Multi-scale modelling for diffusivity based on practical estimation of interfacial properties in cementitious materials

Highlights

• Normal line sampling algorithm is proposed to estimate apparent ITZ thickness.

• Effects of particle shape and PSD on apparent ITZ thickness are investigated.

• Overestimation of ITZ volume fraction and concrete diffusivity are evaluated.

• Relative error between two algorithms is discussed.

Abstract

The interfacial transition zone (ITZ) between matrix and aggregate is a weak region because of its physical nature of relatively high porosity and low rigidity. ITZ microstructural configurations such as its thickness and volume fraction play an important role in transport and mechanical behaviours of cementitious composites. However, such ITZ characteristics are usually overestimated since random sampling planes from conventional sectional plane analysis technologies rarely pass through the normal to the surface of anisotropic aggregates. In this study, we first propose a normal line sampling (NLS) algorithm to evaluate the overestimation degrees of these microstructures for complex geometrical shaped aggregates like Platonic particles. The NLS algorithm applies the sampling lines along the normal of aggregate boundary at the given section plane that is significantly different from the previous methods in the literature. Then, the ITZ volume fraction can be determined by the ITZ thickness according to a generalized formula. We also extend these practical ITZ characteristics to estimate the effective diffusivity of cementitious composites by the classical effective medium approximation. The derived results show that the ratio of the apparent ITZ thickness to the actual ITZ thickness is dependent on the sphericity which is a shape descriptor of particles. Moreover, the overestimation degree of ITZ volume fraction and diffusivity of three-phase composites are governed by aggregate volume fraction, particle size distribution and particle shape. Finally, the results reflect that the aggregate shape has a significant effect on the apparent ITZ thickness, and the sampling process of ITZ thickness by using NLS algorithm is similar to the sampling techniques in experiment.

Introduction

The study of the interfacial transition zone (ITZ) between cement paste and inclusion (such as the aggregate in concrete) is a hot topic in the field of cementitious composites [1], [2], [3], [4], [5]. The ITZ is often thought to be the weakest part of ordinary concrete because penetration of deleterious substances occurs mainly through the ITZ under mechanical, physical and chemical actions [6]. To quantify the ITZ properties such as the ITZ thickness and the volume fraction of ITZ [7], [8], [9], no matter by composite mechanic method [10] or numerical modelling techniques [11], [12], it can help us better understand the influence of the ITZ's microstructure on the macro-performance of cementitious composites, such as mechanical properties [13], [14] and transport properties [11], [15], [16], [17]. However, the actual ITZ thickness t is difficult to measure because the constituents of most composites are opaque. Normally, a backscattered electron (BSE) image of polished concrete sample is used to obtain the morphologies of aggregate, hardened cement paste and ITZ regions [1], [18], [19]. Then, a successive strip delineation of concentric expansion near the aggregate is always employed to study the ITZ's microstructure, as shown in Fig. 1(a). Since a cross-section rarely passes through the normal of the aggregate surface, this leads to the apparent ITZ thickness t' larger than the actual ITZ thickness t, as shown in Fig. 1(b).

Many researchers have been trying to survey the overestimation degree between the apparent and the actual ITZ thickness. Stroeven [20] proposed a formula to quantify the overestimation degree of ITZ thickness around a circular aggregate particle. Chen et al. [21] derived analytical solutions for two-dimensional (2D) rectangular and elliptical aggregates. Later, Chen et al. [3] developed a generalized analytical formula for the overestimation of the interface thickness around convex-shaped grains, as given in Eq. (1).

$$t^{\prime }=2\left(1-k_{\mathit{invalid}}\right)\left(t+\frac{2\pi B_1}{S_1}{t}^2+\frac{4\pi }{3S_1}{t}^3\right)$$

where B₁ and S₁ respectively represent the mean caliper diameter and surface area of a particle, k_invalid is an invalid coefficient which represents the ratio of the volume (area in 2D) of the invalid region to the total volume (area) of the ITZ region, as shown in Fig. 1(b).

The details of k_invalid and the invalid coefficient corresponding to spherical particle can be found in literature [3]. However, it is very difficult to determine the invalid coefficient for non-spherical particle. Recently, based on a process which is similar to the Minkowski sum in mathematical morphology [22], [23], Chen et al. [24] proposed a methodology to perfectly construct the ITZ layer surrounding five kinds of Platonic particles [25]. They further extended the invalid coefficient from sphere to Platonic particles by using a systematic line sampling (SLS) algorithm. In the SLS algorithm, both sampling planes and sampling lines are examined in their study. Taking an arbitrary sampling plane in Fig. 2 as an example, the normal of the sampling plane is determined based on its spatial angle (θ_i, ϕ_j), and the spacing ∆ L_3D between two adjacent sampling planes is used to control the density of the sampling planes. For a specified sampling plane, the sampling lines traversing the entire sampling plane in all orientations are determined by the scanning angle step Δα and the spacing ∆ L_2D between two adjacent sampling lines.

It can be seen in Fig. 2 that a series of parallel sampling lines are oriented in all potential directions in the sampling plane rather than in the normal direction of the aggregate boundary. Moreover, in experiment, the profiles of strip delineation as shown in Fig. 1(a) are always arranged parallel to the outer boundary of aggregate in the given section plane. In other words, this means that the measurement of ITZ thickness can be thought as the distance along the normal of the aggregate boundary. So, the statistical mean value of the apparent ITZ thickness in the SLS algorithm is not consistent with the conventional experiment result by BSE image analysis, because the sampling line in the SLS algorithm is not perpendicular to the aggregate boundary.

Therefore, further studies are still imperative to accurately evaluate the ITZ thickness along the normal of aggregate boundary in an arbitrary section plane. In this study, a normal line sampling (NLS) algorithm is proposed to measure the ITZ thickness. Firstly, we determine the line sampling rule of NLS algorithm for Platonic particles, and the apparent ITZ thickness which is obtained by NLS algorithm for a single particle is discussed. Secondly, the influence of particle shape and particle size distribution (PSD) on the overestimation degree of the interface thickness is concerned. Finally, the overestimation of ITZ volume fraction and diffusivity of concrete are surveyed. This study can clearly indicate that the sampling process of ITZ thickness by using the NLS algorithm is much closer to the experimental situation than that by using the SLS algorithm. Meanwhile, the differences between NLS algorithm and SLS algorithm are investigated in this work.