Mix proportion optimization and early strength development in modified foam concrete: an experimental study

The influence of the single polycarboxylate superplasticizer (PCE) and blending it mixed hydroxypropyl methylcellulose (HPMC) on hardening moulding quality and surface pulverisation of foam concrete was investigated. An orthogonal experimental design was employed to determine the optimum combination of parameters for four property indexes (PIs) in this paper. A multi-index matrix analysis method was used to evaluate the parameter combinations and obtain the overall optimal performance for the PIs. The effect of calcium formate (CaF) on the early compressive strength of modified foam concrete with the most optimal combination in different density grades was also studied. The results indicate that the incomplete cement hydration reaction is the essential cause of pulverisation, which can be alleviated by adding a suitable PCE. Defoaming and precipitation occur when the PCE incorporated exceeds 0.1%, which can be mitigated by the addition of 0.02 to 0.06% HPMC. The orthogonal analysis indicates that the anti-cracking agent has a more substantial effect on the strength of the foam concrete than the thickening agent. The most significant factor of mechanical properties is PP fiber followed by dispersible latex powder (DLP). The optimal combination of foam concrete is 0.06% HPMC, 0.3% DLP, and 0.5% PP fiber. The early compressive strength of foam concrete can be significantly enhanced by increasing the CaF content. However, increasing the density level results in a decrease in the 28-day compressive strength of the foam concrete.


Introduction
Foam concrete, conventionally having a dry density of 300 to 1840 kg·m −3 [1], is a lightweight inorganic thermal insulating material with plenty of closed foams obtained by replacing lightweight aggregate with preprepared foams into the cement paste [2,3]. The elevated-density foam concrete can be used to mould load-bearing components like buffering pads or retaining walls, while low-density (less than 800 kg·m −3 ) foam concrete is usually employed for thermal insulation and noise absorption [4]. Compared with conventional organic thermal insulating materials, such as rigid polyurethane foam (RPUF), expanded polystyrene (EPS) concrete, and extruded polystyrene (XPS) concrete, foam concrete displays better fireproof and waterproof performance [5], and hence gains popularity in the construction industry.
The cellular structure and poor strength of foam concrete limit its extensive use in load-bearing components, rendering it a preferred non-structural thermal insulating material. [6,7]. The compressive strength of foam concrete is one of the leading mechanical properties affected by density [8,9]. It shows a positive correlation between density and compressive strength [10]. Additionally, the mechanical properties and preparation of foam concrete are influenced by a multitude of factors, including but not limited to the type, age, and porosity of the cementitious material [11,12], as well as the water-cement ratio, foaming agent type, curing method, and filler type [13,14]. The compressive strength of foam concrete is mainly derived from the strength of the pore walls [15], which leads to evident stress concentration and reduced strength. The irregular pore structure gravely affected foam concrete's mechanical properties. In addition, low-density foam concrete is superior in insulating, conserving internal heat and producing significant thermal hydration. The result is a large amount of heat generating temperature stress. The porous structure leads to severe evaporation and water loss, and the synergistic effect of temperature stress will result in shrinkage and cracks during the curing process after formation. Therefore, it is a critical method to increase the compressive strength of foam concrete to control the form quality and reduce the generation of cracks.
Studies on the stability of foam concrete have pointed out that densities less than 500 kg·m −3 are more prone to instability. The buoyancy force is proportional to the bubble size and increases at lower densities. When the buoyancy force had overcome the surrounding foam concrete, bubbles displaced the surrounding solids and raised to the surface [16,17]. Under a specific density, a higher water-cement ratio (w/c) will decrease the relative viscosity, and a weaker bubble-maintaining capacity in cement paste [18], foams more easily combine into larger ones and rise to the paste surface, causing an unstable state. Therefore, the rational water-cement ratio and admixture can effectively control the size and distribution of bubbles in concrete [19,20]. Previous researches [21][22][23][24]has demonstrated that hydroxypropyl methylcellulose (HPMC) is a polymer containing both polar and surface-active groups [25], which can regulate the viscosity of the slurry [26], diminish the seepage of foam liquid in foam concrete, and effectively accumulate liquid in the film to enhance toughness and resistance of foam bubbles [27]. It exerts an excellent dual effect on stabilizing foam and preventing segregation, which is conducive to developing slurry strength.
The use of an anti-cracking agent is also an effective means to improve the compressive strength of foamed concrete, such as fiber (polyolefin, polypropylene, polyvinyl alcohol, glass, polyamide, and carbon [28]) could improve the compressive strength of foam concrete by preventing micro-cracks and improving the shear behaviour of foam, the toughness, and flexibility of foam concrete are further improved [29]. Moreover, adding fiber has been proven to reduce dry shrinkage in foam concrete, as fiber retain moisture and delay evaporation, reducing dry shrinkage. The shrinkage is affected by the percentage of fiber, the higher the percentage, the higher the shrinkage reduction [30]. Beyond a certain amount of fiber (5%), the strength also will decrease considerably [31,32].
So far, few scholars have comprehensively studied the mechanism of pulverisation of foam concrete and its effect on compressive strength. The effects of polycarboxylate superplasticizer (PCE) and HPMC on surface pulverisation, defoaming, and material sedimentation are discussed. The reason and prevention measures of pulverisation of foam concrete are revealed by SEM techniques in this study. The mechanism of different mineral admixtures on mitigating the surface pulverisation and cement paste sedimentation of foam concrete was explored, as well as the type and optimal dosage admixtures to solve the above problems was given. Based on computer image analysis technology, a model be calculated by the ratio of major and minor average axial for pores distribution and roundness was proposed. In addition, an orthogonal experimental design was used and a comparison group was established. The effects of thickening agent, anti-cracking agent, and PP fiber on compressive strength, flexural strength, flexural ratio, and density of foam concrete at fixed water-cement ratio were investigated the matrix analysis was used to obtain the optimal mixing proportion. Moreover, the early strength effect of calcium formate (CaF) on the modified foam concrete with the most optimal combination was investigated. This achievement has the potential to positively contribute to advanced high-strength foam concrete.

Materials and methods
2.1. Cementitious materials 2.1.1. Cement Ordinary PO 42.5 Portland cement produced by Jidong Cement Co., Ltd. was used for the experimental tests. The physical properties and main oxide composition are presented in tables 1 and 3, respectively. XRD pattern of Portland cement is shown in figure 1(a).

Fly ash
The fly ash employed in the experiments was Level-I, produced by Henan Hougang Thermal Power Plant Co., Ltd. Its main components and chemical composition are listed in tables 2 and 3, respectively. XRD pattern of fly ash is shown in figure 1(b).

Silica fume
The activity index of silica fumes is mainly determined by the 7d rapid determination method. It is obtained by measuring and calculating the compressive strength of the samples with and without silica fume at 7d curing age  [33]. Three reference and three tested mortars were prepared according to the cement mortar mixing proportions in table 4. The dimension of the mortar specimens was´40mm 40mm 160mm. Before demoulding, the specimens were cured in an environment with a temperature of 20 ± 2°C and a humidity of 95%. After demoulding, the specimens were placed in a curing chamber. After curing in an environment with a temperature of 65 ± 2°C for 6d, they were taken out and cooled to room temperature for compressive strength test. The activity index of silica fumes should be calculated from equation (1).
where A is the activity index of silica fume; R t is the 7 day-compressive strength of tested cement mortar, expressed in MPa; R 0 is the 7 day-compressive strength of reference mortars cement mortar, expressed in MPa. Silica fume employed in experiments was Sichuan Langtian Silicon Power Manufacturing Plant Co., Ltd., the leading performance indicator (see table 5).

Additional materials
The water used in the experiments was common tap water. The water reducer was PCE, the water reducing ratio was 19% for the cement mortar and 32% for concrete, with PCE equal to 0.2% of the weight of the cement    [34]. The foaming agent-to-water ratio was maintained at 1:20, resulting in a foam multiplicity of 33 to 34. The settling distance after one hour was measured at 7.5 mm, while the water secretion rate over the same period was found to be 55.3 ml. Finally, the half and all foam extinction times were determined to be 147min and 275min, respectively. The CaF, manufactured by Tongfarong Industrial Co., Ltd. in Zigong City, was utilized as a hardening accelerator with a specific gravity of 2.023 and bulk density of 950 kg·m −3 . The anti-cracking agent employed was dispersible latex powder (DLP) with an average diameter of 80 μm and a viscosity of 4.0 Pa·s for its 50% aqueous solution. The thickening agent utilized was 100k-unit HPMC, with a specific gravity of 1.21 and bulk density of 470 kg·m −3 . Its aqueous solution viscosity measured at 200,000 Pa·s. The fiber employed in this study was Polypropylene (PP) fiber, featuring an average length of 12 mm and diameter of 18 μm, possessing a density of 910 kg·m −3 as well as tensile strength and Yong Modulus values measuring at 486 MPa and 3.8 GPa, respectively.

Mixing procedure and specimen preparation
The preparation process of the sample (see figure 2) consists of four steps. First, the accurately weighed cement, fly ash, silica fumes, HPMC and DLP were poured into a horizontal blender and mixed at 40 rpm without water for 180 s. In the second step, the PP fiber was rubbed into scattered bulks, added into the blender, and mixed for 90 s to mix it with other dry materials and avoid caking completely. In the third stage, the accurately weighed water, PCE and CaF power (It was used in the early strength testing stage) were added into the blender and mixed with other materials for 120 s to generate an evenly-blended cement paste. Meanwhile, the foaming agent was diluted with water at a ratio of 1:40. The foams produced by the air compressor were added to the cement paste and stirred for 120 s. The paste was poured into the mould when the materials were evenly blended, and no floating foams were left on the surface. After the surface had been scraped, the test specimen was placed for 24h before being deformed. The demolded specimens were put into a curing chamber with the temperature at 20 ± 2°C and the relative humidity at 95% for 28 days.

Mechanical properties 2.4.1. Compressive strength test
The compressive strength (f cu ) of the foam concrete was measured by a microcomputer-controlled electrohydraulic servo compression testing machine under a loading rate of 1500 N/s [34][35][36], the test setup is illustrated (see figure 3(a)).  The test setup is shown (see figure 3(b)). It should be calculated by equation (2).
where R f is the value of flexural strength, expressed in MPa; F is the failure load of the specimen, expressed in N; L is the width of the specimen, expressed in mm; B is the distance between the supporting cylinders at both ends, expressed in mm.

Compressive and flexural ratio
Compressive and flexural ratio (T ) is compressive strength to flexural strength (see equation (3).

Dry density test
The foam concrete's dry density and moisture content were detected according to the Chinese Foam Concrete standard JG/T 266 [34].

Preparation and moulding quality test
Foam concrete with design densities of 400 kg·m −3 , 500 kg·m −3 , 600 kg·m −3 , and 700 kg·m −3 was prepared in the control group without any PCE. Another group with a density of 600 kg m −3 was prepared to examine moulding quality, the PCE was set at 0.1% and 0.2%, and the HPMC is set to 0.02%, 0.04%, 0.06%, and 0.08% in this group, respectively, to study the influence on the hardening moulding quality of 600-density foam concrete containing 0.1% of PCE. The percentage of PCE and HPMC depends on the mass of the cement.

Orthogonal design of experiment
The orthogonal test is an effective method for permuting multi-factor and multi-level tests. A number of representative and typical points from the overall test are selected according to orthogonality to perform the test. The consideration factors are set to four levels, and the L 16 (4 3 ) orthogonal table is used for the test design. The test factors and levels (see table 6). A control group of specimens without any additives was established for comparison. Table 7 shows the proportions of the reference foam concrete.

Hardening accelerator test
The effect of the hardening accelerator on the early compressive strength of the modified foam concrete, the mixing proportion used in this paper, was based on the best primary mixing proportion obtained in previous orthogonal tests. CaF was set in 0.5%, 1%, 1.5%, 2% and 2.5% of the mass of all cement materials, respectively. In addition, a control group without CaF was set in each group.

Range analysis
Range analysis is an intuitive analysis method to determine the different levels of the same factor and the influence of each factor on the experimental indicators, b ij is the average of the test results of the factor i at the level j and is calculated as follows: , is the m calculation result of factor i and level j; n is the number of calculation results under factor i and level j. r i represents the difference between the maximum and minimum values b ij at each level, which can be obtained by equation (5).  Notes: RFC-Reference foam concrete; The water was set at 33% (excluding the water in the foams) in the paper; the density of the foam in this research was 29 kg·m −3 , and the amount of foam incorporated was 1.18 m 3 .

ANOVA
ANOVA is a statistical method applied to experimental results that divides the effect of factors and the experimental error. F-value indicates the ratio of each factor's mean square (MS) to that of the experimental error and is analysed by an F-test. The sum of square deviation (SS) reflects the differences in experimental results caused by changes in every level of each factor or error and is calculated as follows: where x i is the value of the results of each trial, and x is the arithmetic average x .
i MS of each factor is SS divided by the degree of freedom (DF) (see equation (7)).
The critical value of F-value ( a F ) for a different significance can be found in the F-statistic distribution table. When > a= F F , 0.01 the change in the factor level is highly significant to the index and is marked as  the corresponding factor has a significant effect on the index and is marked as * . When < a= F F , 0.10 the corresponding factor has an insignificant effect on the index and is marked as d [38].

Results and discussion
3.1. The effect of PCE and HPMC on foam concrete 3.1.1. Effect of PCE on foam concrete In the specimen preparation of the group without PCE, the paste exhibits excessive viscosity. The moulding quality of foam concrete is shown in figure 4, where it can be observed that specimens with different dry densities (a)-(d) have compressive strengths of 0.56 MPa, 0.44 MPa, 0.34 MPa and 0.25 MPa, respectively. It is evident that all specimens in this group exhibit severe surface pulverisation and insufficient strength due to their loose overall structure. Additionally, with the density increases, the specimen shows more pronounced tendency towards perforation, accompanied by an enlargement of pore diameter and a deterioration of cellular structure, rendering it unlikely to attain the required strength. The microstructure of foam concrete with pulverisation is shown in figures 5(a) and (b) is the microstructure of foam concrete without pulverisation.
As shown in figure 5(a), there are a large number of cracks and connected irregular holes in the pulverised foam concrete. The structure of the hardened pore walls is not dense. The hydration products of the pore wall are mainly composed of a large number of layered CH and a tiny amount of fibrous AFt. Some unhydrated cement particles are also scattered on the surface of the hydration products, which reduces the strength of foam concrete at the macro scale. It is evident from figure 5(b) that there are almost no cracks and holes in the microscopic section of the foam concrete without pulverisation. The pore wall structure is dense and the C-S-H gel in the matrix is significantly increased. The bonding surfaces of these gels overlap and inlay each other, tightly wrapping various hydration products to form a dense continuous phase, and the matrix is smoother and denser.
It can be concluded that there is little free water in the flocculation structure formed by the cement paste without PCE, and that the foam paste exhibits poor workability, making it challenging to pour into the mould. Due to the large physical size of the incorporated foam, it cannot efficiently penetrate inside the flocculation structure, resulting in lubrication effects. Additionally, evaporation of free water in the foam paste leads to precipitation of combined water involved in cement hydration reaction. Incomplete cement hydration results in less formation of C-S-H gel and a dense interwoven structure cannot be established with AFt, leading to macroscopic pulverization and the descent in compressive strength [39]. Thus, incomplete cement hydration is a crucial factor in causing concrete pulverisation. The rapid evaporation of free water during the curing phase, combined with the elevated total porosity and significant proportion of connected pores, can also result in the proliferation of foam concrete.
The profile of the sample block with 0.1% PCE added is shown in figure 6(a). Region I has extra-large pores and Region II has dense minute pores. Figure 6(b) is a binary image of the profile of foam concrete, and the average pore size of the three different positions in the height direction is not consistent, and the higher the height, the larger the pore size. Furthermore, the area of 5 ×100 mm for the foam concrete's binarized image was selected, as shown in figure 6(b). In order to calculate the total number of pores and the average size of pore diameter in each tiny region, the concepts of the average major and minor axis are introduced, as shown in figure 6(c). In the statistical data in this region, the larger the value of axial length, the larger the relative size of holes, and the smaller the difference between the major and minor axis, indicating that the shape of holes is closer to a regular circle. Through image recognition and analysis of figure 6(b), the distribution of the average pore size and the total number of pores in the height direction of the selected area is plotted as shown in figure 6(d). In the range of 65-70 mm in the height direction, the maximum average major and minor axis are 646.95 μm, and 497.77 μm, respectively, there are 29 pores in this range. The minimum average major and minor axis are 319.66 μm and 219.15 μm in the range of 15-20 mm in the height direction, respectively, and there are 103 pores in this range. It can be seen from figure 6(d) that in each micro-unit, the total number of pores are distributed differently in the height direction, and the distribution trend of the average major axis and minor axis are not vertical distribution lines. Therefore, it can be argued that the addition of 0.1% PCE causes the foam concrete to exhibit features with varying pore sizes in the height direction.
As can be seen from figure 7(a), when the proportion of the added PCE reaches 0.2%, plenty of connected pores with huge diameters are distributed in region I, the shape of the holes is irregular, and there is no fixed shape. In Region II, severe paste sedimentation occurred and almost all of the foam overflowed, creating an extremely dense concrete precipitation zone. Figure 7(b) shows that serious thick flakes fall off occurred during the compressive strength test. The foam concrete could not remain intact when a slight load was applied, which had a more significant unfavourable impact on the compressive strength test results. It can be seen from figure 7(c) that there are numerous connected walls of air pores on the outer surface of the foam concrete, which connect the pores of the foam concrete, and the partially enlarged view of the broken surface also shows that the pore walls are incomplete, tiny pores penetrate through the pore walls, thereby forming related areas. The connected walls of the air pores are also a significant cause of the reduced compressive strength of the foam concrete.
Compared to the group without PCE, it can be argued that the inclusion of PCE improves the workability of the foam paste and provides excellent convenience for pouring foam concrete. When the paste is stirred, the PCE releases free water from the flocculated paste to participate in lubrication. As PCE increases, the foam concrete test blocks have varying degrees of paste sedimentation and bubbles floating up. In general, the stability of foam is determined by several parameters in the air at room temperature [34].
Since the densities of the foams prepared in this study are significantly different from those of air, water, and fresh cement paste, the trends of external forces and spatial position motion of foams in different media are quite different. In an air medium, the foam tends to move downward in the direction of its own gravity, while in water, the foam tends to move upward in the direction of buoyancy towards the upper surface of the liquid. However, for foam concrete, the density of fresh cement paste is greatly higher than the density of air, and because of its viscous resistance, the foams in the cement paste medium received a more complex resultant effect, and finally reached a stable equilibrium state of force system, the spatial location distribution is more uniform. Figure 8  shows a schematic diagram of the forces acting on the foam in different media, which must be in equilibrium for the foam to remain stable.
According to figure 8(c), the mixing process of cement paste and prefabricated foam is the process of cement particles attaching to foam wall from cement paste. In the process, Firstly, the foam at the gas-liquid interface is transformed into the foam cement paste at the gas-liquid-solid interface, then the solid-liquid interface and the liquid-gas interface disappear, and finally a fresh solid-gas interface is generated during the curing stage [16]. Therefore, the stability of foam in foam concrete depends on the stability of the gas-liquid interface and the stability of the equilibrium state of the foam from mixing to the initial setting of cement paste [22], which is also the key to the preparation of foam concrete with uniform pore distribution.
Obviously, if the foam stability parameter is kept constant, any factor that affects the equilibrium of the force system of the foam in the fresh cement paste will have a very adverse effect on the distribution of the foam. The paste sedimentation in figure 5 is caused by the breakdown of the foam equilibrium. When the foam is mixed with the cement paste, the foam is separated by the cement paste. Adsorption of cement particles on the surface of prefabricated foam can form a solid-liquid-gas phase (three-phase foam), which is beneficial to increase the liquid film strength [40], although the cement particles attached to the foam surface have hydrated after mixing, the strength evolution of the hydration products is still low at this stage, and the strength of the foam liquid film is still the primary source of support in the foam system [22]. The water-cement ratio is the main factor affecting the fluidity and viscosity of cement paste [18]. The PCE in this paper increases the fluidity of the paste and decreases the viscosity and consistency. The cement paste between the foams is thick when the fluidity of the fresh cement paste is low, which hinders the deformation and size change of the foams. When the excess PCE increases the fluidity, the cement paste between the foams becomes thinner due to the self-weight, and the strength of the pore walls cannot resist the pressure inside the foams, causing the foams to rupture and merge into more giant foams. In addition, the viscous resistance of cement paste to the foam decreases correspondingly, the buoyancy of the cement paste is greatly greater than the resultant force of gravity of foam and the resistance of cement paste to foam upward overflow, eventually, destroying the balance of the force system of the foam. Moreover, the cement paste due to its relatively high liquidity will flow downward, filling the gaps at the bottom caused by the escape of the foam, leading to more severe defoaming and severe paste sedimentation. It may therefore be argued that sedimentation may be avoided by a suitable increase in the strength of the pore walls and the consistency of the paste.

Effect of HPMC on foam concrete
Different content of HPMC on the hardened forming test blocks of foam concrete containing 0.1% of PCE with a design density of 600 kg·m −3 is shown (see figure 9). Figures 9(a)-(c) shows no defoaming or sedimentation when the proportion of HPMC added reaches 0.02% to 0.06%, the foams featuring even distribution and similar diameters. Figure 9 (d) shows that the paste's viscosity increases when the content of HPMC reaches 0.08%, and it is difficult to cast into the mould. No defoaming or sedimentation occurred, but there was slight perforation and an uneven distribution of foam in the test blocks. Figure 10 illustrates the data on pore number and circularity in the foamed concrete profile at varying HPMC concentrations. It can be inferred that when HPMC content ranges from 0.02% to 0.06%, pore distribution along the height direction is more uniform, with a greater resemblance to standard circles. Conversely, at an HPMC concentration of 0.08%, there are significant fluctuations in both gastric distribution and shape.
Relevant studies have shown that HPMC has a high-efficiency thickening effect in concrete [41], and it has the function of stabilizing foams, which is realized by surface tension reduction and covering layer formation on the surface of foam liquid film [42]. A proper amount of HPMC increases the consistency and cohesion of the foam paste, decelerates the coalescence and disproportionation of the foams by adsorbing at the air pore surface and increasing the viscosity of cell wall paste [43], which in time limits the floating of the foam and avoids the defoaming of the paste. Additionally, HPMC has superior water retention [44], which reduces the evaporation of free water in the foam paste, and indirectly provides sufficient water for the whole hydration reaction of the cement, thus avoiding the generation of pulverisation of the foam concrete. However, the excessive HPMC will combine with the free water released by the PCE in the paste, which will considerably reduce the fluidity of the foam paste and eventually lead to the occurrence of pulverisation, which is consistent with the results of the previous analysis. Therefore, in the research of this article, it is believed that 0.02%-0.06% of HPMC can improve the foam distribution and appearance quality of foamed foam concrete, but excessive HPMC (0.08%) harms the forming of foam concrete.

ANOVA for property indexes of foam concrete 3.2.1. Test results of mechanical properties of foam concrete
PIs of all groups in terms of cube compressive strength, flexural strength, compressive /flexural strength ratio, and dry cube density of orthogonal test foam concrete are shown (see table 8). All the values listed are the averages of the test results from three specimens. Figure 11 depicts a comparison of the cube compressive strength and flexural strength for all the specimens.
The Range analysis results are listed in table 9. Table 10 shows the ANOVA results of four PIs of the foam concrete. C s V of all PIs in this paper are smaller than 10%, and the qualities of experiments are all general levels. Each factor's different level trend graphs for pour PIs are plotted (see figure 12).

Compressive strength
As shown in figure 12(a), all the factors show an overall trend of increasing first and then declining to indicate that a reasonable factor mass rate improves cube compressive strength. This is because an appropriate amount of HPMC helps to improve the consistency, viscosity, and super water retention. It can be evenly dispersed in the  cement paste. However, excessive HPMC reduces the fluidity of the foam paste and increases its density and viscosity, making it difficult to blend and leading to an uneven distribution of the foam. The effect of M D on the compressive strength of the cube is not apparent, the broad range of variation is within 3%, and the DLP is not a significant factor. By adding an appropriate amount of PP fiber, it can be evenly dispersed in the concrete and form a specific bonding force with the mortar, resulting in a certain crack resistance, and the non-directional support system formed by the fiber can also bear the load together with the concrete.
When the mixture contains excessive fiber, there is a dispersion problem of the fiber in the fresh cement paste, which leads to the aggregation of PP fiber. A large amount of cement paste is required to cover the fiber when the fiber are in the stirring process, which reduces the fluidity of mixture [45], more internal defects will be formed in the concrete [46], thereby adversely affecting the compactness of matrix. On the other hand, as the fiber content increases, a portion of the cement will be replaced by the increased fiber, and the hydration product C-S-H gel will be inferior. Meanwhile, the strength of fiber reinforced concrete has been demonstrated to depend on the strength of cement matrix and the interfacial transitional zone (ITZ) between cement paste and fiber [47][48][49]. PP fiber is a hydrocarbon polymer material, it results in the formation of a water film at the interface of the fiber and matrix, higher water to cement ratio (w/c) and larger space in the ITZ between fiber and fresh cement paste, which served to produce the larger CH crystal plate. While, due to the presence of crystal

Group
Factor PI Notes: ' * ' represents the most desired results of each PI. In this experiment, the compressive and flexural strength is as high as possible, the ratio of compressive /flexural strength and density are as small as possible.
rich layers of CH, the contact point between C-S-H gel and fiber decreased [50,51], which negatively affected the bonding with cement matrix. Finally, the compressive strength of foam concrete is reduced. According to the ANOVA results for the cube compressive strength of foam concrete in table 10, PP fiber is a highly significant factor for the cube compressive strength, HPMC is a significant factor, and DLP is a negligible factor.

Flexural Strength
As shown in figure 12(b), the M H on the flexural strength of foam concrete is not obvious, and the broad range of changes is within 3%. As the M D and M F increase, the flexural strength of the foam concrete increases by 18.4% and 16.5%, respectively. Because DLP has the effect of filling cracks as the foam concrete hardens and contracts, it reduces the creation of microcracks and thus increases the strength of the pore walls. On the other hand, PP fiber will form a non-directional tensile skeleton inside the foam concrete, the greater the PP fiber mass rate, the denser the tensile skeleton, and the more significant the hindrance to the separation of the concrete components, especially the tension zone of the flexural foam concrete specimen. According to the ANOVA results for the flexural strength of foam concrete in table 10, DLP and PP fiber are highly significant factors for the flexural strength. HPMC is a negligible factor.

Compressive/Flexural strength ratio
As shown in figure 12(c), when the M H increases from 0.02% to 0.04%, the T increases close to 4.0%, while the M H increases from 0.04% to 0.08%, the T is reduced by 8.9%, when the M D increase from 0.075% to 0.3%, the T decreases 14.8%. When the M F increase from 0.2% to 0.4%, the T is not apparent, while the M F increases from 0.4% to 0.5%, and the T decreases to 13.3%. Overall, the influence of the three factors on the compressive/ flexural strength ratio of foam concrete shows a downward trend. The results show that these three factors positively affect the reduction of the compressive/flexural strength ratio and the increase of the toughness of foam concrete.
According to the ANOVA results for the T in table 10, DLP is a significant factor in the compressive/flexural strength ratio. The PP fiber is a significant factor, and the HPMC is a negligible factor.

Dry density
PP fiber has the most significant impact on the dry cube density of foam concrete (see table 9). It can be extracted from figure 12(d) when the M H increases from 0.02% to 0.08%, the dry cube density decreases by 7.4%. The main reason is that the HPMC is greatly effective in increasing viscosity and reserving water, increasing the paste's denseness and viscosity, which pre-empts the foams from floating and avoids defoaming and sedimentation. The prepared foam concrete has pores with a uniform size distribution, which reduces the dry density. This is consistent with the previous conclusion.
On the other hand, the DLP increases the fluidity of the paste, thereby reducing the ability of the material to wrap foams, causing merge and overflow, and the density of the foam concrete increases. When the M F increases from 0.2% to 0.3%, the dry cube density decreases by 3.0%, while the M F increases from 0.3% to 0.5%, and the dry cube density increases by 8.5%. A tiny amount of PP fiber replaces part of the paste, thereby reducing the density of the foamed concrete. However, excessive PP fiber causes the consistency of the paste to increase during the mixing process, the fluidity is reduced, and the phenomenon of fiber agglomeration occurs. The cell walls of the foamed concrete are destroyed, increasing the number of communicating holes in the foam concrete, the gas escapes to form irregular holes or even collapsed holes, and the density increases. On the other hand, DLP increases the fluidity of the paste, thereby reducing the ability of the material to wrap the foam, causing coalescence and overflow, and increasing the density of the foam concrete. When the M F increases from 0.2% to 0.3%, the dry cube density decreases by 3.0%, while when the M F increases from 0.3% to 0.5% the dry cube density increases by 8.5%. A tiny amount of PP fiber replaces part of the paste, thereby reducing the density of the foam concrete. However, the excess of PP fiber causes the coherence of the paste to increase during the mixing process, the fluidity is reduced, and the phenomenon of fiber agglomeration occurs. The cell walls of the foam concrete are destroyed, increasing the number of communicating holes in the foam concrete, the gas escapes to form irregular holes or even collapsed holes, and the density increases.
According to the ANOVA results of the dry cube density (see table 10), PP fiber and HPMC are highly significant factors for the dry cube density, and DLP is an insignificant factor.

Multi-index orthogonal test matrix analysis of foam concrete
The matrix analysis method is introduced to optimise the multi-index orthogonal test design of foam concrete. A three-layer structure model and a four-layer matrix with indicators, factors, and horizontal layers were established. The solution of the index value weight matrix is given. The detailed procedure is listed as follows.

Hierarchical structure model
This paper shows a hierarchical structure model composed of indicators, factors, and levels (see figure 13) for the orthogonal test of three factors and three levels.

Analysis and evaluation matrix of each layer
According to the single-index orthogonal test results, the indicators layer, factor and level layer analysis, and evaluation matrix of each mechanical performance have been set up. B ij is an arithmetic mean of the test results obtained when the factor i at the j level, and when the test evaluation index pursues the more significant, the better, the =

Weight matrix of evaluation indicators
The determination of the weight matrix of each indicator is the focus of the comprehensive optimization of the matrix analysis and multi-indicators. Usually, for four indicators and 12 test groups, the results under the indicator m are transformed into a weight matrix w . m The total weight matrix W of multi-indicators is the average value of the weight matrix of the four evaluation indicators. The calculation of w m and W are shown in equation (11) and (12).
Similarly, the single indicator weight matrix w R f of flexural strength, the w f R cu f of compressive /flexural strength, and the w r of dry density can be calculated as follows.
The four levels of factor A have the largest weighting on the test results compared to A 3 . Similarly, the weights of factors B and C are affected by B 4 and C 4 as the largest (see equation (20)). Accordingly, the multi-index orthogonality of the foam concrete can be determined. The optimal combination tested is found to be A 3 B 4 C 4 , where C being the most significant factor affecting the comprehensive evaluation index of mechanical properties, followed by B, and A having the smallest effect. It can be concluded that the combination of anticracking agents has a prominent effect on improving the mechanical properties of foam concrete, with PP fiber showing the most pronounced enhancement. In contrast, thickening agents have a relatively weak effect. The reason is that PP fiber reduces the creation of microcracks during the curing phase of foam concrete and dramatically increases its toughness through its reinforcement effect. This is consistent with the findings in literature [29][30][31].

Early compressive strength of modified foam concrete
Based on the analytical results of the multi-index matrix orthogonality experiments, the optimal combination is determined in this paper. Based on that, the modified foam concrete test blocks with the dry design density of 550 kg·m −3 , 700 kg·m −3 , and 850 kg·m −3 were prepared, and the early compressive strength of each group at different curing ages were measured (see figure 14). Figure 15 shows the growth trend of the compressive strength of foam concrete for three design densities. It can be seen from figures 15(a)-(c) that the CaF increases from 0.5% to 2.5%, all test specimens with different curing ages showed a rapid growth trend, and the compressive strength of 21 days later increased more slowly. The presence of CaF barely affected the compressive strength enhancement at the 3-day curing age compared to the control group without CaF. However, it significantly affects the increase in compressive strength at days 7 and 14. After 14 days, the growth trend of the compressive strength of the foam concrete tends to moderate as CaF increases. Figure 15(d) shows that the 28-day compressive strength of foam concrete with a design density of 550 kg·m −3 achieves a maximum value of 5.34 MPa when the CaF is 1.5%. The 28-day compressive strength of the foam concrete with the design density is 700 kg·m −3 , and 850 kg·m −3 obtained the maximum value of 8.14 MPa and 11.68 MPa when the CaF was 1.0%. While CaF can promote the growth of compressive strength at early times, excessive CaF directly leads to a decrease in the final compressive strength of the foam concrete.
The compressive strength growth rate in foam concrete is defined as the percentage of compressive strength at various ages achieved to the final 28-day compressive strength. Figure 16 shows the growth rate of the compressive strength of the foam concrete for the different curing ages.
From figures 16(a)-(d), when the CaF is increased to 2.5%, it is compared with the group without any CaF, the growth rate of the 3-day compressive strength increased by 2.78%, 3.34%, and 7.69%, respectively. Furthermore, the 7-day compressive strength growth rate increased by 22.11%, 23.84%, and 25.78%, respectively. The 14-day compressive strength growth rate increased by 8.69%, 8.89%, and 11.02%. Moreover, the 21-day compressive strength growth rate increased by 3.56%, 4.66%, and 5.65%, respectively. The aforementioned results indicate that the compressive strength growth rate of foamed concrete with identical curing age escalates as CaF content increases, with CaF being the most significant contributor to compressive strength growth rate at 7 days and followed by 14 days. The 21-day compressive strength growth rate gradually slowed down, consistent with the previous results. On the other hand, the higher the design density of foam concrete with the same CaF content, the more development of the early compressive strength at different curing ages.
It can be explained that the CaF can increase the concentration of Ca 2+ in the liquid phase in the early curing age, accelerate the dissolution rate of calcium silicate, and the ion effect will speed up the crystallisation, increase the solid phase ratio of the mortar, which is conducive to the formation of cement stone structure. On the other hand, HCOO − ions have a much higher diffusion rate than Ca 2+ , and can penetrate the hydration layer covering the C 3 S and C 2 S grains. Consequently, it accelerates the precipitation of Ca(OH) 2 and the decomposition of calcium silicate [52]. Therefore, the higher the density of the foam concrete, the more cement is contained in the foam paste, and the more pronounced the strengthening effect of CaF.

Conclusions
Preparation and mould quality testing of the foam concrete specimens is accomplished in this paper. The orthogonal test was designed to analyse the effect of the addition of HPMC, DLP, and PP fiber on the four PIs of the foam concrete. The optimal proportion was identified by the multi-index orthogonal matrix analysis. The influence of the CaF on the early compressive strength of optimised foam concrete was studied. The following conclusions have been drawn.
(1)The PCE significantly affects the forming of foam concrete. When PCE is not incorporated, the foam paste becomes less workable and inconvenient for pouring, leading to severe surface pulverization. Additionally, it becomes challenging to reach the design requirements for compressive strength due to insufficient cement hydration reaction. However, increasing PCE content can mitigate surface pulverization.
(2)If the PCE concentration reaches 0.1%, defoaming and sedimentation will occur, which becomes more severe at a concentration of 0.2%. This results in uneven density distribution between the top and bottom of the mixture. However, incorporating 0.02% to 0.06% HPMC into the paste can mitigate these issues and prevent surface pulverization.
(3)The anti-cracking agent exerts a more pronounced influence on the strength of foam concrete than the thickening agent, as it reduces microcracks during curing and reinforces the reinforcement to significantly enhance the toughness of foam concrete, thereby favorably impacting mechanical properties improvement.
(4)The optimal combination of foam concrete is achieved with 0.06% HPMC, 0.3% DLP, and 0.5% PP fiber, resulting in the best comprehensive mechanical properties. Furthermore, PP fiber has the most significant impact on the overall evaluation of mechanical properties, followed by DLP and HPMC with lesser effects.
(5)The early compressive strength of foam concrete can be significantly improved by increasing the CaF content. The growth rate of compressive strength with identical curing age increases as the CaF content increases, and CaF is the most significant contributor to 7-day compressive strength growth rate. The strength is positively correlated with density level. However, excessive CaF leads directly to a reduction in the 28-day compressive strength of foam concrete.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Conflicts of interest
The authors declare that there is no conflict of interest regarding the publication of this paper.