Introduction

For more than two decades, many researchers have focused on carbon nanotubes (CNTs) which have the remarkable Young’s modulus (~1 TPa), strength (~100 GPa) [1–8], chemical/thermal stability, and outstanding flexibility [9–11], motivating their use in composites as a fibrous reinforcing agent. In order to take advantage of the intrinsic properties of individual CNTs, various CNT-reinforced composites [12–14] have been demonstrated to overcome the performance limits of conventional materials. A vast majority of CNT composite studies have focused on polymer matrices [12], while comparatively few investigations have explored inorganic (ceramic and glass) matrices. In the last two decades, there has been growing interest in using CNTs in brittle ceramic monoliths, particularly as toughening agents in these otherwise strong and stiff materials [15–29]. In addition, CNTs may have a significant influence on other properties such as electrical conductivity, thermal expansion coefficient, and hardness [13]. The combination of these properties with the intrinsic advantages of ceramic materials such as high-temperature stability, high corrosion resistance, and light weight makes CNT/ceramic composites to be used as very attractive functional and structural materials for a variety of applications.

Although mechanical properties of CNT-reinforced alumina composites have been widely studied, only a few groups have reported simultaneous strengthening and toughening effects for the composites. Estili et al. reported that the spark plasma-sintered (SPS) MWCNT/alumina composites showed 22 % (413.2 MPa) and 52 % (6.71 MPa m1/2), respectively, simultaneous improvements in bending strength and fracture toughness with 10 vol% MWCNT addition using acid-treatment technique combined with a colloidal-processing method [30]. Zhu et al. reported simultaneous 6.4 % (380 MPa) and 94 % (6.8 MPa m1/2) enhancements, respectively, in bending strength and fracture toughness for a composite containing 4.0 vol% MWCNTs by hot pressing [31]. Similar improvements were also reported by Wei et al. in 3.0 vol% MWCNT-reinforced alumina composite (13 % (410 MPa) in bending strength and 79 % (5.01 MPa m1/2) in fracture toughness) [32]. In addition to the alumina matrix composites, some previous studies have reported CNT/SiO2 [33], CNT/SiC [34], and CNT/MgO [35] composites, and resulting composites have shown the improved mechanical properties. The most recent reviews on these topics were reported by Cho et al. [13] and Zapatas-Solvas et al. [14]. As mentioned above, the bending strength and fracture toughness of the CNT/alumina composites were enhanced by 6–27 and 3–94 % compared with the alumina alone [13]. However, it leaves an open question that whether the composites can be further strengthened and toughened and how far the mechanical properties are increased.

The parameters seemed to influence on the mechanical properties of the composites include CNT contents; dispersibility and mechanical strength of the CNTs; interfacial strengths between CNT and matrix and grain size of the matrix. However, there has been no report of study which has comprehensively investigated the effects of such parameters on the mechanical properties of the composites. In order to make decisions on fundamental material design of composites with CNTs for higher strength and toughness, the understanding of how these parameters influence the mechanical properties is essential.

In this study, alumina composites made with the acid-treated CNTs at up to 3.7 vol% are prepared, and the effects of acid-treatment time on mechanical properties of the composites and the parameters influenced on the mechanical properties including the grain size of the alumina matrix, the dispersibility and the mechanical strength of the CNTs, and the interfacial strength are quantitatively investigated. Dominant factors for the mechanical properties of the composites are identified with the aid of multiple regression analysis.

Experimental procedure

Material preparation

The MWCNT material (supplied from Hodogaya Chemical, Japan) was synthesized by a catalytic chemical vapor deposition method followed by high-temperature annealing. The thermal treatment of the MWCNTs was carried out at a temperature of 2600 °C under argon atmosphere, and the purity was claimed to be 99.5 % by the producer. The pristine MWCNTs were refluxed in 3:1 (volume ratio) concentrated sulfuric acid:nitric acid mixture at 70 °C for various times: 0.5, 1, 2, 3, and 4 h. Then, they were washed thoroughly with deionized water to be acid free, and finally dried in an oven at 60 °C. We have evaluated the influences of the acid-treatment temperature on the composites’ mechanical properties, and demonstrated that the MWCNTs acid treated at 60–80 °C have been more suitable as the strengthening and toughening agents [36].

Mixtures of 0.9, 1.9, and 3.7 vol% MWCNTs and alumina powder were prepared by a precursor method [21–23]. The details of composite preparation can be found in the Ref. [23]. In brief, a certain amount of the pristine MWCNTs or acid-treated MWCNTs were dispersed in 400 cm3 ethanol with the aid of ultrasonic agitation. An aluminum hydroxide (Wako Pure Chemical Industries, Japan) was added to this suspension and ultrasonically agitated. Magnesium hydroxide (Wako Pure Chemical Industries, Japan) was added to prevent excessive crystal growth. The resultant suspension was filtered and dried in an oven at 60 °C. Finally, the product obtained in the previous step was put into a half-quartz tube and was dehydrated at 600 °C for 15 min in argon atmosphere. The composites were prepared by SPS (SPS-1050, Sumitomo Coal Mining, Japan) in vacuum. A graphite die and punchers with 30-mm diameter were used to form samples. The sintering process was carried out under the conditions: 1500 °C, 10 min, and 20 MPa. In order to compare the mechanical properties of the composites, a MWCNT-free alumina sample was prepared under the same processing conditions. X-ray diffraction analyses have been performed in our previous study for the dehydrated product and sintered composite [22], and the phase transformation from aluminum hydroxide to α-alumina via the amorphous phase has been observed.

Characterization

The bending strength was measured by the three-point bending method under an ambient condition, in which the size of the test specimens was 2.0 mm (width) × 3.0 mm (thickness) × 24.0 mm (length). The span length and crosshead speed for the strength tests were 20.0 mm and 0.83 μm s−1 (0.05 mm min−1), respectively. The fracture toughness was measured by the single-edge-notched beam (SENB) method under ambient conditions, in which the size of test specimens was 2.0 mm (width) × 3.0 mm (thickness) × 15.0 mm (length). A notch with depth and width of 0.3 and 0.1 mm, respectively, was introduced in the center part of the test specimen by using a diamond wheel. A span length of 12.0 mm and crosshead speed of 0.83 μm s−1 (0.05 mm min−1) were applied for the toughness tests. All the surfaces of the specimens were finely polished, and the edges were chamfered. Bulk densities of the sintered bodies were measured by the Archimedes method using deionized water as an immersion medium. Theoretical densities were calculated from the rule of mixtures, assuming the following density values: MWCNTs, 2.10 Mg m−3; Al2O3, 3.99 Mg m−3; and MgO 3.65, Mg m−3. The microstructures and nanostructures of the composites were observed using a scanning electron microscope (SEM, Hitachi S-4300, Japan) and TEM (Hitachi HF-2000, Japan). Grain sizes of the alumina matrix were evaluated by using samples that were ground with diamond polishing pastes down to 1-μm particle size and then thermally etched at a temperature of 1400 °C for 1 h in argon atmosphere. The grain sizes were measured in the transverse and longitudinal directions using the line intercept method from 5 SEM images taken at a magnification of ×5000. The dispersion states of MWCNTs in the alumina matrix were quantified by measuring the area of MWCNT agglomerates in 9 SEM images taken at a magnification of ×2000. In order to quantify the dispersibility of the MWCNTs, a dispersion index D was defined and calculated using Eq. (1):

$$ D = \frac{{A_{ 0} - A_{\text{CNT}} }}{{A_{ 0} }}, $$
(1)

where A CNT is the area occupied by MWCNT agglomerates, and A 0 is the total investigated area obtained from the image analysis (=3 × 10−8 m2). According to Eq. (1), the D value of 1 indicates a perfect dispersion of the MWCNT containing no agglomerates, and decreasing dispersibility is reflected in a decreasing the D value. Uniaxial tensile tests of individual MWCNT were carried outwith a manipulator inside the vacuum chamber of SEM (FEI Quanta 600 FEG, US) [7, 8]. Further details of the experimental procedure are described elsewhere [7]. The applied force was calculated from the angle of deflection at the cantilever tip. In this study, nominal (or “engineering”) strength calculated based on the full cross-sectional area including a hollow core was examined, and diameter of the each MWCNT was measured from the TEM images.

Results and discussion

Acid treatment of MWCNTs

Figure 1 shows typical TEM images of pristine MWCNTs used in this research. SEM and TEM observations showed that the average outer diameter, inner diameter and length of the MWCNTs were 61 nm (30–162 nm), 7 nm (2–10 nm) and 8.7 μm (1.1–22.5 μm), respectively. The detailed structural information of the MWCNTs was retrieved from the enlarged TEM images, as shown in Fig. 1b, c. Even though such MWCNTs consist of nested graphitic cylinders that are almost perfectly aligned with the nanotube axis (Fig. 1b), they are observed to possess unevenly spaced lattice fringes and contain local defects such as kinked layers (Fig. 1c).

Fig. 1
figure 1

TEM images of the pristine MWCNTs used in this research

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To disperse the MWCNTs homogeneously in the matrix, we used the acid treatment of the pristine MWCNTs. The rationale behind the acid treatment is to adsorb negatively charged functional groups at the MWCNT ends and along their length. The detailed structural information of the acid-treated MWCNTs was acquired from the TEM images, as shown in Fig. 2. It can be seen that the acid-treated MWCNTs have nanoscale defects on their surface. The nanodefects in these acid-treated MWCNTs had a channel-like structure, as if a ring of material was cut away from the MWCNT around the circumference [23], and the defect density, namely the number of defects per unit of MWCNT surface area increased with the increasing acid-treatment time. In addition to the presence of the nanodefects, peel-off of a few layers in the MWCNT structure was observed for the samples acid treated for 2–4 h, as exemplified in Fig. 2b. The depth of the nanodefects and diameter of the MWCNTs are shown in Fig. 2d against the acid-treatment time. The trends of increasing depths of nanodefects and decreasing MWCNT diameters are observed with the increasing acid-treatment time up to 2 h, and then both sizes level off for the longer treatment times.

Fig. 2
figure 2

TEM images of the MWCNTs acid treated for a, c 2 h, and b 3 h. d Relationship between acid-treatment time and depth of the nanoscale defects and diameter of the MWCNTs

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Next, in order to investigate the effect of the acid treatment on the mechanical strength of the individual MWCNT, the uniaxial tensile tests of single MWCNTs were carried out by means of a nanomanipulator system [7, 8]. By comparing the SEM images of MWCNTs acquired before and after fracture, it has been found that all the nanotubes tested in this study fractured in the so-called sword-in-sheath-type failure mode [7, 8]. Furthermore, the fracture of the acid-treated MWCNTs mostly occurred at the nanodefects. The dependence of the nominal tensile strength of the MWCNTs on the acid-treatment time is shown in Fig. 3a. It should be mentioned that because it was difficult to observe directly the cross-sectional area of the fractured MWCNTs, we could not measure the diameter of the MWCNTs at the fracture position. In this study, the dimension of the net cross sections in the acid-treated MWCNTs was estimated by subtracting the average depth of the nanodefects (Fig. 2d) from the radius of the fractured MWCNTs fixed onto the cantilever tip. As shown in Fig. 3a, the nominal tensile strength of the MWCNTs (σ CNT) acid treated for 2 h shows nearly the same value as that for the pristine MWCNTs. When the acid-treatment time increases further, the σ CNT increases and reaches a plateau value for the longer treatment times of 3 and 4 h. The dependences of the breaking forces and the estimated outer diameters of the MWCNTs on the acid-treatment time are shown in Fig. 3b. Even though there is a significant scatter in the experimental data, the breaking forces of the MWCNTs acid treated for 3 and 4 h appear to be slightly higher than those of the MWCNTs acid treated for 0 and 2 h. On the other hand, the outer diameter is observed to decrease with the increasing acid-treatment time, which probably is due to the peel-off of a few outer layers in the MWCNTs and the introduction of the nanodefects. Thus, the reason of the comparable breaking force may be due to the removal of flows and defects present in the outer layers of the pristine MWCNTs, and the outer layers may have only negligible load-bearing capacity in the MWCNTs. The result shown in Fig. 3 also suggests that the stress concentration due to the nanodefects left may be insignificant. The above discussion may explain the reason why the σ CNT increases with the increasing acid-treatment time.

Fig. 3
figure 3

a Nominal tensile strength of the MWCNTs as a function of acid-treatment time. b Breaking force and estimated outer diameter of the MWCNTs as a function of acid-treatment time

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Composite mechanical properties

The relative densities of the composites are shown in Fig. 4a as a function of the acid-treatment time for the different MWCNT contents. The relative densities of the composites are roughly in the range of 96–98 % (except for the 3.7 vol% pristine MWCNT/alumina composite), and they decrease steadily upon the increase in MWCNT contents in all acid-treatment times. The composites containing the acid-treated MWCNTs tend to have higher density than the composites containing pristine MWCNTs, particularly in the 3.7 vol% MWCNT/alumina composites. This may be attributable to the improved homogeneous distribution of MWCNTs through the acid treatments, as revealed by SEM observations of the fracture surfaces of the composites (describe below in Fig. 6). Figure 4b, c shows the dependences of bending strength and fracture toughness of the composites on the acid-treatment time. The composites containing 0.9 vol% MWCNTs show the highest strength and toughness, whereas further addition of MWCNTs leads to the decreases in the strength and the toughness regardless of the acid-treatment time. Both bending strength and fracture toughness of the composites containing 0.9 and 1.9 vol% MWCNTs increase up to 2 h and then decrease gradually with the increasing acid-treatment time. For the composites with MWCNTs acid treated for 1 h, both composites containing 0.9 and 1.9 vol% MWCNTs show lower bending strength than the composites containing MWCNTs acid treated for 0.5 and 2 h, although no data are available for 1 h in the case of 3.7 vol% MWCNT/alumina composite. A possible reason for this result is discussed later. In the case of the composites containing 3.7 vol% MWCNTs, the mechanical properties increase with the acid-treatment time of 0.5 h and reach a plateau value at the longer acid-treatment time. The bending strength and fracture toughness of the composites are summarized in Table 1. The composite containing 0.9 vol% MWCNTs acid treated for 2 h gives the highest bending strength (689.6 ± 29.1 MPa) and fracture toughness (5.90 ± 0.27 MPa m1/2), respectively. Even though the fracture toughness value of the composite is lower than that of the previous studies [30, 31], the composite shows higher bending strength compared with the results reported in the literature [30–32].

Fig. 4
figure 4

a Relative density, b bending strength, and c fracture toughness of the MWCNT/alumina composites as a function of acid-treatment time

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Table 1 Bending strength and fracture toughness of the MWCNT free alumina and composites
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Quantitative evaluation of composite microstructures

We now discuss the micro- and nanostructures of the MWCNT/alumina composites using SEM and TEM observations. First, we measured the grain size of alumina matrix. The dependence of grain size on the acid-treatment time is shown in Fig. 5. The grain sizes are in the range of 1.0–2.0 μm, and there is no clear correlation between the grain size and acid-treatment time regardless of the MWCNT contents. The composites containing larger amount of the MWCNTs tend to have smaller grains for any acid-treatment time.

Fig. 5
figure 5

Grain size of the alumina matrix as a function of acid-treatment time

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SEM and TEM images shown in Fig. 6 display the morphology of the fracture surface of the composites. The SEM observations show that numerous individual MWCNTs protrude from the fracture surface, suggesting the pullout of the MWCNTs (Fig. 6a, b). Most of the MWCNTs are located in the intergranular phase. A diameter change in the MWCNT structure is observed for a certain percentage of the MWCNTs as exemplified in Fig. 6c regardless of the acid-treatment time, and this morphology is quite similar to the “sword-in-sheath”-type failure as observed in the failure mode of MWCNTs under tensile loading. The protrusion lengths of MWCNTs are plotted in Fig. 7a against the acid-treatment time. Even though some variations in protrusion length are observed, it tends to decrease with the increasing acid-treatment time. This difference may be due to the effect of interfacial strength between MWCNT and alumina matrix. Assuming that all the MWCNTs are pulled out during the crack opening, the relation between the interfacial strength and protrusion length of MWCNTs can be expressed by the following equation under the Kelly–Tyson assumption [37]:

$$ \tau_{i} = \frac{{r\sigma_{\text{CNT}} }}{{L_{\text{c}} }} = \frac{{r\sigma_{\text{CNT}} }}{{4L_{\text{f}} }}, $$
(2)

where r is the radius of the MWCNTs (Fig. 2d), σ CNT is the nominal tensile strength of the MWCNTs (Fig. 3a), L c is the critical length that is the maximum MWCNT length for which the interface shear fracture occurs rather than tensile failure of the MWCNT, and L f is the protrusion length (Fig. 7a) which varies between 0 and L c/2 with a mean value of L c/4 [38]. The calculated interfacial strength is shown in Fig. 7b as a function of acid-treatment time. Here, the interfacial strengths of the composites containing MWCNTs acid treated for 0.5 and 1 h were calculated assuming that nominal tensile strength was 2.1 GPa which was the average value of that of the MWCNTs acid treated for 0 and 2 h. The calculated interfacial strength is in the range of 6.8–25.4 MPa, and it tends to increase with the increasing acid treatment time. Note that such interfacial strengths are expected to be underestimated since the MWCNTs used in this study undergo failure in the sword-in-sheath manner which cannot be observed clearly through the SEM. However, these values are in the range of previous studies [39–41], supporting the validity of this estimation. As mentioned above, the nanodefects are introduced on the surface of the acid-treated MWCNTs, and the density of defects increases with the increasing acid-treatment time. The rough surfaces of the acid-treated MWCNTs are considered to induce the mechanical interlock originated from a nanoscale anchor effect [23] and increase the interfacial strength.

Fig. 6
figure 6

SEM and TEM images of the fracture surface of the alumina composites. Acid-treatment times and MWCNT contents are a 0 h, 3.7 vol%; b 2 h, 3.7 vol%; and c 3 h, 1.9 vol%

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Fig. 7
figure 7

a Protrusion lengths of the MWCNTs and b calculated interfacial strengths as a function of acid-treatment time. The interfacial strengths of the composites containing MWCNTs acid treated for 0.5 and 1 h were calculated, assuming that nominal tensile strengths were 2.1 GPa

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The composites containing 0.9 vol% acid-treated MWCNTs showed a homogeneous MWCNT distribution and trace amounts of agglomerates with the size of a few micrometers. In contrast, in the case of the pristine MWCNTs, a certain number of such agglomerates were observed in addition to the individualized MWCNTs. Even though the agglomerates were observed to increase in size and number with the increasing MWCNT contents regardless of the acid-treatment time, this trend is more noticeable in the pristine MWCNT/alumina composites (Fig. 6a, b). In this study, the dispersion index D defined in the previous section was used to quantify the dispersibility of the MWCNTs in the composites. The dispersion index is shown in Fig. 8 as a function of acid-treatment time. Even though the composites containing acid-treated MWCNTs have higher dispersibility than the composites containing pristine MWCNTs, the value of D decreases with the increasing MWCNT contents. The composites containing 0.9 and 1.9 vol% MWCNTs show that the value of D increases up to 2 h. However, the excessive acid treatment of the MWCNTs results in the decrease in the dispersibility of MWCNTs. In the case of the composites containing 3.7 vol% MWCNTs, with the increasing acid-treatment time from 0 to 0.5 h, the dispersion index markedly increases, and then decreases for the longer treatment time.

Fig. 8
figure 8

Dispersion indices of the MWCNTs as a function of acid-treatment time

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Microstructure–mechanical property relationship

In order to determine the influential parameters on the bending strength and fracture toughness of the composites, the correlation between the mechanical properties and the composite parameters was examined with the aid of multiple regression analysis. A commercial software (Wolfram Mathematica) was used for the multiple regression analysis. The acid-treatment times are viewed as an experimental condition which may affect the dispersion index, MWCNT’s nominal tensile strength, and interfacial strength. Thus, the composite parameters to be considered in the multiple regression analysis were MWCNT content, composite density, dispersion index, grain size, MWCNT tensile nominal strength, and interfacial strength. A full list of the composite parameters examined in this study is summarized in Table 2. The composite parameters listed in the Table 2 are MWCNT content, acid-treatment time, composite density, grain size of the matrix, dispersion index, MWCNT nominal tensile strength, and interfacial strength. Before the multiple regression analysis, examinations for the collinearity were conducted among each composite parameter. Relationships among the composite parameters are shown in Figs. S1, S2, S3, and S4 in Supplementary material. The MWCNT content has the collinearity with the composite density, dispersion index, and matrix grain size (Fig. S1a–S1c). In addition, the composite density and MWCNT nominal tensile strength also correlate with the dispersion index and interfacial strength, respectively (Figs. S2a, S4b). On the other hand, no clear collinearity is observed between the other composite parameters. Based on the examinations, we did not include the MWCNT content, composite density, and MWCNT nominal tensile strength; only dispersion index, grain size of the matrix, and interfacial strength were considered in the multiple regression analysis. To standardize each composite parameter, the difference between the parameter value and its average is divided by the standard deviation for the parameter. In this study, linear approximate equations in bending strength (\( \overline{{\sigma_{\text{b}} }} \)) and fracture toughness (\( \overline{{K_{\text{Ic}} }} \)) are proposed in the first-order approximation, and they are expressed as

$$ \overline{{\sigma_{\text{b}} }} = 0.89\overline{D} - 0.07\overline{d} - 0.03\overline{{\tau_{\text{i}} }} ,\quad R^{ 2} = 0. 7 2 , $$
(3)
$$ \overline{{K_{\text{Ic}} }} = 0.96\overline{D} - 0.10\overline{d} - 0.16\overline{{\tau_{\text{i}} }}, \quad R^{ 2} = 0. 7 2 , $$
(4)

where \( \overline{D} \) is the dispersion index, \( \overline{d} \) is the grain size of the alumina matrix, \( \overline{{\tau_{\text{i}} }} \) is the interfacial strength between MWCNT and alumina, and R 2 is coefficient of determination. As mentioned above, all parameters are standardized in these analyses. Thus, the larger the regression coefficient of the independent variable, the higher its correlation with the dependent variables (\( \overline{{\sigma_{\text{b}} }} \) and \( \overline{{K_{\text{Ic}} }} \)). Based on the analyses, it is clear that the dispersibility of MWCNTs has an overwhelming influence on both bending strength and fracture toughness. In contrast, matrix grain size and interfacial strength have a low impact on the composite mechanical properties compared with the dispersibility of MWCNTs.

Table 2 A full list of the composite parameters examined in this study
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The dependences of the composites’ mechanical properties on the dispersion index and relative density are shown in Figs. 9 and 10, respectively. The relationships between the mechanical properties and the other composite parameters (MWCNT content, grain size of the matrix, MWCNT nominal tensile strength, and interfacial strength) are shown in Figs. S5, S6, S7, and S8 in Supplementary material. As shown in Fig. 9, both bending strength and fracture toughness correlate well with the dispersion index, which is in agreement with the results of the multiple regression analyses as mentioned above. Furthermore, because the MWCNTs agglomerated in the matrix may be similar to that of the pores [22], the composite density depends on the dispersion index (Fig. S2a in Supplementary material), and both bending strength and fracture toughness also correlate with the composite density. It has been shown that the acid treatment employed in this study resulted in the formation of surface carboxylic acid and other groups on the tube surfaces [42]. These kinds of functional groups on MWCNTs dissociate in ethanol and consequently impart negative charge on the MWCNTs. Thus, it can be expected that the larger electrical repulsive forces between nanotubes will facilitate their dispersion and prevent them from tangling and agglomeration, which leads to the improvement of MWCNTs’ dispersibility. Consequently, the homogeneous dispersion of MWCNTs in the alumina matrix suppresses the degradation of composites’ density, which may lead to an improvement in mechanical properties.

Fig. 9
figure 9

Relationship between dispersion index and a bending strength and b fracture toughness of the composites

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Fig. 10
figure 10

Relationship between relative density and a bending strength and b fracture toughness of the composites

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As shown in Fig. 4b, the bending strength of the composites containing MWCNTs acid treated for 1 h shows an irregular variation but gives a lower value than those of the composites containing MWCNTs acid treated for 0.5 and 2 h. In order to examine the reason for this variation, the matrix grain size of the composites containing 1.9 vol% MWCNTs acid treated for 0.5, 1, and 2 h were compared (the dispersion indices of the composites were approximately the same and they were 0.9954, 0.9976 and 0.9982, respectively). As shown in Fig. 5, the composite containing MWCNTs acid treated for 1 h has larger grain size and lower bending strength compared with the composites with the MWCNTs acid treated for 0.5 and 2 h even though it has comparable or better dispersion index than the other two treatment times. The grain growth retardation effect due to the pinning effect of the MWCNTs is expected to improve the bending strength [43]. Thus, the bending strength of the composites prepared in this study may be influenced to some extent by the grain size of the matrix (the reason of the irregular variation in the grain size for the treatment time of 1 h is not clear at the present time).

In addition to the grain growth retardation effect, the energy dissipations by the MWCNT debonding and pullout are also expected to improve the mechanical properties of MWCNT composites [28, 44]. Furthermore, Nozaka et al. reported that tougher composites could be obtained by reducing the interfacial strength so that the MWCNTs pulled out without breakage [45]. As mentioned in the fracture surface observations, MWCNTs undergo failure during crack opening in the composites presumably due to the high interfacial strength (Fig. 6c), which suggests that the contribution of MWCNT pullout is limited. As shown in Eqs. (3) and (4), the regression coefficients of \( \overline{d} \) and \( \overline{{\tau_{\text{i}} }} \) are with minus signs, indicating that with the decreasing matrix grain size and interfacial strength, the mechanical properties increase. This trend is in agreement with the previous studies [43, 45]. The synthesis parameters that influence the dispersibility of MWCNTs may include their flexural rigidity and dimensional parameters (aspect ratio, etc.), in addition to surface modifications such as acid treatment used in this study. In view of the significance of MWCNTs’ dispersibility, further studies should be carried out to examine the effects of the above-mentioned parameters in order to avoid and mitigate the agglomeration of MWCNTs in the matrix.

Conclusions

The alumina composites reinforced with MWCNTs at up to 3.7 vol% were prepared by a precursor method followed by a spark plasma sintering. We systematically and quantitatively investigated the effects of acid-treatment time of the MWCNTs on mechanical properties of the composites, grain size of the alumina matrix, mechanical strength and the dispersibility of the MWCNTs, and interfacial strength between MWCNT and alumina. The main objective of this study is to evaluate the relationships between the above-mentioned composite parameters and mechanical properties of the composites with the aid of multiple regression analyses. We demonstrated that the homogeneous distribution of MWCNTs in the alumina matrix was achieved by the acid treatment, and there was an optimal acid-treatment time for improving both mechanical properties of the composites and dispersibility of the MWCNTs. We also showed that the dispersibility of the MWCNTs correlated well with both bending strength and fracture toughness of the composites. On the other hand, the influences of the other parameters such as the matrix grain size, mechanical strength of the individual MWCNTs, and interfacial strength on the composites’ mechanical properties were small compared with the dispersibility of the MWCNTs. By means of the multiple regression analysis, we quantitatively showed that the MWCNTs’ dispersibility was the most important factor for controlling the mechanical properties of MWCNT/alumina composites. Improved dispersibility of MWCNTs may be useful for the densification of the composites as well as the enhancement of crack-bridging effects by MWCNTs.