Materials and experiments

For the sake of comparison, two cement pastes were prepared at the different water-to-cement mass ratios (w/c) of 0.23 and 0.40. In the specimen preparation process, ASTM type I Portland cement was firstly mechanically mixed with deionized water and then cast into steel molds. After 24 h, the specimens were demolded and cured in moist condition (23 ± 1 °C and 95% relative humidity) prior to testing. Before nanoindentation tests, they were carefully polished to achieve a smooth surface. The polishing procedure can be divided into coarse- and fine polishing processes. Firstly, the small cubes of size approximately 15 mm × 15 mm × 10 mm were cut out at the middle portion. Then, in the coarse polishing process, they were polished carefully with abrasive paper (180, 240, 400, 600, 800, and 1200 grit) in order. After that, oil-based diamond suspensions (3, 1, and 0.25 µm) were used in the fine polishing process. Finally, they were cleaned in an ultrasonic bath to remove debris and suspensions left on the surface of the specimens. Using such polishing procedure, the specimen surface was smooth enough for the nanoindentation tests, as used in the previous work.^[Reference Hu, Han, Gao, Zhang and Li^13]

Instrumented nanoindentation equipped with a three-sided pyramid Berkovich tip was used to investigate the microstructural mechanical properties of the specimens. After nanoindentation was calibrated on standard material (fused quartz), a large number of tests were performed on the polished surface of the specimens. The maximum load used in this study is 2 mN and generates a maximum depth for hydration phases of about 300 nm. The principle of nanoindentation is to push a very sharp tip into the surface of a material and investigate the behavior of the material from the response of indentation tip. Figure 1 shows a typical load–depth curve of nanoindentation test on C–S–H gel, which was generated from the loading history with an initial constantly increasing loading, followed by constant holding and then constantly decreasing loading. Two mechanical parameters can be simultaneously obtained from the initial slope of the elastic unloading in the load–depth curve.^[Reference Pharr and Bolshakov^14] The first one is indentation modulus, M, and the second one is hardness, H, as defined below:

(1)$$M={\left. {\displaystyle{1 \over 2}\left({\displaystyle{{dp} \over {dh}}\sqrt {\displaystyle{\pi \over A}} } \right)} \right\vert _{h={h_{\max }}}}\comma \; $$

(2)$$H={\left. {\left({\displaystyle{p \over A}} \right)} \right\vert _{h={h_{\max }}}}\comma \; $$

where p is the indentation load, h is the indentation depth, and h _max is the maximum indentation depth. A is the projected contact area and can be extrapolated from the indentation depth h through Oliver and Pharr's method.^[Reference Oliver and Pharr^15] The indentation modulus, M, is related to the elastic modulus, E, and Poisson's ratio, v, of the local material through:

(3)$$\displaystyle{1 \over M}=\displaystyle{{1 - {v^2}} \over E}+\displaystyle{{1 - v_{{\rm tip}}^2 } \over {{E_{{\rm tip}}}}}\comma \; $$

where E _tip is the elastic modulus of the indenter tip, and v _tip is Poisson's ratio of the indenter tip. For the diamond tip used in this study, E _tip and v _tip have values 1141 GPa and 0.07, respectively. The hardness is related to the yield strength of the local material, whereas the hardness to yield strength ratio depends on the material type.^[Reference Cheng and Cheng^16] For example, for rigid cohesive plastic solids, Tabor suggested that the hardness-to-yield strength ratio, H/Y = 3.^[Reference Tabor^17] In the work of Cariou et al.,^[Reference Cariou, Ulm and Dormieux^18] for the cohesive-frictional porous materials, the relationship between hardness and yield strength was investigated through the micromechanical approach, where the limit analysis was used to achieve the relationship of the normalized hardness versus the packing density.

Figure 1.

A typical load–depth curve of nanoindentation test.

Results and discussion

As an example, a grid of 400 indent impressions on the w/c = 0.40 hardened cement paste specimen is marked by triangles (enlarged in size for visibility) in the scanning electron microscope–backscattered electron (SEM–BSE) image, as shown in Fig. 2. The residual cement clinker, calcium hydroxide, inner product, and outer product are also indicated in Fig. 2.

Figure 2.

SEM–BSE image of a typical zone showing a grid of N = 400 indent impressions (triangles, enlarged in size for visibility) in the w/c = 0.40 hardened cement paste.

Before the analysis of nanoindentation data, the physically meaningless data stemming from the irregular indentation curves shown in Fig. 3 were eliminated. For instance, Fig. 3(a) shows a flat region at the beginning of the load–depth curve, which was caused by instable contact between the tip and the material surface at the initial loading process, that was a type of irregular appearance. Figure 3(b) shows another irregular appearance, a shoulder appearance (sudden jump or pop-in due to the fracture and collapse of surrounding solids) on the load–depth curve. Physically, the data produced by such irregular curves are meaningless and do not belong to any material phases.

Figure 3.

Irregular load–depth curve representatives due to (a) instable contact between the tip and the material surface at the initial loading process, and (b) displacement sudden jump or pop-in.

To obtain the microstructural mechanical information from grid dataset free of irregular curves, statistical analysis was used to analyze the nanoindentation data. The idea of statistical analysis is briefly introduced here. A large number of nanoindentation tests (i = 1,…,N) were performed on the microstructures of multiphase material composed of n material phases, where each phase occupied a volume fraction, f _j (j = 1,…,n). The basic assumption is that the distribution of the in situ assessed mechanical parameters of each phase can be approximated by the normal distribution or Gaussian distribution, ψ _j, specified with the mean value and SD. The theoretical probability density function (PDF) of nanoindentation data is formulated as:

(4)$$\hbox{PDF}=\sum\limits_{\,j=1}^n {{\,f_j}{\psi _j}} $$

and the volume fractions are constrained by:

(5)$$\sum\limits_{\,j=1}^n {{\,f_j}=1} .$$

Based on the finite Gaussian mixture model, the unknowns were estimated by maximum likelihood estimation through the application of expectation maximization algorithm,^[Reference Dempster, Laird and Rubin^19] and Bayes information criterion (BIC) was used to determine the number of components or phases.^[Reference Schwarz^20] The unknown parameters can be estimated by searching for parameters that maximize the likelihood function, or equivalently, log-likelihood function. BIC was calculated with the prescribed number of phases, n. The larger BIC indicated the stronger the evidence for the existence of material phases in the sample. In this study, the global optimization was searched for by running an iteration procedure for multiple trials with random starting values. Using such analysis method, for the w/c = 0.40 hardened cement paste, the dataset was classified into several groups according to the statistical analysis results. The scatter of indentation modulus and hardness was shown in Fig. 4(a). Accordingly, the map of the identified phases with location of irregular curves is shown in Fig. 4(b). By comparing the map of the identified phases in Fig. 4(b) and the SEM–BSE image in Fig. 2, it was noted that the interfaces of residual cement clinkers were classified into the group of calcium hydroxide. The third phase corresponds to calcium hydroxide or interface of cement clinker, rather than ultra-high-density phase. Thus, for the w/c = 0.40 hardened cement paste, the identified phases were recognized as outer product, inner product, calcium hydroxide (or interface of residual cement clinkers), and residual cement clinkers.

Figure. 4.

Grid nanoindentation on the w/c = 0.40 hardened cement paste: (a) deconvolution of the grid dataset free of irregular curves, (b) map of the identified phases with location of irregular curves.

For the cement paste at low water-to-cement ratio of 0.23, the nanoindentation results are shown in Fig. 5. As indicated in Fig. 5, for the low water-to-cement ratio = 0.23 hardened cement paste, the third phase is mainly the interface of residual cement clinkers rather than ultra-high-density phase.

Figure 5.

Grid nanoindentation on the w/c = 0.23 hardened cement paste: (a) deconvolution of the grid dataset free of irregular curves, (b) map of the identified phases with location of irregular curves.

Finally, for the hardened cement pastes at the different water-to-cement ratios, the nanoindentation results of active individual phases are summarized in Table 1. From Table 1, it was noted that the mechanical properties of the outer product and the inner product present in w/c = 0.23 cement paste are different with those present in w/c = 0.40 cement paste. The previous study using nanoindentation implied that the mechanical properties of C–S–H do not depend on the mix proportions and curing condition, based on which a lot of multi-scale models were proposed to predict the macroscopic engineering performance of concrete and other cementitious materials.^[Reference Constantinides and Ulm^1, Reference Jiang, Zhang, Hu and Li^21] However, as compared in Table 1, the nanoindentation results of this study showed the mechanical properties of hydration products, including outer product and inner product largely depended on the water-to-cement ratio. Meanwhile, it was noted that the average indentation modulus and hardness of residual cement clinkers present in the w/c = 0.40 cement paste are lower than those present in the w/c = 0.23 cement paste, and much lower than those of synthetic cement clinkers reported in Velez et al.^[Reference Velez, Maximilien, Damidot, Fantozzi and Sorrentino^22] The difference can be explained by the effect of porosity, as the similar phenomenon was also noted by Velez et al.^[Reference Velez, Maximilien, Damidot, Fantozzi and Sorrentino^22] using nanoindentation to study the mechanical properties of synthetic cement clinkers.

Table I.

The average nanoindentation modulus M, hardness H (mean ± SD) and volume fraction f of active phases involved in hardened cement pastes

Concluding remarks

Through this study, the mechanical properties of active phases involved in hardened cement pastes at different water-to-cement mass ratios were measured and mapped through the application of grid nanoindentation and statistical analysis. The results obtained from this work suggested that nanoindentation can be used as an effective tool to measure and map mechanical properties of cement paste and can identify the phases include outer product, inner product, calcium hydroxide (or interface of residual cement clinkers), and residual cement clinkers. Meanwhile, the results showed that the mechanical properties of hydration products largely depended on the water-to-cement ratio.