Representative Elementary Volumes for 3D modeling of mass transport in cementitious materials

DRAFT version! For published version please go to http://iopscience.iop.org/0965-0393/22/3/035001/ Modelling Simul. Mater. Sci. Eng. 22 (2014) 035001.

Abstract 1. Introduction.................................................................................................................................................... 2 2. REV and other relevant issues ....................................................................................................................... 3

2.1. 3D Virtual microstructure generation ..................................................................................................... 3 2.2. Effective transport property modeling .................................................................................................... 5 2.3. Boundary conditions ............................................................................................................................... 7 2.4. Pore morphology..................................................................................................................................... 7 2.4.1. Pore connectivity ................................................................................................................................. 7 2.4.2. Tortuosity............................................................................................................................................. 8 2.4.3. Constrictivity........................................................................................................................................ 9 2.4.4. Dead-end porosity and pore-necks....................................................................................................... 9 2.5. Statistical evaluation of REV.................................................................................................................. 9

3. Simulations and performance....................................................................................................................... 10 4. Results.......................................................................................................................................................... 12 5. Discussion.................................................................................................................................................... 12

5.1 Morphological nature of diffusivity ....................................................................................................... 14 6. Conclusions.................................................................................................................................................. 15 Acknowledgments ........................................................................................................................................... 16 References........................................................................................................................................................ 17

1

Representative Elementary Volumes for 3D modeling of mass transport in cementitious materials

N. Ukrainczyk*,1,2 and E.A.B. Koenders1,3

1Microlab, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands. 2Faculty of Chemical Engineering and Technology, University of Zagreb, Marulicev trg 19, 10000 Zagreb, Croatia. 3Civil Engineering Department-COPPE, Federal University of Rio de Janeiro, 68506 Rio de Janeiro, Brazil.

*Corresponding author: Tel.: (+31) (0)15 2781296; Fax: (+31) (0)15 2786383 e-mail: n.ukrainczyk@tudelft.nl; nukrainc@fkit.hr

Abstract. Representative Elementary Volumes (REV) are of major importance for modeling the transport properties of multi-scale porous materials. REVs can be used to schematize heterogeneous microstructures and form the basis of a numerical analysis. In this paper, the most appropriate REV size for numerical modeling for mass transport in hydrating cement paste is investigated. Numerous series (264) of virtual 3D microstructures with different porosities and pore morphologies were generated using Hymostruc, a numerical simulation model for cement hydration and microstructure development. The influence of the initial particle size distribution, hydration evolution, numerical resolution and type of transport boundary conditions (periodic vs. non-periodic) was investigated. The effective diffusivity was obtained by using a 3D finite difference scheme. The connectivity, dead-end porosity, tortuosity, and constrictivity of the capillary pore network was evaluated. Based on a statistical chi-square analysis, it was concluded that the REV size largely depends on the complexity of the pore morphology, which on its turn, primarily depends on the degree of cement hydration, i.e. the porosity of the simulated microstructure, the employed numerical resolution as well as on the initial particle size distribution of the unhydrated cement grains. Furthermore, the results proved that employing periodic boundary conditions can effectively decrease the variability of the calculated effective properties, and, thus, lower the size of a REV.

Keywords: Representative elementary volume; 3D Numerical modeling; Effective transport property;

Cementitious materials; Pore morphology.

1. Introduction

The long term performance of cement-based materials is strongly depending on the resistance it can build-up against the ingress of aggressive species throughout its porous microstructure with time. It is a mechanism that plays a crucial role in the degradation process [1][2] and determines the mechanical and durability performance of these class of construction materials to a large extent. With increasing interest of the ability to assess the durability of cementitious materials, a better understanding of the mass transport phenomenon through an evolving porous microstructure is essential. However, as the matrix of a cementitious material may cover multiple length scales, ranging from nanometer-size C-S-H gel (the main hydration product) towards centimeter-size aggregates [3]-[5], a multi-scale consideration of the porous microstructure should be employed [6]-[13]. When considering the pore structure embedded in a cement paste microstructure, pores cross at least four orders of magnitude, i.e. 10 nm – 100 µm. A multi-scale schematization of the porous medium, therefore, significantly

2

contributes to the accuracy of calculated transport properties. Representative elementary volumes (REVs), together with a homogenization theory, is one of the mostly used multi-scale techniques to handle the vast numerical systems that generally goes along with the schematization of porous media. The REV employed in a multi-scale model for the effective properties of cement-based materials is a lower-bound approach since it is much smaller than the macroscopic body it represents, but it should also be large enough to be representative for its microstructure [14].

In the current article, the Hymostruc hydration model is used to simulate 3D virtual microstructures as a function of the random distribution of cement particles in a predefined representative elementary volume (REV), degree of hydration, particle size distribution, chemical composition of the cement, morphological development, water to cement ratio, and the reaction temperature. Zhang et al. [15] employed both Hymostruc and a finite element method to simulate the effective diffusion of water, and results showed the same order of magnitude with experimental data. Furthermore, in general it turned out that the simulated results captured the experimental trends for diffusivity versus water to cement ratios, and diffusivity versus curing age very well.

In general, the transport properties of cementitious materials are depending on the porosity and pore structure morphology, which is usually characterized by the connectivity, tortuosity, and constrictivity of the pore system (definitions are detailed in section 2.4). A quantitative determination of a REV for this class of multi-phase porous materials is, therefore, very complex and requires considerable efforts. Generally, it can be said that determining the most appropriate REV for transport modeling of porous materials such as in cementitious materials is still incomprehensible to a large extent In the field of engineering mechanics, however, the REV concept is widely used. For instance Gitman et al. [16] proposed a numerical-statistical method to determine the REV size for a mechanical system while considering the effect of the sample size, solid inclusions to volume ratio, and particle size distribution. Recently, Zhang et al. [17] presented a similar numerical-statistical REV approach for transport modeling in cementitious materials. They reported the use of a finite element method to simulate diffusion in flow direction and applied non-periodic boundary conditions for the four remaining side planes. With this 3D system the authors examined, the effect of the water to cement mass ratio (ranging from 0.30 to 0.60) on the REV size of hydrated OPC, whilst keeping the hydration degree constant at a value of 69 %. The current article builds further on that and adds the effect of hydration evolution, the resolution effect on pore structure discretization, and the selected boundary condition (periodic vs. non-periodic) on the transport properties of a 3D virtual cementitious microstructure. Furthermore, the connectivity, dead-end porosity, tortuosity, and constrictivity of the capillary pore network was evaluated. In order to give a detailed report of the work done, the following sectioning has been made; in Section 2, the theoretical basis of the microstructures used for the analyses is addressed. Emphasis is on the REV size in relation to the generated 3D virtual microstructures, the theoretical and numerical schematization of the differential equation to calculate the effective transport properties, characterization of the pore morphology and the REV quantification theory. Section 3 reports the simulation plan followed by Section 4 where the results are presented and discussed.

2. REV and other relevant issues

2.1. 3D Virtual microstructure generation

The 3D virtual cementitious microstructures used in this analysis are simulated as a function of the random positioning of the cement particles inside a predefined REV, the initial particle size distribution (PSD), the degree of hydration, the chemical composition of cement, the morphological development of the hydration products, the water to cement ratio, and the temperature of the reaction process. In this paper simulations are done for a regular Portland cement with a specific surface of 400 m2 kg-1, a water to cement mass ratio of w/c = 0.3 and a isothermal reaction temperature of 20 °C. The progress of the hydration reaction is calculated by means of a step-wise procedure, where the mineralogical composition, the specific surface of the particles and the particle size distribution of cement all affect the kinetics of the four reactive Bogue phases. Besides this, the mineralogical composition also influences the amount of hydration products formed during hydration, and thus the porosity and pore morphology embedded in the cementitious microstructure. For more details on the

3

hydration kinetics reference is made to [11][18]. Therefore, in this paper emphasis is on the morphology of the porous microstructure represented by the porosity and pore morphology of the capillary pore system. As the w/c ratio is the principal parameter that governs the porosity of hardened cement paste [5],[8]-[15], its dependency towards the porosity and pore complexity has to be evaluated, as well as its dependency towards the size of the REV [16]. These dependencies are also supported by the findings reported in this paper, namely that for lower nominal porosities, the variability of the effective diffusivity strongly increases. The REVs are represented by 3D virtual microstructures that have a cubic shape with a rib size of either 100 or 150 μm. The initial state of the microstructures is determined by stacking cement particles that follow a predefined particle size distribution (Figure 1a). For this, periodic boundary conditions were applied to minimize size effects and to comply with the volume balance induced by the water to cement ratio. Particles are stacked based on random selection of locations with equal probability of occurrence. Placing of the particles starts with the largest particles followed by smaller once, while obeying the particle size distribution. This process continues until all particles in the smallest fractions have been stacked. After having generated this initial particle structure, hydration algorithms are invoked that simulate the stepwise evolution of the particle hydration process and associated expansion of the outer shells of hydration products, whilst forming a 3D virtual microstructure (Figure 1b). During this hydration process, the solid volume fractions of the reactants, i.e. the anhydrous cement and free water, decrease, while, in return, the total fraction of formed hydration products increase. The outer expansion of the individual particles is calculated according to the so-called particle expansion mechanism [10][11], that accounts for the geometrical expansion of the expanding particles that overlap smaller particles and are located in the close vicinity. The positions of the solid particles in space are described by means of a vector in a 3D Cartesian coordinate system. This includes the start (0, 0, 0) and end location (x, y, z), the diameter of anhydrous cement grain, and the thickness of inner and outer hydration layers that surround the shrinking core of the anhydrous cement particles.

As a REV should contain sufficient information for the cement paste microstructure it represents, its minimum rib size (rib size = a, and volume V = a3) should be calculated from the internal pore morphology, which primarily depends on the porosity and complexity of the pore structure. Both these parameters are strongly affected by the water to cement ratio of the paste and the particle size distribution of the cement grains. The initial particle structure inside the REV contains of a number of particles per fraction that are governed by the water to cement ratio, the particle size distribution and the size of the finite representative volume V. In general, cement particles are distributed according to the Rosin-Rammler distribution function [11]-[13]:

1 – exp nG d b d (1)

where G(d) is the cumulative weight (in g) of the particles with diameter d (in μm). The constants n and b depend on the fineness of the cement. The number of particles in a certain fraction can be calculated from the weight per fraction and can be determined by differentiating the Rosin-Rammler function with respect to the particle diameter d. Dividing this by the specific density of the cement (ρ) and by the volume of a single particle leads to [11][12]:

1 6 exp /n ndN b n d b d d 3 (2)

In general, the total number of particles in a paste system is strongly dominated by the number of

smallest particles. A large number of small particles claims a huge amount of storage memory and leads to a dramatic increase of the computation time. Therefore, in order to deal with this issue, in general, an upper and lower limit for the particles sizes involved in the particle structure is considered. Whereas the lower limit is often depending on the computer hardware available (RAM, CPU), the upper limit is governed by either the cement fineness or the REV size under consideration. For example, considering a cement, which has according to the grading curve (eq. 1) a maximum particle diameter of 80 μm. Once this cement is used to generate a particle structure (REV) with a w/c = 0.3 (Rosin-Rammler distribution function: b = 0.041 , n = 1.08), and rib size a = 150 μm, 100 μm, 80 μm, 60 μm, the maximum particle size that fits in the REV is 45 μm, 38 μm, 32 μm, 26 μm, respectively.

4

Therefore, imposing a rib size to a REV implicitly defines the maximum particle size possible to be stacked in the initial particle structure (Figure 1a). It, therefore, also affects the porosity and complexity of the pore structure. This means that the REV size will influence the effective transport properties as well. It should also be noted that disregarding particles d < dmin as well as particles d > dmax excludes some particle fractions which also affects the mass balance of the system. This mass loss should be compensated for by means of a correction factor, which can be derived from the Rosin-Rammler function (eq. 1) and can be expressed as (G(dmax)- G(dmin))

-1 [12]. In line with this, it turned out that variations generally observed in the effective transport properties of cementitious materials are mainly caused by the scatter in the geometry of the individual phases and corresponding particles, which results in local morphological and microstructural variations as well [12][12][12]. These variations affect the local generation of hydration products and, with this, induce variations in the pore structure and volume fractions of each reactive phase. Therefore, in order to deal with this issue, and to scrutinize its consequence for the REV size, a series of cement paste cubic samples, with rib sizes a =100 μm and 150 μm, were generated with Hymostruc. Six realizations (reruns) were generated for each sample size and for each initial microstructure defined by its water to cement ratio (nominal porosity) and initial particle size distribution. In total 264 different microstructures were generated in this study. A full simulation plan is given in section 3.

2.2. Effective transport property modeling

The diffusion transport of species in a fluid media is described by Ficks 2nd law according to:

ju t D u (3)

where u is the concentration of the diffusive species, Dj is the diffusion coefficient of the cementitious microstructure for component j, which is representing the phase constituents: pore, hydration products or anhydrous cement grain. At the macroscopic length scale, diffusion transport is generally modeled as:

2effu t D u (4)

where u is the average concentration of the diffusing species, and Deff is the effective macroscopic diffusivity in a porous media. In a steady state condition, when the fluxes are steady in time, Fick’s 2nd law reduces into the Laplace equation. A finite difference (FD) based numerical scheme is derived for solving the steady state transport problem (eq. 18) [18], which is written in the C++ programing language and implemented in the Hymostruc platform. The algorithm starts with a discretization of the 3D virtual microstructure into a regular 3D mesh (e.g. Figure 2a), where each voxel in the mesh is assigned to be either a capillary pore phase or a solid phase, according to its actual position in the microstructure. In this algorithm, identification is done by the center point of a voxel. For each voxel plane that shares with a neighboring voxel in x, y, and z direction, a conductivity coefficient cx, cy, and cz, needs to be assigned, respectively (Figure 2 a). The connectivity of all voxels are stored in three c vectors (whose lengths correspond to the number of voxels in the system, N). A six neighbor configuration was used, representing a situation where the voxel is connected to its neighbors by the six planes of a cubic voxel in x, y, and z direction. The conductivity coefficients of the surfaces that connect a central voxel to its neighbor voxel is calculated from a series connection approach using two conductors according to, Eq. 5 [18][18].

1 / (0.5 / 0.5 / )i ic D i kD (5)

where k = 1, w, or L. With this notation, w represents the number of voxels in a row and L the number of voxels in a layer. Laplace’s equation is solved by a second order finite difference scheme provided in eq. 6 [19]. This equation shows Laplace’s equation in a finite difference form, for node i according to the numbering as shown in Figure 2a.

5

, 1 1 , ,

, , 1 , , , ,

, 1 , , 0

x i i y i w i w z i L i L

x i x i y i y i w z i z i L i

x i i y i i w z i i L

c u c u c u

c c c c c c u

c u c u c u

(6)

Assembling the equations for all (N) FD nodes forms a global system of equations, which can be represented in a matrix notation by Eq. 7.

A u b (7) where u is the voltage vector (size of the total number of voxels in the system, N), A is a sparse and symmetric matrix with 7 diagonals (each voxel has 6 nearest neighbors) that contain information about conductivity coefficients of all the connections among the voxels, and b is the vector of knowns (i.e. boundary condition at the top of the last layer: u(x,y,z=d+1) = 1, Figure 2b). The obtained system of equations (6, 7) is solved by a conjugate gradient algorithm with an optimized matrix-vector multiplication. This has been achieved by multiplying only those elements of the matrix that lie on the 7 diagonals while avoiding multiplication of a very large number of zero elements. Furthermore, since the size of the sparse matrix A is N times N, and may reach huge dimensions, the matrix is not stored explicitly but only by means of the vectors cx, cy, and cz which store the conductivity coefficients of the connections between voxel planes in the x, y, and z directions, respectively. Next, the flux in z direction at each node i, Ji,z was obtained by solving for each node the FD scheme of Fick’s first law. The FD scheme used for this is:

, ,0.5 ( ) ( )i z i L i z i L i i L z iJ u u c u u ,c (8)

The total (effective) steady state flux Jeff is calculated by averaging the fluxes in the total system volume (total number of nodes N = L d).

, , / ( )eff z i zi

J J L d (9)

Then the relative diffusion RDiff was obtained from the calculated effective flux (normalized to the flux through the same system dimensions without any solid inclusions) according to eq. 10.

0Diff eff effR D D J 0J (10)

The relative diffusivity is expressed in terms of the ratio between the effective diffusivity (Deff) of a

diffusing specie in a porous media and its value when diffusing in bulk water (D0), and ranges between 0 and 1. The simulation algorithm for the transport properties is implemented in two different ways regarding the phases that contribute to the porous transport network: 1) transport through capillary pores only; or 2) transport through both capillary pores and hydration products, or more precisely, through the C-S-H gel pores. Non-hydrated cement grains and portlandite (CH) are considered to be impermeable. With respect to the capillary pores that are still percolated [9][10][9][10][19], effective diffusion through the C-S-H gel pores turned out to be negligible since the diffusion coefficient of the capillary pores is about 400 higher than the coefficient of C-S-H gel [5][20][21]. However, in reality transport through a cementitious microstructure is a result of both transport through capillary pores as well as through C-S-H gel pores. This is because of the layered nature of the C-S-H gel nano-structure, which consists of very fine pores that connect the capillary pore network. In this paper, however, focus is only on the transport through capillary pores. A multi-

6

scale modeling approach that includes the transport through C-S-H gel is considered a topic for future research.

2.3. Boundary conditions

The main flow direction of the REV samples simulated in this study is considered to take place in z-direction. The 4 surfaces that are situated parallel to this main flow direction are subjected to two different types of boundary conditions, i.e. periodic and non-periodic. The non-periodic boundaries represent a full sealing off of the four boundary planes that run parallel to the dominant flow direction of the imposed flux (z-direction, Figure 2). This means that no transport can take place through these surfaces at all. On the contrary, periodic boundary conditions are also applied to the REV samples under consideration, and represent a full disclosure of the four parallel boundary planes, which are connected to the planes situated at the opposite side of the sample. In more detail, the four boundary planes are situated parallel to the imposed flux, which crosses the sample in z direction. So, the two surfaces (left and right) in the y-z plane are named yzL and yzR and the two other (top and bottom) in the x-z plane are named xzT and xzB. The voxels in the left y-z surface plane (yzL) are connected with the corresponding (z-layer) voxels in the opposite right y-z surface plane (yzR), and vice versa. Similar, for the x-z planes: voxels in the top surface plane xzT are connected with the voxels in the bottom surface plane xzB, and vice versa. In this way, a flux that leaves the system at a surface side (e.g. yzL), re-enters the system again via the other surface (e.g. yzR), situated on the opposite side of the sample. For example, for the boundary surface plane of the system presented in Figure 2a, the boundary voxels (xzT) 1, 2, 3, 4, 5, 6 and 7 are connected with boundary voxels (xzB) 36, 37, 38, 39, 40, 41 and 42, respectively. Similar holds for the y-z plane, where boundary voxels (yzL) 1, 8, 15, 22, 29 and 36 are connected with the boundary voxels (yzR) 7, 14, 21, 28, 35 and 42. Voxels situated on the edges of the system have two opposite boundary neighbors (figure 2a), in the y-z plane and in the x-z plane. In the example, one can observe that voxel 1 is connected to both voxel 7 and voxel 36. In the numerical algorithm, the boundary conditions are handled as two additional coefficient vectors (px and py), each with a length equal to the number of boundary voxels that form one surface of the system: e.g. length=(h-2)*w, where h is the height and w the width of the surface. These two vectors store the conductivity coefficients for each element at the boundary-surface. If one of the voxels at a surface is a solid (zero conductivity) then the conductivity coefficient of the element it is linking to in the opposite surface plane is zero (no connectivity, and no flux). On the contrary, if a pore element in a surface is connected with a pore element in the opposite surface plane, then a flux is possible in that particular direction. After assembling of the main matrix according to eq. (6), the boundary conditions are applied according to the periodic or non-periodic boundary conditions required for the REV simulations.

2.4. Pore morphology

The relative effective diffusion coefficient RDiff, expressed as a value ranging between 0 and 1, can be related to the open porosity and pore morphology according to eq. (11) [23][24]:

KPPR openopenDiff 2/ (11)

where δ is the pore network constrictivity and τ – the tortuosity, both having a sound physical basis as defined in sections 2.4.3 and 2.4.2, respectively. K is a morphological parameter that combines the constrictivity and the tortuosity and can be used to express the morphology of a system quantitatively. Calculating the parameters RDiff and Popen according to the above mentioned numerical procedure, the K value can be obtained from eq. 11.

2.4.1. Pore connectivity

The 3D capillary pore structure that remains after digitizing the 3D microstructure resembles a special binary structure that can be evaluated with a flood fill algorithm to assess its connectivity. The flood

7

fill algorithm determines the connections of all voxels in a 3D digitized pore structure. Each capillary pore voxel that is part of this pore structure is identified and evaluated upon its position in the system and how it is connected to its neighboring voxels, i.e. whether it borders either to another “pore voxel” or to a “solid phase voxel”. The flood fill algorithm does this for all voxels in a pore structure, and determines how they are connected at the borders. With this algorithm used to evaluate the voxels in a 3D pore structure, the 6 directions configuration was applied for a voxel for evaluating the connections with its neighbors. This 6 direction configuration does not consider the voxels that are situated at the corners and edges of a central voxel to be connected. Therefore, a fully connected voxel share a maximum of six planes with the central voxel. This also holds for the voxels that are situated at the boundary surface planes, where the connectivity accounts for transport from one plane to the other opposite periodic boundary plane. If one of the voxels on either of these surface planes is solid, no connectivity will be identified. However, if the boundary voxels at both boundary surfaces are both pores, then connectivity will be identified, possibly resulting in a higher degree of the overall pore connectivity of the system. The overall system connectivity is calculated from the total number of pore voxels connected divided by the total number of the pore voxels. Note that in this study the porosity represents the total capillary porosity, i.e. consists of both disconnected and connected (open and closed) porosity. The open porosity then equals the (total) porosity times the connectivity.

2.4.2. Tortuosity

The concept of tortuosity (a dimensionless parameter, 0 ≤ τ ≤ 1) describes the effect of tortuous transport paths and variations on the pore path lengths and was introduced in the 1920's. However, until recently there were no available methods to quantify tortuosity directly from a microstructure. Tortuosity was usually indirectly determined from eq. 11, by knowing the measured RDiff, and Popen and associating all other morphological influences such as constrictivity, dead end porosity, and sometimes also connectivity implicitly to the sole morphological parameter named tortuosity (parameter K in eq. 11). Nowadays, digital image analysis techniques enable to quantify tortuosity directly from tomographs [23][23][23][23]. Significant differences are then perceived between indirectly the estimated morphological parameter (K) from eq. 11 and the actual tortuosity obtained through image analysis. Unrealistically high values of K indicate that tortuosity may not be the only morphological parameter to be considered that influence effective transport properties. Thus, additional morphological parameters, all having a sound physical basis, should be described explicitly in a refined model that correlates the effective transport property with microstructural parameters. The definition of tortuosity should capture the precise geometrical interpretation [23][23] of the tortuosity in eq. 11. To quantify tortuosity Dijkstra’s algorithm [23] was employed that enabled to search for the geodesic paths from any departure voxel on the inlet layer to any pore voxel on the outlet layer [23]. Dijkstra’s algorithm computes the geodesic distance from a graph. Analogue to the transport calculation a graph was used whose nodes are the voxels that have the 6-connectity relationship. Some researchers (e.g. [23][23]) obtained a tortuosity distribution by determining the shortest paths only through the medial axis (skeleton) of the pore network. This approach is efficient but restricts transport through a skeleton of pores which might not well-represent a microstructural transport phenomena [23]. Therefore, this approach was modified by applying a 2D skeletonization only to the inlet layer. The mean length, defined as the arithmetic mean on all the arrival nodes, is then divided by the rib size a of the REV to estimate the corresponding mean value of the tortuosity. In a preliminary numerical implementation (conducted in Matlab), the tortuosity algorithm works as follows. (0) The input microstructure is labeled by binary state of voxels: 1 for connected and non-dead-end pores, and 0 for the solids, dead-end and isolated pores. Pore voxels that form a pore skeleton on the 2D inlet layer of the specimen are located with the Matlab skeletonization algorithm, using the bwmorph function. (1) Label one of the skeleton voxels on the inlet layer as the point of departure (first active pore voxel) and other pore voxels located on the outlet layer as the destination outlet voxels. (2) Use a 6-connectivity approach to select unvisited voxels that are directly connected to the active voxels. Selected voxels become part of the active set and the total length of the paths defined by the 6-connected active voxels that are computed. Step (2) is repeated until all targeted outlet voxel become activated. The length of all individual geodesic path connecting the inlet voxel with the outlet layer is recorded and used for the

8

tortuosity calculation. The procedure of the Dijkstra algorithm (steps 1 and 2) is repeated for all inlet skeleton voxels (if there are N inlet skeleton voxels, the algorithm needs to run N times).

2.4.3. Constrictivity

Constrictivity is a dimensionless parameter (0 ≤ δ ≤ 1) that describes a transports’ resistance, which is inversely proportional to the cross section of the pore necks. Mathematical expressions to quantify the pore neck effect were developed only for ideal, simplified geometries [23][23][23]. Petersen [23] modeled the effect of constrictions on diffusion through a single cylindrical pore that has hyperbolic necks so that the cross section varies periodically along its length. He defined a constriction factor as a ratio between the cross section at the neck and the pore body entrance. For random porous medias, a geometrical definition of δ is still lacking and there are also no methods for determining the constrictivity from microstructures directly. This lack of suitable quantifying techniques is the main reason why in most experimental studies the constrictivity effect is tacitly included in (unrealistically high) τ. For an appropriate characterization of the neck effect in random porous microstructures, it is necessary to find a suitable pore shape detection methodology, which should enable not only to quantify the cross-sectional area at the constrictions and at the pore bodies, but also to take into account the connectivity of the network consisting of many pore-neck-pore elements. The elaboration of this methodology is out of the scope of this article. However, Section 5.1 demonstrates the influence of the constrictivity as well as the dead end pores on the effective transport properties. The constrictivity values were estimated indirectly from the measured relative effective diffusivity (RDiff), pore connectivity, total porosity, dead-end porosity and tortuosity according to the extended model for the morphological nature of the effective diffusivity coefficient:

2( )Diff open deadR P P / (12)

2.4.4. Dead-end porosity and pore-necks

The connected pore space, which excludes isolated pores from total porosity, may be further subdivided into pore space that connects both REV boundaries with imposed boundary concentrations, and into pore space where there is no connection between these two boundaries. For the following modeling, the latter is termed “dead-end porosity”. Based on the calculated steady-state concentration distribution, the flux in x, y, and z (forward) direction was calculated according to (eq. 8). In order to investigate pore necks and dead-end porosity, results were analyzed by calculating the 1-norm flux, which is defined as a sum of absolute values of the fluxes in the 3 forward directions x, y, and z. At steady state condition, the 1-norm flux, i.e. concentration difference between the voxel and his forward neighbors, is zero, or very small if those voxels form dead end pores (even at pore necks). In the proposed algorithm a threshold value for the 1-norm flux of 10-10 was used (for this value the estimated dead-end porosity converged). To quantify the number of pore-necks in the system a specific and efficient algorithm was developed. The algorithm first sorts the values of the 1-norm flux in descending order. Then local maxima are determined by comparing the 1-norm flux of the selected voxel, in descending order, to the values of it’s (6 connected) nearest neighbors. Those values of the 1-norm flux that exhibit local maxima, and do not belong to dead end pores, are extracted as pore necks and stored in a list that is available for statistical analysis (whose results are given in section 5.1).

2.5. Statistical evaluation of REV

The REV of a heterogeneous cement paste microstructure defines the size of a sample that must at least be employed for determining the corresponding effective properties of a homogenized macroscopic model. The REV should be large enough to contain sufficient information about the microstructure in order to be representative, however, it should be much smaller than the macroscopic body [6][16][16]. Gitman et al. [16] proposed a numerical-statistical method to determine the size of a REV focusing on the mechanical properties, while Zhang et al. [17] adapted this approach to

9

determine the REV for transport modeling of cementitious materials. The numerical-statistical method employed here is analogue to the procedure followed by Gitman et al. For this, a series of numerical experiments (realizations) with the different parameters were considered. For each nominal microstructure six different generations of the initial cement particle locations were generated and used for the hydration simulation. To find out the most appropriate REV size, the chi-square criterion, Eq. 13, was used to quantify the change of the calculated effective property for each nominal microstructure based on the mean value found for the six different numerical realizations.

aDiff

m

iaDiffiDiff RRR ,

1

2,,

2 /

(13)

where RDiff,i is the investigated effective parameter (Eq. 10), RDiff,a is the average of all RDiff,i, and m is the total number of numerical realizations performed with different initial cement particle locations for a nominal microstructure (characterized by nominal hydration parameters). Moreover, the statistical analyses are applied on results of six numerical realizations (i.e. m = 6). In order to fulfill the dimesionlessness of the chi-square criterion, the normalized approach of Gitman et al. [16] was followed, normalizing RDiff,i with respect to RDiff,a, thus rewriting the Eq. 13 as:

m

iaDiffiDiff RR

1

2,,

2 1/ (14)

which actually eliminates the effect of D0 from the chi-square criterion. If the sample size is too small from being representative, then the variability of the results will be higher. This procedure assumes that the smaller the value of the chi-square, the closer is the volume of the sample under consideration to the expected REV. Under this principle, the true REV may only be obtained for a sample with an infinite volume. Nonetheless, smaller size of samples can normally be used if the value of the chi-square is acceptably low. Generally, 0.1 is regarded to be an acceptable value [16][17] for the chi-square coefficient for a 95 % probability accuracy and two degrees of freedom. The null-hypothesis of the chi-square test is that the sample size is representative, so the variability in results may appear only due to the following two parameters (two degrees of freedom): 1) random distribution of particles and 2) random variability in the sample porosity, e.g. due to a finite hydration time and discretization error. The 95 % probability accuracy means in this case that if the chi-square is accepted there is a 95% probability confidence that the sample size is representative and the variability in effective transport is (95 %) due to the variability of the two considered parameters.

3. Simulations and performance

3D virtual microstructures were generated with the aim to create different nominal porosities and pore morphologies. Simulations are done for ordinary Portland cement with a water to cement mass ratio of w/c = 0.3. The effect of the minimum particle size on the microstructural formation and variability of effective transport simulations is investigated. Two initial particle size distributions were considered, indicated as PSD_fine and PSD_coarse, having a minimum particle size of 1 μm and 2 μm, respectively. Both PSDs follow the Rosin-Rammler distribution curve (eq. 1) with similar parameters, i.e. b = 0.041 , n = 1.08. The maximum particle size for both PSDs was fixed to 38 μm. Fraction width (step) was 1 μm. For the cubic REVs considered, two rib sizes were applied: a =100 μm and 150 μm. For the 100 μm REV system the initial particle structure using the PSD_fine grading consisted of 62248 particles (diameters ranging from 1-38 μm), while for the similar sized REV the initial particle structure with the PSD_coarse grading consisted of only 10684, (ranging from 2-38 μm). For the 150 μm system, the initial particle structure was generated while only applying the PSD_fine grading, leading to a random parking of 191426 particles. Simulations were ran up to different hydration degrees to obtain microstructures with different pore morphologies. In this way, for each realization, six microstructures were generated with porosities of nominal 21 %, 15 %, 10 %, 6 % and 4 %. These nominal porosities were calculated for each of the six numerical realizations and used for the

10

statistical analysis. Three chi-square related parameters were calculated representing the following nominal microstructures, i.e. the effective diffusivity, the pore structure connectivity, and the complexity of the pore morphology (parameter K in eq. 11). In total, 264 different 3D virtual microstructures were generated for this analysis. In order to be able to investigate the effect of the numerical resolution on the effective transport simulation, 3D virtual microstructures were digitized to form a 3D grid of cubic voxels with different resolutions. The employed resolutions for the FD numerical scheme used for the transport simulations were 0.167, 0.25, 0.5 and 1 µm/pixel (is equal to rib size of voxel), corresponding to a FD system size of 6003, 4003, 2003, and 1003 voxels, respectively.

In order to make it possible for a regular PC to handle the generated microstructures the minimum particle diameter were taken 1 μm (PSD_fine) and 2 μm (PSD_coarse). Excluding the smallest particles from the system that have the highest number of particles per fraction reduces the calculation time dramatically [12][14]. Hymostruc and transport simulations with a numerical resolution of 1 μm/pixel and 0.5 μm/pixel were done on a standard desktop computer (Intel(R) Xeon(R) CPU W3565 @ 3.2 GHz). Time for random placing of all particles (w/c=0.3) with the random stacking algorithm (no overlaps are allowed) takes between 16 seconds and 5.5 minutes for a system rib size of 150 µm and with a PSD_coarse and PSD_fine grading, respectively. Time for simulating the hydration and microstructure development for the 150 µm system size up to a hydration degree of around 62 % (w/c=0.3, corresponding to the capillary porosity of 10 %) took about 1 min and 10 min, for the PSD_coarse and PSD_fine grading, respectively. The total sequential time needed to calculate the effective diffusion coefficient (including the input/output operations) of a discretized microstructure with 1003 voxels (REV = 100 μm3), took 57 s, 79 s, 88 s, 116 s and 163 s for porosities of P = 0.22, 0.16, 0.11, 0.07 and 0.04, respectively. The effective diffusion calculation on microstructures discretized with 2003 voxels (REV = 100 μm3) took 831 s, 1001 s, 1279 s, 1578 s and 1678 s for porosities of P = 0.21, 0.15, 0.10, 0.06 and 0.04, respectively. For the REV size of 150 μm3 (resolution of 1 μm/pixel) the calculation times were 276 s, 308 s, 417 s and 570 s for porosities of P = 0.16, 0.11, 0.07 and 0.04, respectively. This indicates that the computation time for solving the effective transport depends primarily on the system size and resolution, and secondarily on the complexity of the pore morphology, which is increasing with progress of hydration or decreasing with the porosity. The porosity as well as the complexity of the pore morphology is influenced by the numerical resolution as demonstrated further in Section 5.

For the higher resolution transport calculations with numerical resolution of 0.25 μm/pixel and 0.167 μm/pixel simulations were done on the Dutch national supercomputer CARTESIUS using 8 cores (2 × 8-core 2.9 GHz Intel Xeon E5-2690 (Sandy Bridge) CPUs/node, 128 GB/node). The parallelization of the sequential transport algorithm was done by OpenMP (Open Multi-Processing) standard API (application programming interface) for multi-platform shared-memory parallel programming [16]. The number of loops needed for matrix-vector and vector-vector multiplications as well as summation operations which are called (many times) as part of the conjugate gradient optimization loop, was firstly minimized and then parallelized by two ‘#pragma omp parallel for’ and two ‘#pragma omp parallel for reduction’ directives. The parallelization performance on the standard desktop computer using 4 cores and on the CARTESIUS supercomputer using 8 cores in parallel resulted in a 3.5 times and a 7.3 times speedup of the sequential algorithm, respectively. For sake of comparison, the (sequential) time needed to do a calculation with a resolution of 0.167 μm/pixel on the standard desktop computer was about 28 h for P = 10 %. Memory requirement was minimized (e.g. 6.6 GB for resolution of 0.167 μm/pixel) by using logical (Boolean) data types for the connectivity vectors and the microstructure matrix while other variables were of type double.

11

4. Results

The effective diffusion coefficient for 3D virtual microstructures are calculated for the different nominal porosities of P = 21 %, 15 %, 10 %, 6 % and 4 % (Table 1, Table 2). The values for the diffusion coefficient are represented as relative values ranging between 0-1, and show how much the effective transport property is reduced, relative to the diffusion in the bulk water (i.e. 100% pore scenario that has no solid inclusions). This reduction in diffusivity is a result of the inclusion of unhydrated cement particles as well as a generation of an additional volume of solid material due to hydration, leading to a reduction of the porosity due to the formation of microstructure. As the random distribution of the initial particle structure is the starting point for the hydration simulation of the microstructure, the effective transport property is a random variable as well. The mean values of all of the calculated effective parameters (P, RDiff, Connectivity, and K) for the microstructures with the system rib size of 100 μm are given in the upper part of Table 1. It can be observed that the porosity of microstructures with the coarser numerical resolution is consistently higher. Due to the finite time steps of the hydration simulation and the discretization error, the chi-square value of the targeted porosity of each nominal microstructure (corresponding to nominal numerical resolution and PSD_type) was allowed to vary within χ2 (P) ≤ 10-4. Table 1 provides an overview of the simulated results for the effect of porosity, type of microstructure (PSD_fine and PSD_coarse), transport boundary conditions (periodic or non-periodic), and the numerical resolution of the mean values and variability of the effective parameters, calculated for the microstructures with a system rib size of 100 μm. Results of the variability of the diffusivity, χ2(Deff) are not shown in Table 1, as they are presented in Figure 3 and Figure 4 for a numerical resolution of 1 μm/pixel and for 0.5, 0.25 and 0.167 μm/pixel, respectively. These figures show the variability of the effective diffusivity against the sample porosity. The curves represent calculation results of effective transport simulations employed with periodic (pbc) and non-periodic (npbc) boundary conditions, with a system rib size of 100 μm and 150 μm, and with a fine (PSD_fine) and coarse (PSD_coarse) grading used for generating the microstructures. Finally, Figure 5 and Figure 6 show the mean degree of pore connectivity as a function of the porosity for both microstructures generated with the PSD_fine and PSD_coarse grading, respectively. In these two figures the error bars represent the maximum and minimum values for the connectivity derived from the statistical analysis of 6 numerical realizations. The studied effect of the various parameters on the calculated effective parameters and their variability will be discussed next.

5. Discussion

From the results shown in Table 1, it turns out that the effective transport simulations that were employed with periodic boundary conditions (pbc) showed consistently higher values for the relative diffusivity RDiff rather than for the simulations employed with non-periodic (npbc) boundary conditions. This increase in diffusivity is attributed to the associated increase in the degree of pore connectivity (Figure 6, a versus b) and to the decrease in pore complexity, indicated by the increasing parameter K (Table 1). Non-periodic boundary conditions assume a zero flux through the four boundary surfaces of the REV cube that are situated parallel to the imposed flux J (Figure 2b) and, thus, does not allow for pore connectivity in those directions. On the other hand, periodic boundary conditions do allow for additional pore connectivity through the four boundary surfaces of the pore structure, where the flux that leaves the system on one side of the sample, enters the system again through the opposite surface (if both connecting boundary voxels are pores). The effect of the type of boundary condition on the diffusivity is becoming more pronounced with progress of hydration, i.e. with reduction of the sample porosity, which leads to an increased complexity of the pore structure, as indicated by mean values of parameter K in Table 1 (lower K represents an increased complexity). At lower porosities the pore network is becoming more complex, which means that the pore paths become more tortuous, exhibit a higher degree of pore constrictivty (i.e. develop bottleneck effects) and show a lower degree of pore connectivity. As shown by eq. 11, the relative effective diffusivity RDiff depends on the connectivity, the open porosity Popen and the pore complexity K. Thus, with this equation it is also possible to assess which of these parameters contribute most to the relative effective diffusivity. In addition to this, it also appears that the connectivity contributes to the relative effective

12

diffusion by a factor (1 – connectivity). Thus, with this, and from the results in Table 1 and 2 it can be observed that the degree of pore connectivity has a relatively smaller impact on the relative effective diffusivity than the other parameters involved in eq. 11. The magnitude of the relative effective diffusivity is strongly governed by the pore structure complexity K as well as the magnitude of the open porosity. However, it must be stressed that the pore structure connectivity does have an indirect link with the pore structure complexity. For increased pore structure complexity (i.e. lower K), lower values for the connectivity of the pore network are observed. In this, the increased constrictivity may be attributed to a more pronounced neck effect of pore necks that connect the pore network clusters (as will be demonstrated in Section 5.1). Analyzing the effect of the particle grading (PSD_fine and PSD_coarse) on the effective transport properties (Table 1), it reveals that for the microstructures generated with the finer PSD_fine grading and a resolution of 1 μm/pixel, all nominal porosities exhibit a consistently lower mean value for the relative effective diffusivity, for the connectivity and also for the morphology parameter K, than for the microstructures generated with the coarser PSD_coarse grading. The chi-square value (variability) of these effective parameters is higher for the PSD_fine microstructures rather than for PSD_coarse. This observation can be related to a lower complexity and higher connectivity of the pore structures generated by the PSD_coarse grading relative to the PSD_fine grading used to generate the initial particle structure. Finer cements result in microstructures with a finer pore size distribution [15] and require a relatively higher resolution. Therefore, lower connectivity of PSD_fine microstructures can be related to a less accurate discretization of the fine pores in case of finer microstructures, which can now add to the lower connectivity within the network of pore voxels as well.

The variability of the effective diffusivity χ2(Deff) as a function of the porosity is provided in Figures 3 and Figure 4 for a resolution of 1 μm/pixel and (0.5 0.25 and 0.167) μm/pixel, respectively. These figures show that the porosity and the employed numerical resolution have significant impact on the variability of the effective diffusivity. With lowering nominal porosities, the chi-square coefficient, and hence the variability of the calculated result for the effective diffusivity, increases strongly. It shows the relevance of an accurate porosity calculation in relation to the calculated accuracy of the effective transport properties, and with this, to the need for an appropriate REV. In this respect, for a rib size of 100 μm and a discretization resolution of 1 μm/pixel (Figure 3) only for those microstructures with porosities that exceed 16 % and 22%, for a PSD_coarse and PSD_fine particle grading, respectively, an acceptable variability in the diffusivity is achieved. The microstructures with lower porosities exhibit an unacceptable high variability, and thus, require a higher rib size for the REV in order to be representative. Figure 3 also shows the effect of the rib size on the variability, where the effective diffusivity declines strongly as the rib size of the sample increases from 100 to 150 μm. Figure 3b shows that because of this increase in rib size the critical porosity of the microstructure with the PSD_fine grading reduces from 22 % to 11 %, which represents an acceptable size for the REV. Microstructures with a porosity below this critical value require a larger rib size to be acceptable as an REV. Table 2 shows that the larger rib size yields in a higher relative effective diffusivity, equal to higher connectivity and higher K values for all nominal porosities. The variability of these parameters was substantially lower for the higher rib size. Furthermore, employing a finer numerical resolution, 0.5 μm/pixel instead of 1 μm/pixel, yielded also into a much lower chi-square coefficient for the effective diffusivity (comparing Figure 3 and 4). Therefore, employing a rib size of 100 μm and a resolution of 0.5 μm/pixel (Figure 4) demonstrates that samples with a nominal porosity of P = 6 % and 10 %, for periodic and non-periodic transport boundary conditions, respectively, still exhibit an acceptable variability for the diffusivity, while for porosities below this value a larger REV is required. When even employing finer numerical resolutions, i.e. 0.25 and 0.167 μm/pixel, a negligible change in the chi-square coefficient for the effective diffusivity was observed for higher porosities. Note that the porosity is different at different resolutions, and by comparing the chi-square coefficients, the values corresponding to 0.25 and 0.167 μm/pixel behave as interpolated values for data achieved for 0.5 μm/pixel. The only observed difference is at the lowest porosities where the chi-square coefficients at the lowest resolutions is significantly lower than the extrapolated value at coarser resolutions. Figures 5 and 6 show the degree of pore connectivity as a function of the porosity for microstructures that were generated with a PSD_coarse and PSD_fine grading, respectively. In these figures the error bars indicate the observed maximum and minimum values for the connectivity as calculated from the

13

statistical analysis. The figures show that periodic boundary conditions result in lower porosity percolation thresholds as well as in a lower variability of the pore connectivity where these results are more pronounced for those microstructures that were discretized with the coarser resolution, i.e. 0.5 µm/pixel versus 1 µm/pixel. Increased refinement of the resolution leads to a more accurate description of the pore system, and especially for the finer pores, which can then be resolved and add to the connectivity of the total pore network. This indicates that pathways that seem to be depercolated at a coarser resolution turn into percolated pathways at a finer resolution. Moreover, part of the changes in the degree of pore connectivity with changing resolution is because of the discretization effect of the voxels that capture the shape of the spherical surfaces in the hydration model [34][35]. Figure 6 also reveals that the effect of the resolution on the degree of pore connectivity (and its variability) is increasingly more pronounced with progress of hydration, i.e. with reduction of the porosity. This also complies with the increasing amount of smaller pores, during hydration, at the expense of the bigger pores, where these smaller pores than require a finer resolution in order to contribute to the connectivity of the entire pore network. For an equal sample size, the variability in connectivity is higher for the coarser resolutions (Figures 5 and 6, and Table 1). Furthermore, the degree of connectivity versus porosity exhibits a sharp increase of the slope near the percolation threshold (the finer the resolution the sharper the change, figure 6a), indicating a higher sensitivity of the depercolation of the pore network towards the porosity near the percolation threshold, and the associated increase in the variability of the connectivity and complexity of a pore network, and, hence calls for a larger size of the REV. The effect of the studied parameters on the variability of the effective diffusivity can be explained in more detail by the degree of connectivity versus the porosity curve (figures 5 and 6). The increase in variability are due to the increased complexity of the pore morphology (characterized by tortuosity constrictivity and dead-end porosity) and the depercolation of the pore network, which, in fact, would require that the REV size should increase as the cement hydration evolves (due to reduction of porosity). Besides this, a refinement of the numerical resolution results in a significant increase in connectivity of the pore structure (figures 5 and 6, Table 1) and K value (lower complexity), while on the other hand, the calculated (total) porosity is decreasing [35]. This opposing effect seems to be partly compensated for when considering the final impact of the numerical resolution on the calculated relative effective diffusion coefficient (Table 1). According to eq. 11, it follows that the variability of the effective diffusivity depends on the variability of the connectivity, the porosity and the pore complexity parameter K. Thus it is possible to quantify the impact (contribution) of the variability of each of these parameters on the variability of the diffusivity. Comparison of the magnitudes of the variability coefficients of these parameters reveals that the variability in the pore structure complexity (χ2(K)) is the prime factor that governs the variability of the effective diffusivity and thus the REV size (Table 1 and 2). A lower connectivity of the pore network also results in a higher complexity of the pore network (i.e. lower K). Furthermore, the higher variability in the connectivity, therefore, also results in a higher variability of the pore complexity. Figures 3 to 6, as well as Tables 1 and 2 show a general scheme that can help to explain the studied effects under consideration on the most appropriate size for a REV.

5.1 Morphological nature of diffusivity

The applied model (eq. 12) for the morphological nature of the effective diffusivity coefficient can be verified only if all parameters can be determined independently. As pointed out in Section 2.4.3, more research is needed to quantify the constrictivity δ independently. Therefore, in this paper, the constrictivity (δ) is calculated indirectly from the other measured parameters according to eq. 12. The morphological parameters, including Pdead, τ and δ, were calculated from microstructures that correspond to only one specific initial state of the unhydrated particles (i.e. position and size), namely PSD_fine and system rib size of 100 μm, but for different porosities (i.e. degrees of hydration) and numerical resolutions as presented in Table 3. In general, the results show that with improving resolution (Pnominal = 6 %) the effective diffusivity continuously increases: being very pronounced from 1 to 0.5 µm/pixel resolution increase and less as the value converges further. As discussed before, this happens at the same time as the value of the total porosity is decreasing and tending to converge, while the pore connectivity is increasing. An interesting observation is the fact that the amount of

14

dead-end porosity is very high for a resolution of 1 μm/pixel, and decreases sharply with resolution refinement, while the constrictivity exhibits an inverse behavior, i.e. it increases sharply and converges. This behavior is in agreement with the increase of the pore network complexity due to depercolation of the pore structure. The tortuosity of the pore structure was found to decrease only slightly with refinement of the numerical resolution. Thus, from Table 3 it can be concluded that the observed lower values of the relative diffusivity at 1 μm/voxel (Table 1 and 3) are mainly due to the higher effects of depercolation of the pore structure that forms significantly more dead-end pores and increases the effect of constrictivity (significantly lower δ). Pore necks can be considered as critical locations that will further strongly depercolate the pore network by only a small additional decrease of the total porosity (induced by a small increase in hydration). The effect of porosity is investigated for a resolution of 0.25 μm/pixel. For higher porosities (Pnominal > 6 %), the tortuosity decreases significantly (30 % for Pnominal = 15 % versus 6 %) while the constrictivity significantly increases (85 % for Pnominal = 15 % versus 6 %). The effect of constrictivity is demonstrated further by analyzing the local flux rates at pore necks. In section 2.4.1 an efficient algorithm was presented to extract pore necks as a list of local maxima for 1-norm fluxes. Calculating the percentiles of 1-norm flux rates at the necks did not yield any meaningful results because the distribution of the flux rates does not follow a normal distribution and the highest rates are behaving as outliers. Therefore, the generated list of local flux rates is divided into 4 categories by sorting their magnitudes into four linear segments. The maximum flux rate (total upper limit), which sets the limits of the flux ranges for each category, varies with resolution, i.e. 0.030, 0.055, 0.054 and 0.055 for a resolution of 1, 0.5, 0.25 and 0.167 μm/pixel, respectively. This irregularity of category limits with increasing resolution refinement is due to the differences it causes in the accuracy of the described pore network, in terms of inclusion of finer pore necks, changes in constrictivity, enhanced connectivity, etc. The categories are named Q1, Q2, Q3, Q4, representing segments with lowest to highest rates of the selected fluxes. For a resolution of 1 μm/pixel and Pnominal = 6 %, the percentage of pore voxels that belongs to the categories Q4, Q3+Q4, and Q2+Q3+Q4 is p4= 0.3%, p3,4=0.8% and p2,3,4=2.4%, respectively. Note that p(Q1+Q2+Q3+Q4)=100% and p1=1-p2,3,4. Thus, Q1 is the main dominating flux rate of the system whose frequency of occurrence is almost 100%. The remaining categories, Q2 to Q4 are considered as critical necks that exhibit significant resistance to transport. Repeating the same statistical calculations for resolutions 0.5, 0.25, and 0.167 μm/voxel (Pnominal = 6 %) results in: p4=0.02 p3,4=0.07 p2,3,4=0.44; p4=0.001 p3,4=0.008 p2,3,4=0.12; p4=0.0005 p3,4=0.005 p2,3,4=0.04, respectively. These results indicate that the number of the critical necks drops drastically with increasing resolution refinement, with a factor of about 0.1 for each step in resolution. The drop in number of necks being again more pronounced at coarser resolutions. Investigating the effect of porosity (Pnominal = 15 %, 10 %, 6 %, 4 %; resolution = 0.25 μm/pixel) gave no significant difference on the statistical results.

6. Conclusions

From the analyses conducted, the following conclusions can be drawn: The smallest possible rib size of the sample that may be considered as a REV depends primarily

on the particle size distribution of the initial cement grains. According to the random particle stacking routine the maximum particle size placed is a function of the system size. In this paper the maximum particle size was fixed to 38 μm, corresponding to a minimum REV of 1003 µm3.

The results indicate a strong dependency between the variability of the calculated effective diffusivity χ2(Deff) and the porosity, and thus with the size of the REV and evolving complexity of the pore morphology during hydration. The porosity and employed numerical resolution have the highest impact on the variability of the simulated effective diffusivity. For lower nominal porosities, the chi-square coefficient for effective diffusivity increases strongly. Employing a system rib size of 100 μm and a numerical resolution of 1 μm/pixel results in porosities equal to and above 16 % and 22 %, for PSD_coarse and PSD_fine microstructures, respectively, which exhibit acceptable values for the variability in diffusivity (χ2(Deff) < 0.1). The variability of the effective diffusivity declines significantly with increasing rib size of the sample from 100 to 150 μm, resulting in a lowering critical porosity, for which the system size can still be accepted as a

15

REV, from 22 % to 11 % for PSD_fine microstructure. A higher REV size yielded higher diffusivity, connectivity and K values, consistently for all of nominal porosities considered.

Employing finer numerical resolutions (0.5, 0.25 and 0.167 μm/pixel versus 1.0 μm/pixel) yielded into a much lower variability of the effective diffusivity. Employing a rib size of 100 μm and a numerical resolution of 0.5 μm/pixel, showed that samples with a nominal porosities exceeding 7 % and 10 %, for periodic and non-periodic transport boundary conditions, respectively, exhibit an acceptable variability for the diffusivity (χ2(Deff) < 0.1), while for lower porosities a larger REV is required.

The observed increase of the REV size with increasing degree of cement hydration is attributed to the associated reduction of the porosity and increasing complexity of the pore morphology K. Results show that the effective diffusivity is governed primarily by the pore structure complexity (K) and porosity, while the degree of pore connectivity has a relatively smaller contribution. Furthermore, the variability in pore structure complexity (χ2(K)) is the prime factor that governs the variability of the effective diffusivity and thus REV size.

Applying periodic boundary conditions results in a lower porosity percolation threshold as well as in a lower variability of both pore connectivity and porosity percolation threshold, relative to applying non-periodic boundary conditions.

Microstructures discretized with a finer numerical resolution exhibit higher pore connectivity as well as a lower variability of the pore connectivity and porosity percolation threshold. The increase in connectivity of the pore structure with increasing refinement of the numerical resolution is attributed to a more accurate description of the finer pores, which can now add to the connectivity within the total pore network.

Degree of connectivity versus porosity curves exhibits a steep slope near the depercolation threshold, which results in an increasing variability in the connectivity and complexity of a pore network, and that could affect the required size of an appropriate REV.

Refinement of the numerical resolution results in both an increasing pore connectivity and pore complexity factor K value (i.e. lower pore network complexity), while on the other hand, the calculated (total) porosity is decreasing. This opposing effect seems to partly compensate for the final impact of the numerical resolution on the calculated effective diffusion coefficient.

The effect of type of grain fineness revealed that the microstructures generated with the PSD_fine grading exhibit for all nominal porosities consistently lower mean values for the diffusivity, connectivity and morphology parameter K, than for the microstructures generated with the PSD_coarse grading. A chi-square calculation of these effective parameters shows a higher variability for the PSD_fine microstructures than for the PSD_coarse ones.

The tortuosity decreases about 30 % for a porosity reduction of Pnominal = 15 % versus 6 %, while the constrictivity increases 85 %.

With increasing refinement of the resolution the number of the critical pore necks drop drastically, the constrictivity increases (meaning lower resistance), and the dead-end porosity decreases. The dead-end porosity can be an important morphological parameter to describe the effective diffusion in porous media.

Further research will focus on an extension of the numerical method for investigating the REVs for the mass transport at mortar and concrete scales, where the percolation effect of the porous interfacial zones between aggregates and cement paste could play a significant role. This will require an additional upscaling of the modeling algorithmization as well as of the computer capacity.

Acknowledgments

The authors acknowledge the financial support by the Marie Curie Actions EU grant FP7-PEOPLE-2010-IEF-272653-DICEM ’’Diffusion of Ions in Cementitious Materials’’ and The Netherlands Organization for Scientific Research (NWO) grant SH-270-13 ’’Effective Transport Properties of Cementitious Materials’’ for calculations on the CARTESIUS Dutch national supercomputer.

16

References

[1] Ukrainczyk N and Ukrainczyk V 2008 A Neural Network Method for Analyzing Concrete Durability Mag. Concr. Res. 60 475-486

[2] Ukrainczyk N, Banjad P I and Bolf N 2007 Evaluating Rebar Corrosion Damage in RC Structures Exposed to Marine Environment Using Neural Network Civ. Eng. Env.l Sys. 24 15-32

[3] Dolado J S and van Breugel K 2011 Recent advances in modeling for cementitious materials Cem. Concr. Res. 41 711–726

[4] van Breugel K 2004 Modelling of cement-based systems—the alchemy of cement chemistry Cem. Concr. Res. 34 1661–1668

[5] Garboczi E J and Bentz D P 1998 Multi-scale analytical/numerical theory of the diffusivity of concrete Adv. Cem. Based Mater. 8 77-88

[6] Bentz D P 2005 CEMHYD3D: A Three-Dimensional Cement Hydration and Microstructures Development modelling Package Version 3.0. NIST Intern. Rep. 7232. http://ciks.cbt.nist.gov/garbocz/NISTIR7232/

[7] Bentz D P 1997 Three-dimensional computer simulation of Portland cement hydration and microstructure development J. Amer. Cer. Soc. 80 3-21

[8] Garboczi E J and Bentz D P 2001 The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration Cem. Concr. Res. 31 1501-1514

[9] Pignat C, Navi P, Scrivener K L 2005 Simulation of cement paste microstructure hydration, pore space characterization and permeability determination Mater. Struct. 38 459–466

[10] van Breugel K 1995 Numerical simulation of hydration and microstructural development in hardening cement paste (I): Theory Cem. Concr. Res. 25 319–331

[11] van Breugel K 1991 Simulation of Hydration and Formation of Structure in Hardening Cement-Based Materials PhD thesis, Delft University of Technology, Delft, The Netherlands.

[12] Koenders E A B 1997 Simulation of volume changes in hardened cement-based materials, PhD Dissertation, Delft University of Technology, Delft, The Netherlands.

[13] Bishnoi S and Scrivener K L 2009 µic: A new platform for modelling the hydration of cements Cem. Concr. Res. 39 266-274

[14] Hashin Z 1983 Analysis of composite materials: a survey J. App. Mech.50 481-505 [15] Zhang M Z, Ye G, van Breugel K 2011 Microstructure-based modeling of water diffusivity in

cement paste Constr. Build. Mater. 25 2046–2052 [16] Gitman I M, Gitman M B and Askes H 2006 Quantification of stochastically stable

representative volumes for random heterogeneous materials Arch. Appl. Mech. 75 79–92 [17] Zhang M Z, Ye G and van Breugel K 2010 A numerical-statistical approach to determining the

representative elementary volume (REV) of cement paste for measuring diffusivity Mater. de Construcc. 300 7–20

[18] Ukrainczyk N, Koenders E A B, and van Breugel K 2012 Multicomponent Modelling of Portland Cement Hydration Reactions: Proc. 2nd Int. Conf. on Microstructural-related Durability of Cementitious Composites (Amsterdam, April, 2012, RILEM Proceedings pro083, Bagneux, France, Rilem Publications s.a.r.l.) ed Ye G, van Breugel K, Sun W and Miao C W, 8 pp.

[19] Garboczi E J 1998 FE and FD programs for computing the linear electric and elastic properties of digital images of random materials NIST Intern. Rep. 6269. http://ciks.cbt.nist.gov/~garbocz/manual/

[20] Ye F 2005 Percolation of capillary pores in hardening cement pastes Cem. Concr. Res. 35 167-176

[21] Bentz D P, Quenard D A, Baroghel-Bouny V, Garboczi E J and Jennings H M 1995 Modelling drying shrinkage of cement paste and mortar Part 1. Structural models from nanometres to millimetres Mater. Struc.28 450-458

[22] Garboczi E J and Bentz D P 1996 Modelling of the Microstructure and Transport Properties of Concrete Constr. Build. Mater. 10 293-300

17

[23] García-Gutiérrez J L, Cormenzana T, Missana M and Mingarro M 2004 Diffusion coefficients and accessible porosity for HTO and 36Cl in compacted FEBEX bentonite Appl. Clay Sci. 26 65–73

[24] van Brakel J and Heertjes P M 1974 Analysis of Diffusion in Macroporous media in terms of Porosity, a Tortuosity and a Constrictivity factor Int. J. Heat Mass Transfer 17 1093-1103

[25] Gommes C J, Bons A-J, Blacher S, Dunsmuir J H and Tsou A H 2009 Practical Methods for Measuring the Tortuosity of Porous Materials from Binary or Gray-Tone Tomographic Reconstructions AIChE J. 2009 55 2000-2012

[26] Cecen A, Wargo E A, Hanna A C, Turner D M, Kalidindi S R and Kumbur E C 2012 3-D Microstructure Analysis of Fuel Cell Materials: Spatial Distributions of Tortuosity, Void Size and Diffusivity J Electrochem Soc 159 B299-B307

[27] Lindquist W B, Lee S, Coker D A, Jones K W and Spanne P 1996 Medial axis analysis of void structure in three-dimensional tomographic images of porous media J. Geophys Res. 1996 101 8297–8310

[28] Sun W C, Andrade1 J E and Rudnicki J W 2011 Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability Int. J. Numer. Meth. Engng 88 1260-1279

[29] Dijkstra E W 1959 A note on two problems in connexion with graphs Numerische Mathematik 1959 1 269–271.

[30] Petersen E E 1958 Diffusion in a Pore of Varying Cross Section, AIChE J. 4 343-345 [31] Michaels A S 1959 Diffusion in a pore of irregular cross section, AICh E J. l5, 270-271 [32] Currie J A 1960 Gaseous diffusion in porous media, Parts 1 and 2, Br. J. Appl. Phys. 11 314-324 [33] Barbara Chapman, Gabriele Jost and Ruud van der Pas, Using OpenMP - Portable Shared

Memory Parallel Programming, Scientific and Engineering Computation Series, The MIT Press, Cambridge, Massachusetts, 2008.

[34] Chen W and Brouwers H J H 2008 Mitigating the effects of system resolution on computer simulation of Portland cement hydration Cem. Concr. Comp. 30 779-787

[35] Ukrainczyk N and Koenders E A B 2012 Multi-scale Resolution Refinement Method for Transport Properties of Cementitious Materials, submitted to Modelling and Simulation in Materials Science and Engineering

List of Tables: Table 1. Mean values and variability of the effective parameters (χ2(Deff) depicted in Fig 3 and 4)

calculated for the microstructures with the system rib size of 100 μm: effect of the porosity, microstructure type (PSD_fine and PSD_coarse), the transport boundary conditions and the numerical resolutions.

Table 2. Mean values and variability of the effective parameters calculated for the microstructures: the effect of system size (PSD_fine microstructures with system rib size of 100 μm and 150 μm discretized with numerical resolution of 1μm/pixel employing periodic boundary condition for the transport calculation.

Table 3. Effect of numerical resolution and porosity on the relative effective diffusivity and pore morphology obtained from a selected initial microstructure (i.e. results from sole numerical realisation, PSD_fine system rib size of 100 μm). Constrictivity (δ) is calculated indirectly from the measured relative effective diffusivity (R), pore connectivity, total porosity (P), dead-end porosity (Pdead) and tortuosity (τ) according to eq. 12.

18

List of Figures: Figure 1. a) 3D simulated microstructure (grey-cement, red-inner hydration product, yellow-outer

hydration product): a) initial unhydrated cement P = 0.5 (w/c = 0.3); b) after hydration at P = 0.3.

Figure 2. a) FD implementation, position and size of coordinates: width (x), height (y), and depth (z). Each sharing surfaces between neighboring voxels in x, y, and z directions has an assigned conductivity coefficient cx, cy, and cz, respectively. b) Steady state flux (J) across z-axis with (periodic or non-periodic) boundary conditions (b.c.) employed on 4 side faces parallel to the imposed flux.

Figure 3. Variability of effective diffusivity vs. sample porosity: the effect of transport boundary condition (npbc-non-periodic, pbc-periodic), the system size (rib = 100 μm or 150 μm) and the microstructure type (PSD_coarse or PSD_fine): a) full range of variability; b) inset of Fig a. The calculations employ numerical resolution of 1 μm/pixel.

Figure 4. Variability of the effective diffusivity vs. the sample porosity: effect of transport boundary condition (npbc – non-periodic, pbc – periodic) and the microstructure type (PSD_coarse or PSD_fine) employing numerical resolution of 0.5, 0.25 and 0.167 μm/pixel.

Figure 5. Degree of pore connectivity as a function of porosity for the PSD_coarse microstructures discretized with numerical resolutions of 1 μm/pixel: the effect of transport boundary condition (npbc – non-periodic, pbc – periodic) (mean, maximum and minimum values are presented based on statistical analysis of 6 numerical realizations for each porosity). Note that the connectivity curves only show the last part of the complete curve that starts from 1 (fully connected) at initial porosity of un-hydrated cement particles P = 48.6 % (for w/c = 0.3). Emphasis is, therefore, on the range near the percolation threshold where the variability of the results are significant and relevant for the REV size.

Figure 6. Degree of pore connectivity as a function of porosity for the PSD_fine microstructures discretized with different numerical resolutions (mean, maximum and minimum values are presented with error bars based on statistical analysis of 6 numerical realizations for each porosity) employing: a) periodic boundary conditions and b) non-periodic boundary conditions for transport calculations.

19

Appendix A. Embedding graphics in a Microsoft Word document

a) b)

Figure 1. a) 3D simulated microstructure (grey-cement, red-inner hydration product, yellow-outer hydration product): a) initial unhydrated cement P = 0.5 (w/c = 0.3); b) after hydration at P = 0.3.

a) b)

Figure 2. a) FD implementation, position and size of coordinates: width (x), height (y), and depth (z). Each sharing surfaces between neighboring voxels in x, y, and z directions has an assigned conductivity coefficient cx, cy, and cz, respectively. b) Steady state flux (J) across z-axis with (periodic or non-periodic) boundary conditions (b.c.) employed on 4 side faces parallel to the imposed flux.

20

4 6 8 10 12 14 16 18 20 220

1

2

3

4

5

6 (D

ef )

system rib size = 100 m npbc, PSD_fine pbc, PSD_fine npbc, PSD_coarse pbc, PSD_coarse

system rib size = 150 m: pbc, PSD_fine

Porosity

1 m/pixel

a)

4 6 8 10 12 14 16 18 20 220,0

0,2

0,4

0,6

0,8 Acceptable level: (D

eff ) < 0.1

system rib size = 100 m: npbc, PSD_fine pbc, PSD_fine npbc, PSD_coarse pbc, PSD_coarse

system rib size = 150 m: pbc, PSD_fine

(Def )

Porosity

1 m/pixel

b) Figure 3. Variability of effective diffusivity vs. sample porosity: the effect of transport boundary condition (npbc – non-periodic, pbc – periodic), the system size (rib = 100 μm or 150 μm) and the microstructure type (PSD_coarse or PSD_fine): a) full range of variability; b) inset of Fig a. The calculations employ numerical resolution of 1 μm/pixel.

21

2 4 6 8 10 12 14 16 18 20 220,00

0,05

0,10

0,15

0,20

0,25

1

2

3 (D

ef )

system rib size = 100mPSD_fine:

npbc, 0.5 m/pixel pbc, 0.5 m/pixel pbc, 0.25 m/pixel pbc, 0.167 m/pixel

PSD_coarse: pbc

Acceptable level: (Deff

) < 0.1

Porosity

Figure 4. Variability of the effective diffusivity vs. the sample porosity: effect of transport boundary condition (npbc – non-periodic, pbc – periodic) and the microstructure type (PSD_coarse or PSD_fine) employing numerical resolution of 0.5, 0.25 and 0.167 μm/pixel .

22

2 4 6 8 10 12 14 16 18 200,0

0,2

0,4

0,6

0,8

1,0D

egr

ee o

f por

e c

onne

ctiv

ity

Porosity

PSD_coarse, 1 m/pixel pbc npbc

Figure 5. Degree of pore connectivity as a function of porosity for the PSD_coarse microstructures discretised with numerical resolutions of 1 μm/pixel: the effect of transport boundary condition (npbc – non-periodic, pbc – periodic) (mean, maximum and minimum values are presented based on statistical analysis of 6 numerical realisations for each porosity). Note that the connectivity curves only show the last part of the complete curve that starts from 1 (fully connected) at initial porosity of un-hydrated cement particles P = 48.6 % (for w/c = 0.3). Emphasis is, therefore, on the range near the percolation threshold where the variability of the results are significant and relevant for the REV size.

23

2 4 6 8 10 12 14 16 18 20

0,0

0,2

0,4

0,6

0,8

1,0

PSD_fine, pbc 0.167 m/pixel 0.25 m/pixel 0.5 m/pixel 1 m/pixel

Deg

ree

of p

ore

conn

ectiv

ity

Porosity

a)

2 4 6 8 10 12 14 16 18 200,0

0,2

0,4

0,6

0,8

1,0

De

gre

e o

f po

re c

on

ne

ctiv

ity

Porosity

PSD_fine, npbc 0.5 m/pixel 1 m/pixel

b)

Figure 6. Degree of pore connectivity as a function of porosity for the PSD_fine microstructures discretized with different numerical resolutions (mean, maximum and minimum values are presented with error bars based on statistical analysis of 6 numerical realizations for each porosity) employing: a) periodic boundary conditions and b) non-periodic boundary conditions for transport calculations.

24

Table 1. Mean values and variability of the effective parameters (χ2(Deff) depicted in Fig 3 and 4) calculated for the microstructures with the system rib size of 100 μm: effect of the porosity, microstructure type (PSD_fine and PSD_coarse), the transport boundary conditions and the numerical resolutions.

Parameter

Boundary condition

PSD type

Numerical resolution, μm/pixel

Mean values and variability

Pnominal n.a.! 0.5 21 15 10 6 4 0.167 19.7 12.7 8.6 5.2 3.1 0.25 20.1 13.3 9.0 5.5 3.3 0.5 21.2 14.9 10.2 6.2 3.8

fine

1 22.6 16.1 11.2 7.1 4.3

P

n.a.!

coarse 1 22.5 16.5 11.6 7.8 4.7 0.167 0.0450 0.0279 0.0118 0.0026 0.0014 0.25 0.0448 0.0272 0.0115 0.0026 0.0014 0.5 0.0443 0.0262 0.0112 0.0026 0.0012

fine

1 0.0364 0.0154 0.0045 0.0006 0.0007

periodic

coarse 1 0.0450 0.0185 0.0074 0.0012 0.0014#

0.5 0.0440 0.0257 0.0103 0.0025 0.0008+ non-periodic

fine 1 0.0354 0.0131 0.0036 0.0003 2.7e-4*

Relative diffusivity RDiff

coarse 1 0.0440 0.0176 0.0054 0.0009 0.0010+

0.167 0.2284 0.2197 0.1372 0.0505 0.0470 0.25 0.2229 0.2045 0.1278 0.0478 0.0530 0.5 0.2208 0.1881 0.1200 0.0484 0.0451

fine

1 0.1751 0.1083 0.0492 0.0130 0.0326

periodic

coarse 1 0.2128 0.1232 0.0733 0.0195 -# 0.5 0.2351 0.2176 0.1316 0.0668 -+ fine 1 0.1914 0.1132 0.0513 0.0136 -*

K

non-periodic

coarse 1 0.2208 0.1298 0.0616 0.0210 -+

0.5 6e-6 1.4e-5 6.5e-4 2.2e-3 1.3e-2 fine 1 0.0009 0.0031 0.0050 0.080 0.216

periodic

coarse 1 4.7e-5 7.7e-4 3.2e-3 4.7e-3 1.2e-2#

0.5 7.1e-6 1.7e-5 8.2e-5 2.1e-3 20+ fine 1 0.0011 0.0038 0.0070 1.09 3.35*

χ2 (Connectivity)

non-periodic

coarse 1 2e-5 8.6e-4 3.5e-3 8.2e-2 1.4+ 0.5 0.0013 0.0047 0.0156 0.0775 2.977

fine 1 0.0189 0.0896 0.1932 0.4913 1.42 periodic

coarse 1 0.021 0.0636 0.2017 0.7528 -#

0.5 0.0067 0.0130 0.0487 0.1487 -+ fine 1 0.0921 0.2021 0.3497 1.0349 -#

χ2(K)

non-periodic

coarse 1 0.029 0.0719 0.2061 0.8979 -+ ! - not applicable * - 3 numerical realizations resulted in no connected pores (i.e. zero connectivity) + - 2 numerical realizations resulted in zero connectivity # - 1 numerical realization resulted in zero connectivity

25

26

Table 2. Mean values and variability of the effective parameters calculated for the microstructures: the effect of system size (PSD_fine microstructures with system rib size of 100 μm and 150 μm discretized with numerical resolution of 1μm/pixel employing periodic boundary condition for the transport calculation.

Parameter

System rib size, μm

Mean values and variability

150 16.4 11.3 7.0 4.3 Popen 100 16.1 11.2 7.1 4.3 150 0.0163 0.0051 0.0008 0.0012 Relative diffusivity

RDiff 100 0.0154 0.0045 0.0006 0.0007 150 0.9003 0.8791 0.8325 0.7518 Connectivity 100 0.883 0.817 0.65 0.5 150 0.1102 0.0511 0.0137 0.0383 K 100 0.1083 0.0492 0.0130 0.0326 150 9e-6 2e-4 8e-4 4e-3 χ2(Connectivity) 100 0.0038 0.0070 1.09 3.35 150 0.0069 0.0476 0.0878 0.29 χ2(K) 100 0.0896 0.1932 0.4913 1.42

Table 3. Effect of numerical resolution and porosity on the relative effective diffusivity and pore morphology obtained from a selected initial microstructure (i.e. results from sole numerical realisation, PSD_fine system rib size of 100 μm). Constrictivity (δ) is calculated indirectly from the calculated relative effective diffusivity (RDiff), pore connectivity, total porosity (P), dead-end porosity (Pdead) and tortuosity (τ) according to eq. 12.

Numerical resolution, μm/pixel

Pnominal, %

RDiff P, % Connectivity Pdead, % τ δ

1 0.00054 6.99 0.703 2.25 2.64 0.141 0.5 0.00228 6.21 0.939 0.67 2.63 0.306

0.25 0.00272 5.48 0.984 0.48 2.62 0.380 0.167

6 0.00298 5.16 0.990 0.15 2.61 0.410

15 0.0274 13.3 0.997 0.16 1.83 0.701 10 0.0112 9.0 0.995 0.22 2.08 0.555

0.25

4 0 3.3 0 0 - -