comb copolymers, identified as P c7,4 (54 monomers per polymer) and P c

7,10 (102

monomers per polymer). They have the same backbone linear charge and side chain grafting

density, but differ with respect to the side chain length. In addition to the comb copolymers,

a block copolymer, notated P b10,30, was investigated, consisting of one block of 10 charged

monomers linked to another block of 30 neutral monomers, see Figure 2b. Note that the

described polymers are smaller than polymers typically used for industrial applications in

order facilitate the simulations, but keep the qualitative behavior. The C-S-H platelets were

modeled as discs (7 nm in diameter) decorated by 169 sites (σsite = 0.5 nm) arranged in a

hexagonal pattern, see Figure 2c. Each particle carries 308 negative charges, spread evenly

over the sites. The resulting surface charge density σs amounts to roughly -640 mC/m2 in

agreement with the literature.23 More details of the platelet and copolymer models are given

in the Supporting Information.

Simulation conditions

To balance the negative charge of the platelets, 154 divalent counterions, i.e. Ca2+, per

platelet were added to the systems. In addition, the electrolyte solution always contained

an excess of 2:1 salt, corresponding to a saturated calcium hydroxide solution (≈ 20 mM).

Two sets of simulations were performed in the canonical (NVT) ensemble:

• Firstly, pair potentials between two non-rotating parallel charged platelets with a face-

to-face configuration in contact with the various copolymer solutions, were measured

as a function of the (center of mass) inter-platelet distance, D, see Figure 1a. These

5

simulations were carried out with molecular dynamics (MD) simulations (GROMACS

version 4.5.424). An in-house software (Monte Carlo (MC) method), was also employed

to confirm the validity of some of the MD free energy calculations. The MC method

uses the standard metropolis algorithm25 to sample ion configurations, with platelets

and ions confined in a closed cylinder.

• Secondly, static and kinetic properties (radial distribution functions, tactoid sizes and

auto rotational correlation functions) of many-platelet systems (on the same level of ap-

proximation with freely rotating platelets) were analyzed, see Figure 1b. For this pur-

pose we again used MD simulations with a cubic box geometry and three-dimensional

periodic conditions. Due to the heavily computer power demanding simulations, the

multi-platelet systems were followed during a limited period of time of up to 100 ns.

This corresponded to performing more than 10M time steps. The simulations were run

in parallel on 12 processors, and took about 1 month to complete in real time.

For more details about the description of the methods and input parameters used, see Sup-

porting Information. All the simulated systems in this study, referred to as sj, with j being

the simulation index, are summarized in Table 1.

Results and discussion

Force calculations: fixed parallel plates

In Figure 3a the calculated mean force of interaction between two parallel platelets as a

function of their center of mass separation Fp,z(D), is shown for systems s1, s2, s3 and s4

(see snapshot in Figure 1a, illustrating the setup). In the simulations the platelet volume

fraction was set to 1 %. The bulk concentration (approximated as the concentration at the

box boundary) of divalent ions, Ca2+, was roughly 19 mM. The number of copolymers was

adjusted for each system to give a bulk concentration of charged monomers Cbp, equal to

6

Table 1: Simulation details. The columns give the Simulation identifier (ID), Copolymertype (Ptype), Number of platelets (Nplate), Total charge per platelet (Qplate), Number ofcopolymers (Npol), Number of divalent ions (NX2+), Number of monovalent ions (NX−) andBox dimensions (Lx/Ly/Lz). The superscripts in the third column stand for rotating (r) orparallel (p) platelets.

ID Ptype Nplate Qplate/e Npol NX2+ NX− Lx/Ly/Lz (nm)s1 - 2p -308 0 608 600 30/30/30s2 P c

7,4 2p -308 30 818 600 30/30/30s3 P c

7,10 2p -308 24 776 600 30/30/30s4 P b

10,30 2p -308 60 908 600 30/30/30

s5 - 50r -84 0 2100 0 35/35/35s6 - 50r -168 0 4200 0 35/35/35s7 - 50r -308 0 8200 1000 35/35/35s8 P c

7,4 50r -308 100 8900 1000 35/35/35s9 P c

7,4 50r -308 300 10300 1000 35/35/35s10 P c

7,10 50r -308 300 10300 1000 35/35/35s11 P b

10,30 50r -308 300 9700 1000 35/35/35

about 15 mM.

20 30 40 50 60 70 80 900

0,5

1

1,5

2

10 15 20 25 30 35D / Å

-90

-60

-30

0

30

60

βFp,

z Å-1

s2 (Pc

7,4)

s3 (Pc

7,10)

s4 (Pb

10,30)s1 (MD)s1 (cyl-MC)

a)

-15 -10 -5 0 5 10 15z / Å

0

5

10

15

20

Cp(z

) / M

D = 16 ÅD = 17 ÅD = 19 ÅD = 24 ÅD = 28 ÅD = 30 ÅD = 35 Å

-50 -25 0 25 50

z / Å

0

5

10

Cp(z

) / M

D = 35 Å b)

Figure 3: (a) Net mean forces Fp,z(D), exerted on two charged (σs = -640 mC/m2) parallelplatelets (oriented face-to-face), in the presence of the polyelectrolytes P c

7,4, Pc7,10 and P b

10,30.The inset shows the side chain induced barrier at longer range. A comparison between MCand MD force calculations for the polymer-free system (s1) is also included. (b) Densityprofiles (along the z-axis, see Figure 1) of the charged monomers in the block copolymerfor system s4 (P b

10,30). In particular, we show the profile behavior between the plateletsfor a selection of platelet-platelet separations. The upper inset emphasizes the fact thatadsorption occurs on all four platelet surfaces.

7

Polymer-free systems

When the bulk electrolyte contains nothing but a calcium hydroxide salt in saturated con-

ditions, a short-ranged attraction, with a pronounced minimum at about 9 A, is found (see

system s1). The attraction is due to ion-ion correlations between dense layers of adsorbed

calcium ions on the platelet surfaces. This result is in agreement with previous atomic force

measurements, as well as simulation calculations, and was identified to be the main cause

of the strong cohesion in hydrated cement systems.15 For recent reviews on ion-ion correla-

tions see e.g.14,26–28 A comparison of system s1, solved with Monte Carlo simulations in the

cylindrical cell model, is also included, showing excellent agreement.

Response to copolymer addition

When the bulk solution also contains copolymers, see systems s2, s3 and s4, the obtained

force curves are qualitatively different. However, at short separations the force curves are

practically identical for all four systems. This simply means that the platelet-platelet sepa-

ration is physically too small to incorporate any copolymers between the platelets, with the

dominating force being the correlation forces induced by the calcium surface counterions. As

the separation is increased to about 13 A, copolymers enter the gap between the surfaces,

adsorbing to the high-density calcium layers outside the platelet surfaces, with a concomi-

tant pronounced platelet-platelet repulsion. As the separation is further increased, the force

drops and actually becomes attractive in the region [15 < h < 20], due to a copolymer bridg-

ing mechanism. This attractive bridging force, diminishes with increasing side chain length

and, concurrently, the repulsive barrier, at h ≈ 14 A, also becomes smaller. At even larger

separations the force becomes repulsive, due to overlapping layers of adsorbed side chains

on each surface. The repulsion is seen to have a maximum at around h = 30 A (systems s2

and s3), decaying to zero at large separations (see inset). The corresponding interaction free

energies (See the Supporting Information for more details) are shown in Figure 4, where the

long ranged repulsive barrier, due to adsorbed side chains, is more clearly seen.

8

20 30 40 50 60 70 80 90 1000

10

20

30

10 20 30 40 50 60 70 80 90 100D / Å

-200

-150

-100

-50

0

50

βWp,

z(D)

s2 (Pc

7,4)

s3 (Pc

7,10)

s4 (Pb

10,30)s1 (MD)

Figure 4: Potentials of mean force Wp,z(D) for systems s1, s2, s3 and s4. The inset zooms inat longer range, revealing a substantial free energy barrier due to side chain excluded volumeeffects.

Block vs. comb copolymers

Quite different from systems s2 and s3 (branched copolymers) is system s4, containing the

block copolymer composed of one block with charged monomers attached to a block of neutral

monomers. From Figure 3a, we see that the bridging force in the interval [15 < h < 20] is

stronger in comparison with the comb copolymer systems, which is partly due to a higher

backbone linear charge density. Moreover, the charged block of the block copolymer can

enter the gap between the plates, without incorporating the neutral monomers, which is not

the case for the comb copolymers. The two effects combined, lead to the pronounced local

free energy minimum at around 15 A, see Figure 4. By the same reasoning, the repulsion

starting at around 13 A, is also stronger for system s4. The implication of this difference

will be further discussed in a later section.

One can also notice an oscillating force profile for system s4. This behavior is reminiscent

of the force profile obtained between two charged parallel flat surfaces in thermodynamic

equilibrium with an (infinite) bulk solution containing a polyelectrolyte solution with oppo-

sitely charged homopolymers.29 A further indication of that similarity, is the observation of

the same type of non-monotonic behavior of the mid-plane density of the charged monomers

with increasing D, see Figure 3b.

9

Comparisons with force measurements

Our finding of a long ranged repulsive barrier, corresponding to the overlap distance between

adsorbed sidechain layers, is in good agreement with recent experimental data by Flatt et

al.30 In that study the force between a single flat C-S-H crystal and a colloidal tip cov-

ered with C-S-H crystals, was studied by atomic force microscopy (AFM) measurements in

saturated calcium hydroxide solutions containing various copolymers.

After one has accepted that the measured force in such a geometry is equivalent to a free

energy (via the Derjaguin approximation31), a remarkable agreement with the simulated

repulsion range is found. Indeed, the measured value of 7.2 nm with polycarboxylates P c27,13,

see Table 2 in ref.,30 compares very well with the value of 7 nm obtained with our model

polycarboxylates P c7,10, which have the same grafting density and almost the same side chain

length, see Figure 4. However, given the size of our model platelets, the magnitude of the

forces cannot be directly compared. It should also be mentioned that at high copolymer

dosage, purely repulsive curves was measured by AFM, while a short ranged attraction was

always found in our simulations as discussed above, see Figure 3a. This apparent contra-

diction could be explained by i) kinetically trapped copolymers (implying a non-equilibrium

situation) in the gap between the surfaces in the AFM experiment ii) the usage of a low spring

constant for the AFM-tip (0.6 Nm−1), which may prevent the copolymers to be pushed out

in the surrounding bulk solution at short distances.

On the other hand, the short ranged attraction found in our simulations is fully compat-

ible with the recent observation (by small angle X-Ray scattering20) of the formation of a

gel network in diluted dispersions of C-S-H particles in aqueous comb copolymer solutions

when sufficiently aged, and with the well characterized problem of fluidity/workability loss

of admixed cement paste with time.32–34 The latter is a phenomenon that cannot be fully

explained by the reactivity of the system, i.e. by the gradual consumption of admixture

by the hydration products. Nevertheless, our result brings new light on the effect of comb

copolymers in freshly hydrated cement paste, since contrary to the common idea, we here

10

demonstrate that such polymers do not suppress the attractive interparticle forces at short

range. At least not under the condition of full equilibrium between the bulk and the gap

between the surfaces. In other words, the results presented here, suggest that the colloidal

stability observed in such systems is mostly kinetically driven. The consequence of this find-

ing will be further investigated in the next section, by means of simulations of many-platelets

systems

Dynamics and structure: multi-platelet systems

Radial distribution functions

We begin by analyzing the polymer-free systems (s5, s6 and s7) under saturated Ca(OH)2

conditions, see Table 1. The difference between these systems is the platelet surface charge

density. In Figure 5, the trend of going from a dispersed system (s5) to a completely

aggregated system (s7) is shown by i) center-center platelet radial distribution functions g(r),

main figure ii) single-platelet auto rotational correlation functions (upper rightmost inset)

and iii) simulation snapshots. We see that unless the surface charge density is extremely

high, the ion correlation forces alone are not sufficient to aggregate the system. Above a

threshold value (also dependent on the concentration of 2:1 salt), regularly spaced peaks

in g(r) appear (system s6), which is a signature of the formation of tactoids, in which

the platelets stack face to face in columns. For the lowest surface charge density, such

tactoids are absent, while for the highest surface charge, the system has collapsed into one

aggregate, comprising all platelets in the system (see color coded snapshots). Accordingly,

the rotational autocorrelation functions show the successively slowed down relaxation of the

systems with increasing surface charge densities. Note the non-monotonic behavior of the

rotational autocorrelation for system s6, indicating correlations of rotational modes between

different tactoids, which is a signature of liquid crystal formation.

The dispersive effect of copolymers in such a system is illustrated in Figure 6 where the

radial distribution functions (sampled during the last 15 ns of the simulations) are shown

11

0 1000 2000 3000t / ps

0

0,5

1

C(t

)

0 5 10r / nm

0,01

1

100

10000

g(r)

s5, σs = 175 mC/m2

s6, σs = 350 mC/m2

s7, σs = 640 mC/m2

Figure 5: Radial distribution functions g(r), with respect to platelet centers of masses in thebulk for polyelectrolyte free systems. Three surface densities are shown: σs = -175 mC/m2

(system s5), σs = -350 mC/m2 (system s6) and σs = -640 mC/m2 (system s7). For thetwo lowest surface charge densities, no excess divalent salt was added. Note the logarithmicscale on the y-axis. The upper right inset shows the corresponding platelet auto correlationfunctions. Color coded simulation snapshots (upper left) are also shown to guide the eye.Note that the snapshot corresponding to the highest surface charge density (green) includesall platelets in the system.

for systems s9, s10, and s11 in comparison with the polymer-free system s7. In accordance

with the free energy calculations in Figure 4, we see that adding copolymers gives rise to

a substantial stabilization of the system which otherwise forms large tactoids and finally

aggregates. Already for system s9 (red curve), which contains comb copolymers with short

side chains, a bulk region is found at large r, i.e., g(r) = 1. The presence of peaks up to

D ≈ 80 A shows, however, that relatively large tactoids with a inter-platelet distance equal

to around 7.5 A are still present. When the side chain length of the comb copolymer is

increased, i.e. system s10, a repulsive g(r) for the whole plotted r-range is observed (green

curve). In full agreement with the free energy calculations, replacing the comb copolymer

with a block copolymer with the same neutral block length, s11 (black curve), leads to

a somewhat more attractive system characterized by the formation of small independent

tactoids. This is manifested by a shift in the maximum of g(r) at larger r. Although

illustrative, the radial distribution functions plotted here should be treated with caution

12

40 60 80 100 120 140 160r / Å

0

1

2

3

4

5

g(r)

s7s9s10s11

Figure 6: Radial distribution functions g(r), with respect to platelet centers of masses inthe bulk for the systems s7, s9, s10 and s11 (sampled during the last 15 ns of the simulation).σs = -640 mC/m2.

since, as we will see below, the kinetics towards equilibrium of the dispersions is greatly

affected by the presence of copolymers.

Cluster and time correlation analysis

As shortly discussed in relation to the calculated free energy curves, displaying deep global

minima at short separations and a long range repulsion, one would expect a slowdown of

the platelet aggregation in the presence of copolymers and not a thermodynamic stable

suspension of individual platelets. Our simulations of many platelets dispersed in aqueous

solution of Ca(OH)2 and of various Ca(OH)2/copolymers provide such an evidence. The

slowdown in the rate of the tactoid formation in copolymer systems as compared to the

polymer-free system is illustrated in Figure 7 which gives the time evolution of the platelet

clustering in dispersions containing copolymers (systems s8, s9 and s10), in comparison

with the corresponding polymer free dispersion, i.e. system s7. Indeed, the rate of cluster

formation is found to be much slower for the copolymer systems and the maximum cluster

size is reduced as compared to the polymer free system. A platelet was defined to belong to

a specific cluster, if its center of mass was separated no more than a cut-off radius, rcut from

13

0 20 40 60 80 100t / ns

0

10

20

30

40

50N

umbe

r of

indi

vidu

al c

lust

ers s7 (r

cut = 1 nm)

s7 (rcut

= 4.5 nm)s8 (r

cut = 1 nm)

s9 (rcut

= 1 nm)s10 (r

cut = 1 nm)

a)

0 10 200

10

20

30

40

50

s7 (rcut

= 4.5 nm)

0 20 40 60 80 100 120 140t / ns

0

4

8

12

16

20

Num

ber

of p

late

lets

in la

rges

t clu

ster

s7 (rcut

= 1 nm)s8 (r

cut = 1 nm)

s9 (rcut

= 1 nm)s10 (r

cut = 1 nm)

b)

Figure 7: a) Number of observed platelet-clusters and b) size of the largest platelet-clusteras a function of the simulation time, for systems s7, s8, s9 and s10. The inset in figure b,shows the fast and complete aggregation of system s7 for a cluster cut-off, rcut = 4.5 nm.

any platelet center of mass of that cluster.

The most efficient stabilizing comb copolymer is P c7,10 (system s10), which presents the

longest neutral side chains. For example, at 50 ns, s10 consists of roughly 30 individual

platelet clusters (tactoids), see Figure 7a, with the biggest one consisting of only 4 platelets,

see Figure 7b. In comparison, lowering the side chain length for the same copolymer structure

(system s9) leads to a higher maximal cluster size and a somewhat lower value in the number

of clusters. This is again in line with the potential of mean forces, see inset in Figure 4,

which show a larger and more long ranged barrier for the comb copolymer with the longest

neutral side chain length, i.e. P c7,10. With a lower copolymer concentration (system s8), with

on average 2 copolymers/plate, the brush layers of copolymers on the platelet surfaces create

less steric sidechain repulsion and, consequently, a faster aggregation into bigger tactoids is

found. For example, at t = 50 ns, system s8 displays only 10 individual clusters, with up to

14 platelets per tactoid.

Note that the difference in clustering for system s7 with respect to the choice of rcut, comes

from the aggregation between tactoids stacked together and arranged in a perpendicular

fashion, as seen in the snapshot of Figure 5. On the contrary, for the copolymer-containing

systems, no marked difference between the two different choices of rcut was observed, meaning

14

that no tactoid-tactoid aggregation was observed during these simulations (not shown).

Another way to look at the stabilizing effect of the copolymer is to calculate time rota-

tional autocorrelation functions for the different system studied. This is what is presented

in Figure 8 for systems s7 - s11, sampled at large t. We see that the lowest relaxation

time is obtained for system s10, indicating faster rotational modes, which is coherent with

Figure 7, displaying the largest amount of individual clusters (containing few platelets per

tactoids) for system s10. The polymer free system (s7) is also shown in Figure 8. Due to the

0 200 400 600 800 1000t / ps

0

0,2

0,4

0,6

0,8

1

C(t

)

s7s8s9s10s11

Figure 8: Platelet rotational autocorrelation functions for systems s7-s11.

formation of a large aggregate, the rotational relaxation time is very slow, and we observe

an almost immobile cluster. We also see that loading the system with more polymers cre-

ates a faster relaxation time (compares system s8 and s9). System s11, containing the block

copolymer shows a relaxation time somewhat lower than for system s9, indicating fairly good

stabilization.

Discussion in regard to experiments

Unfortunately, a direct comparison between these simulation predictions and experiments is

difficult since similar experimental investigations does not exist. However, indirect macro-

scopic measurements of viscosity and dynamic rheology in cementitious systems,35–39 reveal

the same trends. That is, the elastic modulus and viscosity of a freshly hydrated cement

15

paste, when mixed with hydrophilic comb copolymer, are generally observed to increase

with a decreasing degree of polymerization of the polyhydrophilic neutral side chains and

when lowering the copolymer dosage. This is in full agreement with the simulated aggre-

gation rates (and free energy barriers). On the other hand, the slowdown of the simulated

aggregation rates may in part clarify the mechanisms involved in the dormant period, so

far not well understood, observed during the hydration of cement when mixed with comb

copolymer solutions. Indeed, very similarly to the behavior of the aggregation rates reported

here, this period of slow hydration kinetics is observed to be increasing as the stabilizing

effect of the polymer becomes more efficient, see e.g.35 Although the growth of the individ-

ual C-S-H particles is not accounted for in our simulations, which is known to be greatly

affected by the presence of comb copolymers,20,40 this assumption deserves, to our opinion,

further theoretical and experimental investigations since it is generally accepted that the

C-S-H nanoplatelet networks formed during cement hydration,41 is the result of the contin-

uous aggregation/germination-growth of new C-S-H platelets in contact with previous ones.

A slowdown of the C-S-H aggregation kinetics should therefore impart cement hydration

kinetics. Finally, one should also mention that for a similar copolymer to platelet size ratio,

the same aggregation mechanism was observed from SAXS measurements20 on diluted C-S-

H/copolymers dispersions, i.e., the growth of individual C-S-H tactoids, see Figure 17, case

(b), in ref.20

The snapshots in Figure 9 show the largest tactoid and the surrounding divalent ions

and copolymers formed in presence of comb polymers (system s9 at t = 50 ns) and block

copolymers (system s11 at t = 100 ns). We see that the comb copolymers are located on the

surfaces of the terminal platelets of the tactoids, creating comb copolymer end-caps, pre-

venting rapid aggregation of tactoids. The stepwise increment of the maximum tactoid size,

seen in Figure 7b, shows that the tactoids grow by merging smaller (copolymer end-capped)

tactoids, and not by single platelet addition. In the case of the block copolymer (system

s9), polymer intercalation between platelets in the tactoids is found. The intercalation is ex-

16

Figure 9: Simulation snapshots showing the largest tactoid, with its neighboring copolymersand divalent ions for a) system s9 (comb copolymers) at t = 100 ns , and b) system s11 (blockcopolymers) at t = 50 ns. The platelet sites are shown in dark grey; divalent ions in lightgreen; copolymer backbones in red and neutral sidechain monomers in light grey.

plained by a bridging mechanism characterized in Figure 4 by a significantly deep minimum

in the free energy between two plates at around h = 15 A. At t = 100 ns, roughly 75 copoly-

mers (spread over 13 separate intercalation sites) were found to be intercalated, thus, giving

on average 6 incorporated polymers per intercalation site. At the same time 25 individual

tactoids were found with a maximum tactoid size of 4. This means that roughly 50 % of the

tactoids (13/25) contained intercalated copolymers. Instead, no polymer intercalation was

observed for any of the systems containing comb copolymers as a result of a too large loss

in the internal entropy of the neutral side chains as compared to the gain in the backbone

bridging contribution to the energy. This prediction is in full agreement with experimen-

tal observations, see e.g. ref,42 and allows to rationalize the long standing question of why

comb copolymers never intercalate in like-charged C-S-H tactoids while they apparantly do

in oppositely charged platelet systems, like e.g. aluminate hydrates.43 Indeed, in the latter,

the bridging contribution to the energy should be much greater due to the direct polymer

adsorption to the bare charged surface of the platelets. What is more, the intercalation of

block copolymer into like-charged platelet tactoids predicted here is in full agreement with

the recent experimental finding of hybrid C-S-H mesocrystals in presence of similar block

copolymers.21 From our previous simulation studies on clays one can reasonably argue that

17

at higher simulated particle volume fractions, one should also found conditions for which

hybrid nanoplatet mesocrystals are formed. In any case, this work constitute the first brick

toward the understanding of the self-assembly of C-S-H nanoplatelet in like-charged block

copolymer solutions.

Conclusions

To summarize, we have studied the behavior of highly negatively charged nano-platelets in

a divalent (2+) salt solution, both in the absence and presence of different types of neutral-

anionic copolyelectrolytes. For high enough platelet surface charge density (in polymer-free

systems) the platelets stack face-to-face in tactoid clusters, which eventually aggregate, a

phenomenon driven by ion-ion correlations at the origin of the cohesive properties of C-S-H

particles in a cement slurry. When mixing copolymers into such systems, polymer adsorption

occurs on the platelets, via a calcium-mediated adsorption mechanism which dramatically

slows down the kinetics of platelet aggregation. This behavior was rationalized by the long

ranged free energy barrier, appearing in response to overlapping brushes of neutral side

chains, and by the short ranged ion-ion free energy attraction found in potential of mean

force calculations between two platelets in a parallel (face-to-face) configuration. The stabi-

lizing effect of the copolymer was found to be all the more important as the dosage and the

degree of polymerization of the neutral side chain of copolymers was large, in full agreement

with experiments. Tactoids are found to grow in a step-wise fashion,44 by merging smaller

(polymer end-capped) tactoids into bigger ones, keeping a polymer end-capped tactoid con-

figuration, a phenomenon also observed experimentally for sufficiently large copolymers. Un-

like comb-copolymers, block copolymers intercalate in between charged nanoplatelets of the

tactoids, due to a bridging mechanism. This last finding shed more light on the mecanisms

at play in the recent synthesis of a cementitious hybrid mesocrystal.

18

Acknowledgement

Financial support from the Wenner-Gren Foundation and the support of the CRI from

the university of burgundy (https://haydn2005.u-bourgogne.fr/CRI-CCUB), to access their

computer facilities are gratefully acknowledged. The authors also thank Prof. Bo Jonsson

(Lund university), Dr. A. Picker, Prof. H. Colfen (university of Konstanz) and Dr. L.

Nicoleau (BASF) for valuable discussions.

Supporting Information Available

Simulation details are included. This material is available free of charge via the Internet at

http://pubs.acs.org/.

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Graphical TOC Entry

Platelet separation

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Comb co-polymerBlock co-polymer

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