Introducing a New Cement Hydration and Microstructure Model

This paper presents a new cement hydration model (HYDCEM) to predict the microstructure evolution of hydrating tricalcium silicate (C3S). The model is written in MATLAB and employs the continuum approach and integrated particle kinetics relationships to show the change in C3S and the growth of Calcium Silicate Hydrate (C-S-H) and Calcium Hydroxide (CH) in the pore space over time. Cement hydration is a highly complex process. While hydration models should never completely remove experimental analysis, they are an aid to better understand cement hydration and microstructure development by providing a method to analyse a large number of pastes with different cementitious make-ups in a relatively short time. This model uses spherical particles to represent the C3S with customizable input files such as cumulative weight distributions (CWD), to determine the particle size distributions, PSD), w/c ratio, C3S, C-S-H and CH phase densities, kinetics rates, stiochiometries and enthalpy values. The current study presents simulated microstructures and demonstrates the versatility of the model, while still in the development stage, to simulate cement hydration and microstructure development over 100 days. With further development, it can become a flexible tool for both academia and industry that can easily incorporate the inclusion of supplementary cementitious materials etc. *Corresponding author: Holmes N, School of Civil and Structural Engineering, Dublin Institute of Technology, Dublin 1, Ireland, Tel: +353 1 402 4039; E-mail: niall.holmes@dit.ie Received July 22, 2018; Accepted August 27, 2018; Published September 07, 2018 Citation: Holmes N (2018) Introducing a New Cement Hydration and Microstructure Model. J Material Sci Eng 7: 480. doi: 10.4172/2169-0022.1000480 Copyright: © 2018 Holmes N. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.


INTRODUCTION
Cement hydration and microstructure development is a complex process. However, the advances in computing power and range of programmable software in recent years has made the modelling of cement hydration achievable. While computer modelling should never completely replace experimental analysis, it does provide valuable insights into the process particularly with the increased use of supplementary cementitious materials. Model outputs can now be directly compared with experimental analysis provided accurate input data is used throughout. A complete and accurate hydration simulation would enable materials engineers not only to predict the performance of concrete in the field, but also to troubleshoot performance problems and even to help design new cementitious materials. Despite significant effort and progress, the ability to perform such a complete simulation has not yet been developed, mainly because cement hydration is one of the more complex phenomena in engineering science.
This paper presents a new model written in MATLAB® to undertake cement hydration and microstructure simulations over any time-period with all input parameters changeable by the user. Changeable parameters include cumulative weight distributions (CWD), cement properties (including w/c ratio, cement phase density and enthalpy) and hydration and microstructure kinetic rates. The model employs the vector approach, which permits a large range of cement phase diameters to be modelled computationally effectively while 1 Cement chemistry notation: C=CaO, S=SiO2, A=Al2O3, F=Fe2O3, H=H2O taking advantage of the inbuilt mathematical functions within MATLAB. The vector approach has been used in several previous hydration models [1][2][3][4][5][6][7][8].
This paper presents the initial design of the model for the hydration of tricalcium silicate (C3S) 1 which represents the greatest contribution of the four cement phases towards strength development. The approach used to develop the model are presented along with results from a number of simulations using different particle size distributions (PSD) as well as the versatility and flexibility of the model for the user.

CURRENT CEMENT HYDRATION AND MICROSTRUCTURAL MODELS EMPLOYING THE VECTOR APPROACH
The development of cement hydration and microstructure models have used either the discretization or the vector approach. [9] gave a comprehensive review of the most wellknown models employing the vector approach and brief overview of these models is presented below.

Jennings & Johnson microstructure simulation model
This model used spherical particles within a cubic volume with the reducing diameter of C3S and increasing inner and outer C-S-H predicted as hydration continued over time [5]. The model also determined the number of new CH particles and placed them in the pore space. The overlapping of particles, and how the hydration and microstructure is affected, was also simulated. Due to the limited computational power at the time, the model was not developed further but it did provide a methodology for future programmes using this approach. Figure 1 shows an image from their model, which consists of 1,000 C3S and 1,000 CH crystals after 50% hydration.

Navi and Pignat Model
In 1990s, the integrated particle kinetics model was developed by Navi and Pignat [3,[6][7][8]. This model simulated the hydration of spherical C3S particles using a similar vector approach as the Jennings and Johnson model [5]. This model used kinetic laws and relationships that simulated every particle. Hydration was simulated in three phases. Initially an Avrami-type equation was used, followed by a phase-boundary analysis which depended on the particles surface area and finally a diffusion-type where the rate of C3S consumption was inversely proportional to the thickness of hydrates deposited on the surface of cement particles. This approach was found to be very computationally expensive and as a result, the number of particles had to be limited to ~ 10,000. An example of their models output is shown in Figure 2.

HYMOSTRUCT model
The HYdration, MOrphology, and STRUCtural (HYMOSTRUC) model was developed by van Breugel [2] and simulates the 3D hydration of spherical particles. Reacting particles form concentric hydrating layers around the original cement grains, which grow and intersect with each other (see Figure 3). It employs various mass and volume balance rules to accommodate microstructural changes from the dissolution and precipitation of various phases. HYMOSTRUCT, however, does not accurately predict the evolution of the cement microstructure nor does it account for the changing chemistry or transport properties. Recent developments include for random packing of particles and the calculation of autogenous shrinkage. Also, not every individual particle is allowed to hydrate as the authors feel the computation time would be too large. Therefore, a statistical approach to reaction rates is used that assumes the reaction rate of each particle depends only on its size. As the particle interactions or overlaps are not analysed, the microstructure information is not available. The HydratiCA simulation model The HydratiCA model has been developed by Bullard at NIST [10] and is based on more fundamental principles of hydration kinetics. HydratiCA directly simulates the (1) dissolution and growth of mineral phases, (2) diffusion of mobile species in solution, (3) complexation reactions among species in solution or at solid surfaces and (4) nucleation of new phases. The principles of detailed balances and mass action are used to unify these disparate kinetic processes.
HydratiCA makes more detailed predictions of the kinetics of phase changes and microstructure development as a function of solution chemistry and temperature than other microstructure-based models of hydration. However, it is very computationally intensive with simulations requiring small time steps of about 0.2 milliseconds to ensure numerical stability. As a result, approximately 18 million steps are required to simulate just 1 hour of hydration. HydratiCA has been used to investigate various mechanisms of hydration of C3S and tricalcium aluminate (C3A).

The μic microstructure model
The μic (pronounced "mike") microstructural modelling platform [11] is a continuation of the work done by Navi and Pignat [3,[6][7][8], which was, in turn, based on the original Jennings and Johnson model [5] discussed above. μic has been written using object-oriented programming in Java and simulates the growth of spherical particles to mimic hydration. As such, it makes significant advances on the limitations of previous methods due to inadequate computational power at the time. The support libraries within μic allow for the hydration modelling of millions of cement particles (see Figure 4), which more realistically represent real PSD's.
The main advantage of μic is the fully customizable nature of the simulations by the user including the initial PSD, defining the phases and their composition and the stoichiometry of the reactions to be simulated. While the large number of customizable elements to be set up can be tedious, its flexibility makes μic a valuable hydration study tool.

CEMHYD3D hydration model
While not specifically employing the vector approach, no review of cement hydration models is complete without Figure 4: Typical μic microstructure simulations [11] including the CEMHYD3D model [12]. CEMHYD3D is one of the most widely used and well known hydration and microstructure models and employs the discrete or pixel approach and the cellular-automata (CA) method. The microstructure is presented as a grid of discrete three dimensional cubic elements ( Figure 5), called volume-pixels, each representing an anhydrous or hydrate phase or pore.
In CEMHYD3D, CA is used to simulate the diffusion of species through the pore-water leading to reactions and precipitations which either dissolves or reacts and forms a product of hydration. However, due to the maximum pixel size of 1μm, particles below that are not included.

MODEL DESIGN
The model was created to employ the problem solving speed of MATLAB utilising the full suite of inbuilt functions and the inherent stability of the programme to set-up and simulate the hydration of tens of thousands of cement particles within a 100μm volume. The model is also setup to be as user friendly as possible with key properties changeable before starting hydration. Unlike μic, this model is designed to undertake the analysis with minimal user input values. Two *.txt files are loaded into the programme which are described in detail below. Many different formats are possible to input for MATLAB with *.txt chosen as a personal preference. Furthermore, MATLAB is a reasonably easy language to learn and apply as it's based on matrix mathematics. The use of comments throughout improve the readability of the code.

User customisable input
As mentioned above, users have flexibility on input via two *.txt files. The first describes the CWD, which the model uses to determine the PSD and number of particles. The CWD input is read into the model via two row vectors with no limit on the number of input columns. The second input file allows the user to customise the cement properties including w/c ratio, volume size, C3S and CH density (Mg/m3), time step size (hours) and duration of analysis (days). This file also contains hydration kinetic, reaction stoichiometry, CH growth constants and C3S enthalpy values. Examples of the input data will be discussed in more detail in the next section by means of a worked example.

Materials
The model, using user defined kinetic and stoichiometries, will output the volume of reaction products from the original C3S particles, namely C-S-H and CH. Each material has its density, reaction stoichiometry defined. For instance, what proportion of C-S-H should be defined as inner C-S-H and outer C-S-H and subsequently deposited on the C3S particle with their particular density defined and volume determined. Four types of materials are defined and simulated ( Figure 6).

WORKED EXAMPLES
The following section describes examples of analysis undertaken on C3S particles with different CWD based on rules for hydration described by Pignat et al. [3,[6][7][8] including hydration and microstructure kinetic rates. These rules are fully customisable in the model by the user before the analysis begins. The results demonstrate the ease of use of the model and speed of calculations. The following sections provide an overview of the steps undertaken during the analysis with input and output shown in figures and tables.

Particle size distribution
Two CWD's shown in Table 1 were used for this example for a 0.4 w/c cement within a 100x100x100μm volume. The two CWD's produces 2,628 and 96,579 C3S particles (Figure 7). The model places the largest diameter particle (40.4μm here) first and determines the next particle diameter using the CWD above. This repeats until no volume is remaining. The model then places each particle within the volume ensuring there are no overlaps.   Table 1 Hydration and microstructure calculations The input data for the second *.txt file is shown in Table 2. The value for the density of C3S varies amongst researchers so the value used (3.15 g/cm3) is based on the published value given by [11]. A time step of 1-hour was chosen for the analysis for 100 days, giving 2400 steps in total.

Particle Kinetics
The hydration of cement has been described by previous modellers [3,[6][7][8]11] as three mechanisms over different time scales. The first is nucleation and growth which is the subject  of much discussion amongst researchers in this area. There is widespread disagreement between the precise cause of very rapid heat production and reduction within 1-2 hours. Some believe this is due to the formation of a meta-stable layer of hydrates around the reacting particles [13,14]. Others [15] believe it is based on a geochemical approach where, during cement hydration, dissolution is initially dominated by the formation of etch pits on surfaces and later becomes limited to step retreat from such pits As this period of hydration is quite short (1-2 hours) and contributes little to the final strength, it has been left out of the analysis here. The second mechanism is commonly referred to as 'the phase-boundary controlled reaction' [11]. Here, the hydration productions of C-S-H inner and outer are formed around the C3S particle and CH is formed in the pore space. The growth of the thickness of the inner product is given by Equation 2 [3,[6][7][8]11] where K1 is in units of μm/hr. The degree of hydration α(t) (for a spherical particle with an original radius R) is given by Equation 3 [3,[6][7][8]11] where rin is the changing C3S radius over time.

Equation 3
The third mechanism is controlled by diffusion which begins when the radius of the outer C-S-H (routrin) reaches a critical value, typically 4μm. The rate of hydration is then given by Equation 4 [3,[6][7][8]11] where rout is the outer C-S-H radius As shown in Table 2, the values for K1, K2 and routrin are 0.012μm/h, 0.006μm2/hr and 4μm respectively. The above hydration rates determine the C-S-H growth on the C3S particles, either growing inwards (inner C-S-H) our outwards (outer C-S-H). As shown in Equation The CH is allowed to form in the free pore space using a random number generator that defines a location after checking for the availability of a free space. The number of CH particles generated (n(t)) is given by Equation 6 where nmax is the maximum number of CH particles and assumed to be 20% of the total number of C3S particles [3,[6][7][8]11] but is changeable by the user. α is another constant which has been reported to be 0.213 but also changeable by the user. The volume of C3S converted to CH as shown in Equation 5 and the density of CH are 0.593 and 2.24 g/cm3 respectively, and are both adjustable by the user.
( ) = (1 − − ) Equation 6 As shown in Equation 5, the volume of the hydration products is less than the volume of the reactants. This follows Le Chatelier's contraction [16] and leads to chemical shrinkage, which is also calculated by the model together with the change in water volume, porosity and the heat release during hydration using a enthalpy value of 500 J/g ( Table 2).

The simulations
The model was run for the two cements (C1 & C2) in Table  1 with a w/c ratio of 0.4. The pre-processor time required to run a complete analysis for a range of particle distributions ( Figure  8) is quite short and approximately linear in n, compared with the processing time required of n2 and n.log(n) (where n = number of particles) reported by Navi and Pignat [3,[6][7][8] and Bishnoi [11] respectively. Figure 9 shows the 2D microstructure development at t = 0, 7, 28 and 91 days (0, 60.3, 91.7 & 99% hydration) for the 2,628 particle distribution in Figure 6(a). The lesser number of particles were chosen for clarity. The red, green and dotted black lines circles represents the decreasing C3S, the outer C-S-H and the original C3S radius respectively. At each step in the analysis (1 hour here), the degree of hydration, heat release during hydration, chemical shrinkage and porosity are calculated and shown in Figure 10. Figure 11 presents a phase assemblage diagram for the C3S, C-S-H and CH.

DISCUSSION AND CONCLUSION
A new model has been developed to simulate cement hydration and microstructure development over time. The simulations of several thousand particles in a short period of time above give reasonable predictions. Future work includes calibrating against experimental analysis and comparing with other hydration models in the area. The model is currently focussed on C3S hydration. However, work is ongoing to include the other cement phases, namely C2S, C3A and C4AF.
The model has been developed with customisable user input in mind and has the potential to be a useful tool for the simulation of cement hydration that can include, in time, supplementary cementitious materials (SCM's) and supplement experimental analysis of new cement products as the industry continues to move towards sustainable cement.