A parametric study was conducted to explore the effect of the interfacial transition zone (ITZ or interphase) on the overall elastic modulus of the CNTreinforced cement. The effect of the ITZ properties on the elastic modulus of the CNTreinforced cement was investigated using a fourphase axisymmetric model consisting of a single CNT aligned at the center of composite unit cell, an interface, an ITZ (or interphase), and a cement matrix. The CNT and cement matrix were assumed fully elastic while the interface was modeled using a cohesive surface framework. The width and mechanical properties of the ITZ and the interface were found to affect significantly the elastic modulus and the behavior of the composite material.
Due to their excellent physical, mechanical, thermal, and electrical properties, carbon nanotubes (CNTs) are able to produce composites with outstanding properties. For example, single walled CNTs have a modulus of elasticity of approximately 1 TPa (i.e., slightly lower than that of diamond), a yield strength between 20 and 60 GPa, thermal conductivity almost two times that of diamond [
The utilization of CNTs has attracted researchers in all scientific disciplines. For example, CNTs were used as sensors in human and medical science [
Previous experimental results have showed marginal improvement in the properties of CNTreinforced cementitious materials. On the other hand, several theoretical studies have predicted higher enhancement levels. The difference between the experimental and theoretical findings may be attributed to (1) a poor dispersion of carbon nanotubes, (2) a weak interfacial bonding between the matrix and the CNTs, and (3) an agglomeration and/or defect of CNTs.
The region surrounding the CNT has a significant influence on the mechanical properties of the bulk composite. Thus, the properties of this region need to be included in the modeling of CNTreinforced cementitious materials. The region is characterized by an interfacial transition zone (ITZ or interphase) and an interface. The ITZ or interphase region is defined as the layer of finite thickness and distinct mechanical properties between the CNT/cement interface and the cement matrix. On the other hand, the interface is defined as the cohesive surface with zero thickness lying between the CNT and the interphase.
A limited number of experimental studies have been conducted to investigate the interfacial properties in CNTreinforced cementitious materials as compared to those performed on CNT/polymer composites [
The effect of the interphase, or ITZ, on the mechanical properties of CNTreinforced cementitious materials is also significant. For example, it is widely accepted that the properties of the ITZ in concrete (without fibers) need to be considered for a better estimation of the overall mechanical properties. However, very little research has been done to characterize and provide quantitative measurements of the interphase. Interphase widths of less than 100 nm with quantitative mechanical measurements at each position have rarely been reported in the literature. This is due to the lack of experimental techniques that can measure the mechanical properties in nanoscale spatial resolution (Nair et al. [
Researchers have recently used Atomic Force Microscopy (AFM), nanoindentation, and nanoscratching to investigate the interphase in fiber/polymer composites. Williams et al. [
Nair et al. [
Pakravan et al. [
This study represents a portion of the work done by the authors to investigate the elasticity of nanocomposite cement paste. In this paper, a fourphase axisymmetric unit cell model has been developed for a single aligned CNT embedded in the cement matrix with appropriate boundary conditions. The model has been used to conduct a parametric study to explore the effect of the mechanical properties of the ITZ (or interphase) and those of the interface on the elastic response of CNTreinforced cementitious materials. However, the fiber waviness and random distribution within the matrix are the main features that limit their effectiveness and may result in composite with mechanical properties lower than composite with aligned fibers. Another study should be directed towards investigating the effect of fiber waviness and distribution on the elastic modulus of the nanocomposite cement paste using 3D and axisymmetric models assuming different distributions of fibers (uniformly or randomly distributed straight and curved CNTs). A novel methodology should be followed to simulate the 3D random distribution using the axisymmetric models and compare the results. As the focus of this study is to investigate the effect of the properties of the ITZ on the overall elastic behavior of the nanocomposite cement paste, the simple axisymmetric model with one CNT aligned at the center of the matrix is used in this study without considering the orientation and random distribution of the fibers. The effects of random distribution and waviness of CNTs on the elastic properties of the nanocomposite cement are presented in Abu Taqa et al. [
An axisymmetric model has been developed to investigate the effect of ITZ on the overall elastic modulus of the CNTreinforced cement paste. Several models were found in the literature to simulate the behavior of CNTs as space frame [
Mechanical properties of the cement matrix and fibers.
Elastic modulus ( 
Poisson’s ratio ( 


GPa  N/(nm)^{2}  
Cement matrix  30 

0.20 
CNT  1000 

0.35 
The representative volume element (RVE) of the CNTreinforced cement paste was first divided into unit cells having hexagonal cross section and containing a single carbon nanotube in its center. Then, the hexagonal cross section was replaced by an equivalent circular cross section to form an axisymmetric unit cell. In the study, the axisymmetric model consisted of four phases, namely, CNT, interface, interfacial transition zone (ITZ or interphase), and cement matrix as shown in Figure
Fourphase axisymmetric model.
The unit cell length was taken equal to
The distribution of the ITZ (or interphase) around the CNT was assumed to be uniform (i.e.,
Different mesh sizes were investigated to check the sensitivity of the analysis results to the meshing size. A 1 nm × 1 nm mesh size generated a sufficient accuracy.
The cohesive surface framework used to describe the CNT/interphase interface was based on the tractionseparation behavior, which allows for both normal and tangential separations. It can be modeled in Abaqus software as a cohesive interaction because (1) the interface thickness is very small (i.e., approaching zero) and (2) the overall behavior of the composite is investigated rather than the exact stresses within the interface. As the displacement across the cohesive interface increases, the interface traction first increases until it reaches a maximum and then decreases to zero, if the friction between the CNT and the matrix is ignored.
The available tractionseparation model in Abaqus assumes an initial linear elastic behavior followed by the initiation and evolution of damage. Once a damage initiation criterion is met, a material damage can occur according to a specified damage evolution law.
The elastic behavior is written in terms of an elastic constitutive matrix that relates the nominal stresses to the nominal strains across the interface as follows:
If an uncoupled behavior between the normal and shear components is desired (as considered in the calculations here),
When the stresses and/or strains at the interface point satisfy the specified damage initiation criteria, the process of degradation of the response of that point begins. Several damage initiation criteria are available in Abaqus. However, the one used in this work depends on the maximum nominal stress criteria which assumes that the damage initiates when the maximum nominal stress/traction ratio reaches a value of one. This criterion can be represented using the following equation:
The rate at which the material stiffness is degraded once the corresponding initiation criterion is reached is described by the damage evolution law. The overall damage in the material is represented by a scalar damage variable,
In Abaqus, the damage evolution can be described based on the displacement at failure or based on the fracture energy dissipated as a result of the damage process. The damage softening can be linear, exponential, or in tabular form. Linear damage softening based on the failure displacement is used herein in which the damage evolution variable
In this work, the friction between the CNT and the cement matrix after debonding is ignored because the surface of the pristine CNT is very smooth. All stresses in the normal direction are ignored. Hence, the deformation is assumed to be purely tangential.
As previously noted, the interphase in fiberreinforced composite is a transition region, which extends from nanometers to microns. The mechanical properties change from the bulk properties of the fiber to those of the matrix in the interphase. In carbon nanofiberreinforced polymer composites, nanoindentation and AFM were used to determine the property gradient such as the reduced elastic modulus within the interphase. It is widely accepted that the interphase has a heterogeneous nature. However, no attempts were done to estimate the interphase thickness relative to the fiber diameter or fiber volume fraction within the polymer. Also, no specific distribution of the interphase elastic modulus was reported in the literature (i.e., linear distribution, nonuniform, heterogeneous, etc.). Studying the ITZ in the CNTreinforced cementitious materials is not an easy task due to the brittle nature of the cement paste. Limited research work was reported in the literature. Hence, a parametric study was carried out to investigate the effect of the interphase volume fraction relative to the cement matrix assuming a uniform distribution around the CNT. Since the precise spatial variation in the interphase is not known, the distribution of the elastic modulus within the interphase varied. The following distributions have been considered: (1) a uniform distribution with a specific average elastic modulus value, (2) a nonuniform linear distribution (increasing or decreasing) assuming the interphase has the modulus of the matrix material at the interphase/matrix boundary and has a specified modulus value at the interphase/CNT boundary, and (3) stochastic elastic modulus values within the interphase which follow a normal distribution. The later distribution may be the most realistic one.
No effort has been put yet towards investigating the ITZ mechanical properties and width in the nanocomposite cement paste. The new techniques, which are used to investigate the properties of the ITZ in composites (such as nanoindentation and AFM), are not widely utilized in the nanocomposite cement paste due to the challenges associated with investigating this phase and/or locating the nanotubes in brittle material such as cement paste. On the other hand, from economical (time and cost) point of view, performing straightforward macromicroexperimental measurements on a number of samples of various particle sizes, types, shapes, volume fractions, and dispersions/distributions under different loading conditions is hardly a feasible task. However, at the present time, it is still not possible to perform atomistic or molecular dynamics simulations (MD) on realistic time scales and structures. Therefore, in this study, a fourphase axisymmetric model was developed to carry out a parametric study considering all possible ITZ volume fractions (widths) and elastic modulus distributions within the ITZ to explore the effect of those parameters on the overall elastic properties of the composite. The results of this study can be used as a guide for the experimental program that should be adopted to investigate the ITZ properties according to the degree of impact of each parameter on the overall behavior.
Four ITZ volume fractions were investigated, namely,
Average elastic modulus versus ITZ volume fraction: (a) uniform
The following four assumptions for the distribution of the ITZ elastic modulus were investigated:
A uniform distribution with an average elastic modulus value equal to 75% of that of the cement matrix; that is,
A nonuniform linear distribution with
Nonuniform linear distribution with
Stochastic normal distribution with an average elastic modulus value equal to 75% of that of the cement matrix (
As aforementioned, a limited number of attempts to investigate the interfacial properties in nanocomposite cementitious materials could be found in the literature. The interfacial shear strength values between the steel fiberreinforced concrete (SFRC) and cementbased matrices are in the range from 1.4 MPa to 9.6 MPa as reported by Naaman et al. [
In this parametric study, interfacial shear strength
The effect of the ITZ volume fraction and the mechanical properties of the interface (modeled as cohesive surface) on the longitudinal elastic modulus were studied. The effect of the friction between the CNT and cement matrix after debonding was ignored herein. The purely tangential deformation was also not considered herein. The CNT volume fraction and the aspect ratio were fixed to
Tables
Elastic modulus for different interfacial strengths and ITZ volume fractions (case 1: uniform ITZ elastic modulus = 22.5 GPa).
Interfacial shear strength  ITZ volume fraction (%)  

0 (no ITZ)  1  10  50  99.5  
Perfectly bonded  34.312  34.226  33.578  30.641  26.709 

34.193  34.111  33.464  30.522  26.605 

34.033  33.956  33.309  30.351  26.46 

33.598  33.524  32.871  29.885  26.047 

30.836  30.76  30.073  26.959  23.375 
Elastic modulus for different interfacial strengths and different ITZ volume fractions (case 2: increasing ITZ elastic modulus from 15.0 GPa to 30.0 GPa, average of 22.5 GPa).
Interfacial shear strength  ITZ volume fraction (%)  

0 (no ITZ)  1  10  50  99.5  
Perfectly bonded  34.312  34.250  33.762  31.737  29.385 

34.193  34.135  33.652  31.615  29.295 

34.033  33.980  33.505  31.446  29.161 

33.598  33.548  33.075  30.971  28.738 

30.836  30.785  30.291  28.186  25.960 
Elastic modulus for different interfacial strengths and different ITZ volume fractions (case 3: decreasing ITZ elastic modulus from 60.0 GPa to 30.0 GPa, average of 45.0 GPa).
Interfacial shear strength  ITZ volume fraction (%)  

0 (no ITZ)  1  10  50  99.5  
Perfectly bonded  34.312  34.413  35.316  38.796  43.795 

34.193  34.293  35.195  38.671  43.649 

34.033  34.129  35.032  38.517  43.478 

33.598  33.693  34.607  38.137  43.055 

30.836  30.937  31.909  35.716  40.365 
Elastic modulus for different interfacial strengths and different ITZ volume fractions (case 4: stochastic elastic modulus within ITZ with average of 22.5 GPa and standard deviation of 3.75 GPa).
Interfacial shear strength  ITZ volume fraction (%)  

0 (no ITZ)  1  10  50  99.5  
Perfectly bonded  34.312  34.22  33.663  31.396  28.415 

34.193  34.106  33.554  31.279  28.323 

34.033  33.952  33.406  31.124  28.189 

33.598  33.521  32.975  30.666  27.775 

30.836  30.758  30.185  27.77  25.041 
Figures
The results show that assuming an interfacial shear strength of 20 MPa gives elastic modulus values, which are very close to those of the perfectly bonded case regardless of the distribution of the elastic modulus within the ITZ. Hence, a shear strength value of 20 MPa at the CNT/cement interface has a considerable effect on the mechanical properties of the composite.
Figures
In Figure
As noted previously, the cracking and plasticity of the matrix including the ITZ were not taken into account herein because the work focuses on the elastic response of the composite. This may cause further reduction in the composite elastic modulus when “cracked” in the elastic range.
Figures
Plots of the average elastic modulus versus ITZ volume fraction: (a) perfectly bonded interface, (b) interfacial strength = 20 MPa, (c) interfacial strength = 10 MPa, (d) interfacial strength = 5 MPa, and (e) interfacial strength = 1 MPa.
Figure
Let us first investigate the results of the perfectly bonded case and compare them to the upper and lower limits of the theory of mixtures for composite materials. It is worth noting that the theory of mixtures considers that the CNT and the matrix are springs acting on parallel and on series. Let us consider the case where the ITZ volume fraction was assumed equal to 99.5%. This case corresponds to the entire matrix being interphase with a uniform average
Let us consider the case where the ITZ volume fraction was assumed to be 1% with a uniform average
Applying the same calculation procedure to the case where the elastic modulus is increasing linearly within the ITZ, the composite elastic modulus was found to be equal to
As noted above, the assumption of the spatial distribution of the mechanical properties within the ITZ is essential and has a considerable effect on the elastic response of the composite. Hence, nanoindentation needs to be used to study the mechanical properties within this layer in order to be able to have a valid computational finite element model with real assumptions.
Moreover, the use of stochastic normally distributed
The cohesive surface was assumed to be between the CNT and the ITZ in all previous models. However, it was observed from the pullout test carried out for CNTs embedded in the cement matrix that part of the hydration components were attached to the surface of pulledout CNTs. Hence, the insertion of a cohesive surface at the boundary between the ITZ and the cement paste needs to be investigated.
Table
Elastic modulus for different locations of cohesive surface (50% ITZ volume fraction).
Interfacial shear strength  Distribution of elastic modulus within ITZ  

Uniform, 
Increasing, average 
Stochastic, average  
Location of cohesive surface  
Between CNT and ITZ  Between ITZ and cement  Between CNT and ITZ  Between ITZ and cement  Between CNT and ITZ  Between ITZ and cement  
IS = 20 Mpa  30.522  27.732  31.615  28.733  31.279  28.384 
IS = 10 Mpa  30.351  27.693  31.446  28.648  31.124  28.314 
IS = 5 Mpa  29.885  27.418  30.971  28.241  30.666  27.958 
IS = 1 Mpa  26.959  22.676  28.186  22.854  27.77  22.792 
The results summarized in Table
The paper studied the effect of ITZ and interfacial properties on the elastic properties of the CNTreinforced cement paste using a fourphase axisymmetric model. The following conclusions could be drawn.
For the same ITZ volume fraction and the same distribution of the mechanical properties within the ITZ, increasing the interface strength increases the elastic modulus of the nanocomposite until it reaches the maximum value, which corresponds to the perfectly bonded case.
The results also show that assuming an interface strength value of 20 MPa generates elastic modulus values which are very close to the perfectly bonded case. Hence, achieving this shear strength value at the CNT/cement interface has a considerable effect on the mechanical properties of the composite.
Assuming that the ITZ is weaker than the cement paste, the elastic modulus of the composite decreases by increasing the volume fraction (thickness) of the ITZ for a specific interface strength (regardless of the distribution of the elastic modulus within the ITZ). On the contrary, increasing the width of the ITZ increases the overall elastic modulus of the composite if the ITZ is assumed to be stronger than the cement matrix (elastic modulus is linearly increasing within the ITZ).
Generally, considering more than 50% as volume fraction of a “weaker” ITZ results in a composite elastic modulus less than the matrix elastic modulus. Hence, it is important to investigate the mechanical properties and width of this layer in order to study the effect of the CNTs on the matrix properties.
The increasing or decreasing trends of the composite elastic modulus due to an increase of the width of the ITZ is almost the same for all interface strengths.
The effect of the distribution of the elastic modulus within the ITZ becomes more significant as the width of the ITZ increases.
Assuming a uniform distribution within the ITZ gives elastic modulus values less than those obtained when assuming a linearly increasing trend within the ITZ. The difference is larger for wider ITZs.
The assumption of normally distributed
The location of the cohesive surface significantly affects the elastic modulus of the composite.
Future experimental work may be directed towards exploring the elastic properties of the nanocomposite cement paste using the static compression test. The thickness of the ITZ in the CNTreinforced cement paste and its relation to other parameters may be also investigated using the AFM technology, for example. Moreover, nanoindentation may be carried out to study the local nanomechanical properties of the nanocomposite cement paste and within the ITZ in order to be able to have valid computational finite elements model with real assumptions.
The statements made herein are solely the responsibility of the authors.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was made possible by a National Priority Research Program award (NPRP 411422440) from the Qatar National Research Fund (a member of The Qatar Foundation).