Evolution of microstructures of cement paste via continuous-based hydration model of non-spherical cement particles

Composites Part B 185 (2020) 107795

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Evolution of microstructures of cement paste via continuous-based hydration model of non-spherical cement particles Zhigang Zhu a, b, Wenxiang Xu a, *, Huisu Chen b, **, Zhijun Tan c a

College of Mechanics and Materials, Hohai University, Nanjing, 211100, China Jiangsu Key Laboratory of Construction Materials, School of Materials Science and Engineering, Southeast University, Nanjing, 211189, China c Department of Materials Science and Engineering, University of Sheffield, Sheffield, S1 3JD, United Kingdom b

A R T I C L E I N F O

A B S T R A C T

Keywords: Cement hydration Non-spherical cement grains Degree of hydration Porosity Simulation

As the soul of cement-based materials, hydration is a complex process involving many chemical reactions, which is one of the intrinsic driving forces for the evolution of the microstructure and macroscopic properties of cementitious composites. The predominant lines of previous researches on cement hydration models are domi­ nated by spherical particles. However, the actual shape of cement grains is non-spherical. Herein, a continuousbased hydration model of non-spherical cement particles (HYD-NSP) is developed, in which the hydration ki­ netics strongly depend on the morphological features of cement grains. Subsequently, the proposed HYD-NSP model is validated by experimental results. By applying the HYD-NSP model, we further investigate the evolu­ tion of microstructures of cement paste including icosahedron-shaped cement grains over time. Results reveal that the influence of the shape of cement particles on the degree of hydration and the porosity of cement paste is mainly reflected by their specific surface area.

1. Introduction Hydration is an extremely complex reaction process over time to form a miraculous microstructure in cement-based materials, and further, the macroscopic properties (such as mechanical [1–6] and transport properties [7–10]) of cementitious composites are dependent on the microstructure imposed by the evolution of materials due to hydration reactions at different time and space scales [11–14]. In the past century, the consumption of cement has consistently increased, causing the rising threat of global environmental impacts. Thus, better understanding of the hydration mechanism [15–19] can provide a so­ lution to incorporate higher amounts of supplementary cementitious materials [20–23] or to develop new clinkers [24–27]. Simulating the cement hydration and its microstructure by numerical modeling faces a significant challenge. Many researchers have proposed a number of computer-based models in the past decades. According to the simplifi­ cations of hydration models, two main approaches could be classified, viz. discretization approach and continuous approach. In the discretization approach, one of the most famous models is called CEMHYD3D which was launched by Garboczi and Bentz [28,29]. The features of this model are digital image basis, and the cement

particles are approximated as a digitized collection of voxels. After­ wards, Bullard [30] developed a novel model HydratiCA which considered hydration kinetics, and local probabilistic rules based on cellular automaton are applied on a regular computational lattice to simulate the hydration reaction. In imaging techniques, the irregular-shape particles can be used to produce three-dimensional digitized cement paste microstructures. Bullard et al. [31] studied the hydration of cement paste microstructures by using VCCTL (Virtual Cement and Concrete Testing Laboratory), they found that cement particle shape indeed affected early age behavior. Recently, Liu et al. [32] also used CEMHYD3D model to investigate the influence of cement particle shape on the hydration process by a central growth model. They found that the effect of particle shape on hydration of Portland cement and microstructure of cement paste is significant due to its surface area. Although the discretization models have achieved a lot of success, it should be noted that the discretization models have a noticeable resolution-dependent limitation and huge time-consuming during the hydration simulation. In the continuous approach, an early and influential review of cement hydration kinetics research published by Kondo and Ueda [33] in 1968. They proposed a mathematical model where the tricalcium

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (W. Xu), [email protected] (H. Chen). https://doi.org/10.1016/j.compositesb.2020.107795 Received 3 July 2019; Received in revised form 23 November 2019; Accepted 18 January 2020 Available online 22 January 2020 1359-8368/© 2020 Elsevier Ltd. All rights reserved.

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Composites Part B 185 (2020) 107795

silicate (C3S) particles are modelled as spheres. At the same time, Frohnsdorff et al. [34] proposed the idea of using computers to simulate the hydration process. This idea was implemented in cement hydration kinetics model proposed by Pommersheim and Clifton [35]. Jennings and Johnson [36] developped a computer program which was based on the study of Wittmann et al. [37] to simulate the development of microstructure during the hydration of C3S. In their model, the cement hydration is considered as nucleation and growth of spherical particles in three-dimensional space. This method has been widely used as the basis of cement hydration continuous-based model in subsequent research, such as HYMOSTRUC [38,39] and Integrated Particle Kinetics Model (IPKM) [40,41]. In these two models, the hydrating cement particles are represented as expanding spheres, the growing layers of new hydration products will interconnect to form the microstructure of hardening cement paste [42]. To evaluate the early-age properties of hardened concrete, such as the heat release and thermal conduction, pore structure formation and moisture equilibrium, Meakawa et al. [43] proposed a DuCOM model. Cement particles in DuCOM model are assumed to be monosize spheres. In 2009, Bishnoi and Scrivener [44] developed a new platform for modeling the hydration of cement called μic. This model is also limited to spherical particles in order to obtain high computational efficiency. However, it is evident that the actual shape of cement particles is irregular, because the grinding of cement clinker is a process of breaking larger particles into smaller ones [38]. It is a significant but unsolved issue on the non-spherical cement particles on the hydration microstructure by using a continuous-based approach. Therefore, this study aims to develop a continuous-based hydration model of non-spherical cement particles. Since the shape of cement particles is specific angular and faced constituents close to polyhedra, this paper adopts icosahedron to portray the actual shape of cement particles. The hydration kinetics and the construction of the hydration products of the non-spherical cement particles will be presented. And further, the evolutions of the microstructure (including the thickness of hydration products, degree of hydration and porosity of cement paste) in hydrated cement paste are investigated.

the formation of the microstructure of cement paste and the develop­ ment of the mechanical properties of cement-based materials [48]. 2.1. Definitions and general points For cement powder, the volume-based cumulative particle size dis­ tribution G(Deq) is commonly described using the Rosin-Rammler function [38,49,50], as given in Eq. (1). � � � (1) bDneq G Deq ¼ 1 exp where Deq is equivalent sphere diameter which is the diameter of a sphere with the same volume as the given cement particle, b and n are two constants of cement. The volume-based probability density function can be expressed by differentiating Eq. (1), as given in Eq. (2). � � � � dG Deq g Deq ¼ (2) ¼ b � n � Dneq 1 � exp bDneq dDeq The number of cement particles in the range of (Deq dDeq, Deq] is � � � n 1 bDneq dDeq � mc =ρc � g Deq dDeq mc � b � n � Deq � exp . . N Deq ¼ ¼ πD3eq 6 � 10 12 ρc � πD3eq 6 � 10 12 (3) where mc and ρc are the mass and density of cement, respectively. For a given water cement ratio wc, which is the ratio of weight of water to the weight of cement, i.e., wc ¼ mc/mw, the volume fraction of cement φc is derived by wc, as given in Eq. (4). φc ¼

mc =ρc mc =ρc 1 ¼ ¼ 1 þ wc � ρc =ρw mc =ρc þ mw =ρw mc =ρc þ mc � wc =ρw

(4)

where mw and ρw are the mass and density of water. In this study, ρc ¼ 3.15 g/cm3, ρw ¼ 1 g/cm3. To model the hydration of cement particles, we need to define a spatial cell as a calculation space unit. Similar to the concept of the cell in HYMOSTRUC model [38], the definition of the cell is a cubic space in which the largest cement particle has a diameter Deq, and further con­ sists of 1/N(Deq) times the original water volume and of 1/N(Deq) times the volume of all particles with diameter smaller than that of particle Deq, as shown in Fig. 2. The volume of the cell is given by Eq. (5).

2. Hydration model of non-spherical cement particles In cement chemistry, Portland cement comprises four main types of minerals, i.e., alite (C3S), belite (C2S), aluminate (C3A) and a ferrite phase (C4AF). Hydration of cement is essentially the process in which cement particles react with water to form hydration products. The hy­ dration products of C3S and C2S are C–S–H gel and calcium hydroxide (CH). The hydration products of C3A and C4AF are more complex because the chemical reactions of C3A and C4AF depend on the actual contents of gypsum and ettringite in the system. According to the density of hydration products, the dense C–S–H gel corresponds to so-called inner products. The loose C–S–H gel corresponds to outer products of cement hydration [45–47]. We assumed that the growth of new layers of hydration products are deposited on the surface of unhydrated cement particles, resulting in a homogeneous distribution of hydration products, as shown in Fig. 1. It is necessary to point out that the composition of cement is of great significance to the generation of hydration products,

� Vw þ V�Deq � I Deq ¼ N Deq

(5)

where Vw is the volume occupied by the initial water, i.e., Vw ¼ mw/ρw. V�Deq is the volume of cement particles not larger than the central particle Deq in the cell, as given in Eq. (6).

Fig. 1. Schematic diagram of the hydration of a cement particle.

Fig. 2. Schematic representation of the cell. 2

Z. Zhu et al.

V�Deq ¼

mc

ρc

Composites Part B 185 (2020) 107795

� G Deq

�

(6)

� � jAA’�� ¼

To measure the proportion of cement particles in the cell, as shown in Fig. 2, the ratio of the volume of cement particles in the cell to the total volume of the cell is defined as the cell density, which is given in Eq. (7). � � � � V�Deq N Deq mc ρc � G Deq � � � ξDeq ¼ ¼ mw =ρw þ mc ρc � G Deq I Deq (7) 1 � � ¼ 1 þ wc � ρc G Deq

δin;Deq ;j ¼ u1 δin;Deq ;j sinðθ=2Þ � sin ϖ

(9)

where θ is dihedral angle between two intersecting faces, ϖ is the angle between an edge and the vector from the center of the vertex (i.e. ϖ ¼ ∠PAA’, as shown in Fig. 3), u1 is a constant depending on the shape of cement particle. For an icosahedral particle, θ � 138.1897� , ϖ � 58.2825� and u1 � 1.2584. In this study, the size of icosahedral particle is determined by its equivalent sphere diameter Deq. For the sake of simplicity, the rela­ tionship between the circumscribed sphere diameter Dcr of the icosa­ hedral particle and the equivalent sphere diameter Deq is derived and given in Eq. (10). � � �� rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 π cosðπ=5Þ Dcr ¼ � tanðπ = 5Þ � tan arcsin � Deq ¼ u2 Deq 5 s cotðπ=3Þ sinðπ=3Þ

There is only one central particle with the largest diameter of Deq in the cell, as shown in Fig. 2. Surrounding the central particle, the remaining space can be defined as a shell. The volume of the shell is I (Deq) – v(Deq), where v(Deq) is the volume of the central particle with diameter Deq. To determine the amount of cement particles in the shell, the ratio of the volume of cement particles in the shell to the total vol­ ume of the shell is defined as shell density, as given in Eq. (8). � � v Deq ξD � I Deq � � ξDeq ;sh ¼ eq (8) v Deq I Deq

(10) where s is the sphericity of icosahedral particle, u2 is a constant. Herein, s ¼ 0.939 and u2 ¼ 1.1821. On the particle level, the degree of hydration of an individual icosahedral particle is described by the fraction of cement that has reacted. It can be calculated based upon the thickness of the hydrated layer, δin;Deq ;j , as given in Eq. (11).

In this section, a theoretical model, called HYD-NSP, the abbrevia­ tion for continuous-based HYDration model of Non-Spherical cement Particles, is proposed. The hydration kinetics of an individual cement particle is derived via its morphological features. For multi-sized cement particles, the interaction between the outer hydration products layer is considered for adjacent particles in the reaction process. The penetra­ tion rate of cement hydration depending on the rate-controlling mech­ anism will be obtained in the following.

�

π D3eq αDeq ;j ¼

π Dcr

6

2jAA’j u2

6

�3

πD3eq 6 � u2 Deq

¼1

2.2. Kinetics of an individual cement particle

(11) 2u1 δin;Deq ;j u2 Deq

�3 ¼1

� 1

2

u1 δin;Deq ;j u2 Deq

�3

Similarly, if we know the degree of hydration of the single cement particle, then the penetration thickness of the inner product can be determined by Eq. (12).

In the view of the grinding process of cement clinker, the morphology of cement particle should be approximated as a polyhedral structure. On the other hand, the surface of icosahedron is composed of twenty of the simplest congruent triangles. For these reasons, icosahedron-shaped cement is selected as an example to describe the hydration of non-spherical particles. For an individual icosahedral cement particle, the volume of cement particle involved in the reaction depends on the penetration depth, δin;Deq ;j , i.e., the thickness of the inner products, which means the distance from the original suface of the particle to the new surface of unhydrated core, as shown in Fig. 3. Ac­ cording to this definition, the distance between two vertices from the original particle to the unhydrated core, i.e., |AA’| in Fig. 3, can be deduced, as given in Eq. (9).

δin;Deq ;j ¼

Deq � � 1 2

αDeq ;j

1

�1=3 �

(12)

On the overall level, the degree of hydration of all cement particles at time tj is given in Eq. (13).

αj ¼

1 G Deq

� max

DX eq max x¼Deq

(13)

αx;j � gðxÞ

min

where Deq_min and Deq_max represent the minimum and maximum equivalent spherical diameter of particles, respectively. The volume of the outer products vou;Deq ;j that corresponds the degree of hydration can be obtained as follows vou;Deq ;j ¼ ðυ

1Þ � αDeq ;j �

πD3eq

(14)

6

where υ is the volume expansion ratio of hydration products and reacted cement at the curing temperature T (Celsius degree) [51], as given in Eq. (15). The effect of temperature-induced morphological and structural changes on reaction rate has been reviewed by van Breugel [38].

υðTÞ ¼ 2:2 � e

2:8�10

(15)

5 �T 2

And further, the thickness of the outer products is δou;Deq ;j ¼

� � �3 �1=3 vou;Deq ;j Deq þ 4π =3 2

Deq 2

(16)

2.3. Spatial model of non-spherical cement particles

Fig. 3. Schematic diagram of the thickness of the inner product for an icosa­ hedral cement particle.

With progress of the hydration process, many cement particles may 3

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Composites Part B 185 (2020) 107795

become embedded in the hydration products. This will affect the hy­ dration process and the morphology of hydration products. In reality the outer shell is partly filled with embedded cement particles, as shown in Fig. 4. The volume of cement in the outer shell with thickness δou;Deq ;j can be determined by multiplying the shell volume vou;Deq ;j with the shell density according to Eq. (7), as given in Eq. (17).

where ki is a constant, depending on the rate-controlling mechanism, cement composition and the degree of hydration, i ¼ 0 and 1 represent phase boundary reaction (i.e. hydration product thickness smaller than transition thickness δtr. In this stage, k0 is assumed as an average value of reaction rate, μm/h.) and diffusion controlled reaction [53] (i.e. hy­ dration product thickness bigger than transition thickness, in this stage, k1 ¼ k0(δtr)c, the default value of c is 2 in this model based on the cali­ bration result [52]), respectively. β1 is a constant which is determined from suitable tests. δDeq ;j is total thickness of product layer of particle with diameter Deq at the end of time step tj. In HYMOSTRUC model, van Breugel [38] pointed out that the basic rate factor k0 turned out to correlate directly with the C3S content of cement: k0 ¼ 0.02 þ 6.6 � 10 6 � [C3S%]2, and the transition thickness only a slight correlation with the C2S content: δtr ¼ 0.02 � [C2S%] þ 4. Herein, C3S% and C2S% are the percentage of clinker minerals in cement mass. The curing temperature not only affects the rate of the reaction, but also the morphology and structure of the hydration products. The effect of temperature on the rate of chemical and physical reactions has been commented in HYMOSTRUC model [38]. A temperature function F1, as given in Eq. (21), is adopted to consider the effect of the curing tem­ perature on the kinetics by means of the activation energy. The tem­ perature T ¼ 20 � C is defined as a reference temperature, i.e. F1(T20) ¼ 1.

(17)

vem;Deq ;j ¼ vou;Deq ;j � ξDeq ;sh

Meanwhile, those embedded particles in the outer products layer also undergo hydration, which indirectly increases the depth of the outer products layer. The increased depth of the outer products layer in turn causes another extra embedded cement. This process at time tj finally reaches an equilibrium. Next, the key point is to obtain the final front of the outer products layer at time tj. To solve this issue, the overlapping volume between hydration products layers of two particles will be redistributed. Based on the shell density mentioned above, the volume of the final outer prod­ ucts of particle v’ou;Deq ;j can be derived by the volume of all embedded particles and the expanded volume of all hydrated cement particles in this outer layer [52], as shown in Fig. 4, which can be expressed by Eq. (18). � � v’ou;Deq ;j ¼ vou;Deq ;j þ v’ou;Deq ;j � ξDeq ;sh � α
AEðT20 Þ

F1 ðTÞ ¼ eR�ð273þ20Þ � e

where α
α
1 ;j

¼

1 � G Deq 1

Deq

X1

x¼Deq

αx;j � gðxÞ

π D3

1

(22)

where R is gas constant (8.31 � 10 3 kJ/mol⋅K), AE is the activation energy of cement paste at temperature T which is affected by the C3S content and the degree of hydration. The activation energy can be approximately expressed by Eq. (23) [38]. � 33:5 kJ=mol ; T > 20� C AEðTÞ ¼ (23) 33:5 þ ð20 TÞ kJ=mol; T � 20� C

(19)

min

For the morphology and structure related component of the tem­ perature effect, to be accounted for using the temperature function F2, as described in HYMOSTRUC model. It is assumed that in the diffusion stage of a hydration process temperature-induced densification of the diffusion layer allows for a reduction of the rate of penetration [38]. � � � �β2 F2 T ¼ υðTÞ= (24) υðT20 Þ

There is only one unknown in Eq. (18), the function can be solved with Eq. (20). v’ou;Deq ;j ¼

AEðTÞ R�ð273þTÞ

αDeq ;j � 6eq � ðυ 1Þ � � ξDeq ;sh � α
(20)

For an incremental increase of the penetration depth Δδin;Deq ;jþ1 of tj, the basic pene­ particle Deq during a time increment Δtjþ1 ¼ tjþ1 tration rate of cement particle hydration is employed based on HYMOSTRUC model. � � � Δδin;Deq ;jþ1 ki � F1 ðTÞ � ½F2 ðTÞ�i � Ω 1 Deq ; αDeq ;j � Ω 2 αj � Ω 3 αj ¼ � �i �β1 Δtjþ1 δDeq ;j

where T is the mean temperature in the hydration domain, β2 is a con­ stant to be derived from adequate experimental data. The majority of test results could be approximated satisfactorily for β2 ¼ 2 [38]. There are many embedded and still incompletely hydrated particles in the outer shell of the central particle. The extra expansion of the outer shell will cause a decrease of the diffusion rate. This implies that water is

(21)

Fig. 4. Schematic diagram of the embedded particles and the final front of outer product layer. 4

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Composites Part B 185 (2020) 107795

more difficult to enter the unhydrated core of the central particle. For this phenomenon, a reduction factor Ω1(Deq, αDeq ;j ) is introduced to consider water withdrawal effect due to the hydration of embedded particles in the outer products layer of central particle Deq. �

Ω 1 Deq ; αDeq ;j ¼

ΔwDeq ;j ΔwDeq ;j þ Δwem;Deq ;j

3. Simulation of the hydration process and microstructure of cement paste In the simulation of hydration process, the shape of cement is rep­ resented as an icosahedron. This section describes a methodology of the construction of the inner products, outer products and unhydrated cores for icosahedral cement particles. Then, the degree of hydration of a simulated cement paste is calculated and compared with experimental results to verify the reliability of the proposed HYD-NSP model.

(25)

where ΔwDeq ;j is the amount of water needed for an increase of the degree of hydration ΔαDeq ;j of particle Deq during Δtj, as given in Eq. (26). ΔwDeq ;j ¼ 0:4 � ΔαDeq ;j � vDeq �

ρc ρw

3.1. Hydration simulation of a single icosahedral cement particle

(26)

In this study, the hydration process is considered to be outward expansion to form outer products and inward shrinkage to form inner products with a constant thickness layer, as shown in Fig. 5, which is similar to Minkowski addition and difference in mathematical morphology [54,55], respectively. The construction method of hydra­ tion products is described as follows. For an icosahedral cement particle, the original shape is composed of vertices, edges and faces [56]. The inner hydration products are the region between the original surface of cement and the unhydrated core, i.e., the volume of cement as the reactant. Geometrically, the shape of unhydrated core can be determined by Eq. (9), which remains in a similar shape to the original shape for convex polyhedral particle. The inner products layer can be considered as a sphere (radius is the thick­ ness of the inner product) moving on the surface of cement particle. Fig. 5 shows the construction processes of the outer products, inner products and unhydrated core. The thickness of the inner products δin;Deq ;j is controlled by the penetration rate of cement particles, as given in Eq. (21). Next, we want to know the thickness of the outer products δou;Deq ;j around the icosahedral particle. The outer hydration products are thought to be uniformly dispersed around the original particle surface due to the expanded volume of the hydration products. It can be rep­ resented by a structure with an uniform-thickness shell around cement particle, as shown in Fig. 5. In our previous study, this shell is con­ structed by spherical arcs at vertices, and cylinder arcs at edges, and translational planes at faces [57]. The volume of the uniform-thickness shell can be obtained according to Steiner formula [58,59], as given in Eq. (34). . 2 3 vou;Deq ;j ¼ Sδou;Deq ;j þ 2π Bδou;D 3 (34) þ 4πδou;D eq ;j eq ;j

And Δwem;Deq ;j in Eq. (25) is the amount of water that was required for further hydration of the embedded particles in time step Δtj, it can be obtained by Eq. (27). Δwem;Deq ;j ¼ 0:4 � vem;Deq ;j �

ρc � α�Deq ρw

1;j

α�Deq

� 1;j 1

(27)

During the hydration process, the water in the capillary pores gradually becomes difficult to mobilize for further hydration. To ac­ count for the assumed phenomenon, a reduction factor Ω2(αj) is pro­ posed by van Breugel [38], as given in Eq. (28). � Dwat;αj Ω 2 αj ¼ Dpor;αj

D0 Dpor;αj � D0 Dwat;αj

(28)

where D0 is the size of the smallest capillary pore (usually set to 0.002

μm), Dpor;αj and Dwat;αj are the maximum pore size and the largest pore

completely filled with water, which are given in Eq. (29) and Eq. (30), respectively. �� � � � Vpor αj Dpor;αj ¼ Dmax αj ¼ D0 exp (29) a where a is a constant which depends on the type and fineness of cement and w/c ratio. �� � Vfr αj Dwat;αj ¼ D0 exp (30) a The volume of pores at a degree of hydration αj is given in Eq. (31) according to van Breugel’s work [38]. � Vpor αj ¼

ρc � wc ρw þ ρc � wc

� 0:3375 � αj � V

(31)

where S and B are the surface area and the mean calliper diameter of the original cement particle [57], respectively. For an icosahedral particle, S and B are given in Eq. (35).

where V is a volumetric unit. The volume of free capillary water Vfr ðαj Þ means removing adsorbed water from the total capillary water, it can be obtained from Ref. [38]. Z Dpor;α � � ρc Γ � D Γ2 Vfr αj ¼ � wc 0:4 � αj � V 4aV � dD ρw þ ρc � wc D3 D0

(32)

where Γ is the thickness of the adsorption layer. It is usually an integer multiple of the thickness of a mono-molecular layer with a thickness of about 3 Å. It should be noted that the rate of hydration depends not only on the availability of water immediately at the pore wall surfaces, but also on the total amount of water in the pore system. However, the amount of water involved in the reaction with cement decreases with the hydration process progresses. In Eq. (21), a reduction factor Ω3(αj) is employed to measure the reduction of the amount of water in the hydrating mass (overall effect), as given in Eq. (33). � wc Ω 3 αj ¼

0:4 � αj wc

(33) Fig. 5. Schematic diagram of the hydration of an icosahedral cement particle. 5

Z. Zhu et al.

pffiffi �! pffiffi 5 3 þ 5 S¼5 3 2π

Composites Part B 185 (2020) 107795

D2eq � 3:3445D2eq

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi �! �pffiffi u u 1þ2 5 5 3þ 5 pffiffi � B ¼ acost � π 2π 3 2 þ 3 5 10 15

selected, the calculated four main constituents are shown in Table 1. The degree of hydration in the experiment was obtained by measuring the hydration heat release within 14 days using an isothermal calorimeter. The hydration process and the microstructural develop­ ment of a 100 � 100 � 100 μm3 cubic sample of cement paste is simulated with w/c ratio 0.4 at the curing temperature 20 � C. The shape of cement is assumed to be spherical particle, and the minimum and maximum diameter of cement particles are 2 μm and 45 μm, respec­ tively. The actual particle size distribution and particle number are shown in Fig. 7. According to the aforementioned input parameters, the degrees of hydration of spherical and icosahedral cement particles simulated by the HYD-NSP model are shown in Fig. 8(a). It can be found from Fig. 8(a) that the degree of hydration of icosahedral cement particles is slightly higher than that of spherical, but overall, both simulation results display a good consistency with the experimental results, it indicates that the proposed HYD-NSP model of non-spherical particles is reliability. Next, the problem is the representative volume element (RVE) of the specimen size. We examined three sizes of specimen: 100 � 100 � 100 μm3, 1000 � 1000 � 1000 μm3 and 10000 � 10000 � 10000 μm3. Taking spherical cement grains as an example, w/c ratio is 0.4, particle size distribution obeys the Rosin-Rammler distribution (b ¼ 0.0517 and n ¼ 1.0145 from van Breugel [38]), the minimum and maximum diameter are 1 and 100 μm, respectively. The phase transition thickness δtr ¼ 2.6 μm and the rate constant k0 ¼ 0.03 μm/h. The degrees of hy­ dration of three simulated cement paste are shown in Fig. 8(b). It is evident from Fig. 8(b) that there are no much difference among them in the degree of hydration. Thus, the specimen size 100 � 100 � 100 μm3 can be considered to meet the requirements of RVE.

2=3

(35)

1=3

Deq � 1:0826Deq

Meanwhile, according to the thickness of the inner products δin;Deq ;j , the degree of hydration αDeq ;j and the volume of the outer products around icosahedral particle are calculated by Eq. (11) and Eq. (14), respectively. The thickness of the outer products δou;Deq ;j is determined by combining Eq. (14) and Eq. (34). 3.2. Hydration simulation of multi-sized icosahedral cement particles Before simulating the hydration process of multi-sized cement par­ ticles, it is necessary to generate a cement particle packing structure according to particle size distribution function in Eq. (1). At the initial stage, cement particles are randomly distributed in a cubic space com­ bined with the random sequential addition procedure [60], and there are no overlaps between them. For icosahedral particles, the separation axis algorithm [61–63] is employed to accomplish the overlap detection. When cement particles react with water, the hydration products are generated in the vicinity of cement particles in the early stage of hy­ dration. At this time, phase boundary reaction dominates the hydration process. The thickness of the hydration products will increase with the hydration of cement. Afterwards, the outer products around a particle will encapsulate adjacent smaller particles in the process of volume expansion. If the embedded small cement particles are not fully hydrated yet, they will continue to consume water and further reaction. When the thickness of hydration products reaches to transition thickness, the re­ action will be controlled by diffusion. In consequence, the rate of reac­ tion of the bigger particles will further decrease, and the final outer layer can be calculated by Eq. (20). Fig. 6 visualizes a microstructure of cement paste with icosahedral cement particles under the conditions: the container size is 100 � 100 � 100 μm3 in a periodic boundary conditions, w/c ¼ 0.4, b ¼ 0.0517 and n ¼ 1.0145 [38], δtr ¼ 2.6 μm and the basic rate factor k0 ¼ 0.03 μm/h, the minimum and maximum di­ ameters of cement particles are 1 and 41.9 μm, respectively.

4. Development of microstructure in hydrating cement paste In this section, the evolution of the microstructure (including the thickness of hydration products, degree of hydration and porosity of cement paste) in hydrating cement paste is investigated. Spherical and icosahedral particles are evaluated by the following input parameters: Table 1 Constituents of cement [64].

3.3. Benchmark test To verify the reliability of the HYD-NSP model, the experimental results from Ye’s work [64,65] are collected to compare with the HYD-NSP model results. In their work, CEM I 32.5R Portland cement is

Phases

Weight (%)

C3S C2S C3A C4AF

63 13 8 9

Fig. 6. Visualization of microstructure of cement paste with icosahedron-shaped particles. 6

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Fig. 7. Particle size distribution and the number of cement particles (a) Particle size distribution (b) Number of particles.

Fig. 8. Degree of hydration of cement paste as a function of hydration time (a) Simulation and experimental results on the degree of hydration (b) Effect of the specimen size on the degree of hydration.

cement with two constants: b ¼ 0.0517 and n ¼ 1.0145 [38], Deq_min ¼ 1 μm and Deq_max ¼ 100 μm, water cement ratio w/c ¼ 0.4, 0.5 and 0.6, curing temperature 20 � C, the transition thickness δtr ¼ 2.6 μm, and the basic penetration rate k0 ¼ 0.03 μm/h. According to this assumption, in a 100 � 100 � 100 μm3 cubic container, the cumulative probability and the total number of cement particles are shown in Fig. 9 for different w/c ratios. It can be seen obviously from Fig. 9 that the decrease of w/c ratio will encounter an increase of the number of cement particles. 4.1. Thickness of hydration products During the hydration process, the adjacent hydration products will contact or overlap to build up a three-dimensional network structure around the unhydrated cores. The formation and development of the microstructure of cement paste are closely related to the rate-controlling mechanisms. According to the size distribution of cement in Fig. 9, six values of diameter (Deq ¼ 1.00, 2.03, 4.02, 7.95, 16.19 and 23.90 μm) are taken as an example, the evolution of the inner and outer products thickness of icosahedral cement particles over time are shown in Fig. 10. It appears from Fig. 10 that the hydration of cement exhibits two stages with marked differences in the rate of reaction. In the early stage of hydration, the duration and rate-controlling mechanism is the phase boundary reaction. The thickness of hydration products will rapidly increase with time. And later, hydration products layer coated on the

Fig. 9. Particle size distribution in the simulation of cement particles in a 100 � 100 � 100 μm3 container.

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Fig. 10. Evolution of the thickness of hydration products for different particle diameters (a) Thickness of the inner products (b) Thickness of the outer products.

surface around a hydrating cement particle will inhibit their further reaction. When the thickness of hydration products layer reaches to the transition thickness, the reaction will be controlled by the diffusion mechanism. It also can be found from Fig. 10(a) that the cement parti­ cles with diameter Deq ¼ 1.00, 2.03 and 4.02 μm will be hydrated completely at about 16, 32 and 67 h, respectively. The thickness of the inner products after complete hydration reaches their theoretical value, i.e., u2/u1*Deq � 0.94*Deq which can be derived from Eq. (11). Simul­ taneously, the corresponding thickness of the expanded outer products can be found in Fig. 10(b). Besides, the cement particles with diameter Deq ¼ 7.95, 16.19 and 23.90 μm in Fig. 10(a)–(b) show that the rate of penetration turns out to decrease with increasing particle diameter. The reason is that the amount of embedded cement will increase with the increase of the shell thickness. As a result, the products layer hampers the hydration of particles due to the limited access of water for the hydrating particles.

4.2. Degree of hydration The overall progress of the hydration reactions is described by the degree of hydration. The degrees of hydration of the six values of diameter (Deq ¼ 1.00, 2.03, 4.02, 7.95, 16.19 and 23.90 μm) of icosa­ hedral cement particles are shown in Fig. 11. It can be found from Fig. 11 (a) that the overall tendency of the degree of hydration reduces with the increase of particle diameter by the viewpoint of linear-based curves. Much more variation details at the early stage of hydration are shown in Fig. 11(b) by logarithm-based time. It indicates that small particles quickly reach complete hydration, while the rate of reaction of large particles becomes slow with time. Compared to spherical cement particles, the degrees of hydration of spherical and icosahedral particles are investigated using the proposed HYD-NSP model. It can be found from Fig. 12(a) that the degrees of hydration of two types of cement shape increase with the increase of the hydration time, and the degree of hydration of cement paste with icosahedral cement particles is slightly higher than that with spherical particles. We believe that the reason for this phenomenon is due to their

Fig. 11. Degree of hydration of an individual icosahedral cement particle in the HYD-NSP model (a) Hydration time with linear-based (b) Hydration time with logarithm-based. 8

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Fig. 12. Influences of cement shape and water cement ratio on the degree of hydration (a) Effect of cement shape on the degree of hydration (b) Effect of w/c ratio on the degree of hydration.

specific surface area (SSA), which is defined as the surface area per unit weight, i.e., the total surface area of all particles with diameters Deq_min � D � Deq_max, as given in Eq. (36). The specific surface area is a function of both the particle shape (s, sphericity) and the particle size distribu­ tion. It is mainly the fineness of cement that forms a major percentage of the specific surface area. DX eq max

SSA ¼ D¼Deq

min

πD2 NðDÞ

(36)

s � mc

Herein, the specific surface area of spherical and icosahedral cement particles are calculated to be 346 and 368 m2/kg, respectively. Icosa­ hedral cement particles possess the larger specific surface area, which makes it easier to contact with water and further rapid reaction. Therefore, we can draw a conclusion that the degree of hydration of cement increases with the decrease of the sphericity (the increase of the SSA) of cement particles, as shown in Fig. 12(a). By the way, Bullard et al. [31,66] studied the real-shaped cement particles with spheres on the degree of hydration and concluded that cement particle shape mainly affected early age behavior but mattered less at later time stages. Next, the influence of water cement ratio on the degree of hydration is examed. Three values of water cement ratios are taken into account, i. e., w/c ¼ 0.4, 0.5 and 0.6. Fig. 12(b) shows the simulation samples with different w/c ratios that there is no obvious effect on the degree of hy­ dration at the early stage (hydration time within 48 h) of the hydration among them. But later, the degree of hydration increases gradually with the increase of water cement ratio. This is because high w/c ratio can provide more water to further participate in the process of hydration.

Fig. 13. Comparison of the porosity of cement paste with spherical and icosahedral cement particles.

on the porosity is analyzed. For icosahedral and spherical cement, the porosity occupied by water is about 0.557 before hydration (at initial state), as shown in Fig. 13, this is because the volume fraction of cement φc is 0.443 when w/c ¼ 0.4. It also can be seen from Fig. 13 that the porosity of cement paste decreases with the increase of hydration time, which is contrary to the trend of the degree of hydration versus hydra­ tion time in Fig. 11. The reason is the volume of large pores significantly decreases with reaction proceeding, whereas the proportional amount of the small pores dramatically increases. In addition, the porosity of cement paste with icosahedral cement particles is slightly smaller than that with spherical particles, this is attributed to the identical specific surface area between them as mentioned above. Besides, water cement ratio is also the main factor affecting porosity. Fig. 14 shows the evolution of porosity at various of w/c ratios. It can be found that the porosity reduces with the increase of hydration time and degree of hydration, and high w/c ratio will increase the porosity of cement paste. The strongest negative correlation is found between porosity and the degree of hydration of cement paste. As the hydration process progresses, hydration products contact with each other, forming a three-dimensional network structure around the unhydrated cores. This results in a decrease in the total porosity of

4.3. Porosity The porosity is an important parameter to measure the macroscopic properties of porous media [67,68]. In the hardening process of cement paste, there are many pores appeared in the cement paste due to chemical shrinkage, which is because the volume of hydration products is smaller than that of the original reacted cement and the consumed water. Actually, there are three different types of water in the cement paste, viz, chemically bound water, physically bound water and capil­ lary water. The porosity can be calculated based on the degree of hy­ dration, as given in Eq. (37). φpore ¼

ρc � wc ρw þ ρc � wc

0:3375 � αj

�

(37)

Next, the influence of cement particle shape and water cement ratio 9

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Fig. 14. The development of the porosity at various of w/c ratios (a) Hydration time (b) Degree of hydration.

cement paste. A general trend is that the volume of the large pores de­ creases, while the proportional amount of the small pores dramatically increases [38]. We should bear in mind that the macroscopic behavior of cement paste is closely linked to its structural formation. The porosity of cement paste significantly affects mechanical, durability, and dimen­ sional stability behavior of hardened concrete [69]. In addition to porosity, curing conditions, chemical composition and the fineness of cement, water cement ratios and admixtures are important factors in determining the macroscopic behavior of cement-based materials.

significantly different, it will directly reflect in the degree of hydration and porosity. Besides, in the early stage, there is no obvious effect on the degree of hydration for different w/c ratios, but in the later stage, the degree of hydration increases gradually with the increase of w/c ratios. Results also indicate that there is the strongest negative correlation between porosity and the degree of hydration of cement paste. The purpose of this study is an attempt to explore a theoretical framework for non-spherical cement particles in hydration model. We expect that it will contribute to understand the influence of cement particle shapes on cement hydration and the development of the microstructure of cement paste, and further on the macroscopic prop­ erties of cementitious composites.

5. Conclusions In this work, a continuous-based HYDration model of Non-Spherical cement Particles (HYD-NSP) is proposed to address the problem that most of cement hydration models are dominated by spherical particles. In the proposed HYD-NSP model, the hydration kinetics for an individ­ ual cement particle and for multi-sized cement particles are derived to simulate the reaction process of non-spherical cement particles. The penetration rate of cement hydration is controlled by two types of ratecontrolling mechanisms, i.e., phase boundary reaction and diffusioncontrolled reaction, which is depended on the thickness of hydration products layer. With the hydration reaction proceeding, the outer products layer will encapsulate other adjacent particles. The thickness of the final outer layer is deduced by considering all embedded particles and the expanded layer of all hydrated cement particles. Afterwards, taking an icosahedral cement particle into account, the constructions of the inner products, outer products and unhydrated cores are described in hydration simulation. To verify the reliability of the HYD-NSP model, the degrees of hydration of cement paste with spherical and icosahedral cement particles are obtained by the proposed HYD-NSP model. Both simulation results are in good agreement with the experimental results. Then, the evolutions of the thickness of hydration products layers and the degree of hydration of simulated cement paste over time are inves­ tigated for icosahedral cement particles with different diameters. Results show that the penetration rate and the degree of hydration turn out to decrease with increasing particle size, because it is more difficult for water to enter and further react with the anhydrous cores. Finally, the influences of cement particle shape and water cement ratio on the de­ gree of hydration and the porosity are analyzed. The results show that the degree of hydration of cement paste with icosahedral cement par­ ticles is slightly higher than that with spherical particles. On the con­ trary, the porosity of icosahedral particles is slightly smaller than that with spherical particles. This is because their specific surface areas are very similar, corresponding to 368 and 346 m2/kg, respectively. It can be inferred that if the specific surface area of cement particle shape is

Declarations of competing interest None. CRediT authorship contribution statement Zhigang Zhu: Conceptualization, Methodology, Software, Writing original draft, Visualization. Wenxiang Xu: Supervision, Writing - re­ view & editing, Resources. Huisu Chen: Conceptualization, Writing review & editing. Zhijun Tan: Methodology, Investigation. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2018B17214), the National Natural Science Foundation of China (Grant Nos. 11802084, 11772120, 11402076, 51878152 and 51461135001), the Project funded by China Postdoctoral Science Foundation (Grant No. 2019M651668), the Open Research Fund of Jiangsu Key Laboratory of Construction Materials (Grant No. CM2018-05), the Natural Science Foundation of Jiangsu Province (Grant No. BK20170096), Jiangsu Planned Projects for Post­ doctoral Research Funds (Grant No. 2018K052C), and the Ministry of Science and Technology of China “973 Project” (No. 2015CB655102). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.compositesb.2020.107795.

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