Optimization of properties and structure of lightweight materials using digital methods

Original article

Бессонов И.В., Жуков А.Д., Боброва Е.Ю., Горбунова Э.А., Говряков И.С. Оптимизация свойств и структуры особолегких материалов с применением цифровых методов. Нанотехнологии в строительстве. 2025;17(2):109-118. – EDN: WXZIJJ.

Lightweight materials (LWM) are used in the process of implementing systems both related to thermal insulation of building structures and in acoustic systems: soundabsorbing or sound-insulating. Materials of this group can be divided into two main subgroups: LWM based on foamed polymers and LWM based on mineral components. The first subgroup includes polystyrene foam, extruded polystyrene foam, polyurethane foam, polyisocyanurate foam, and polyethylene foam. Products based on foamed plastics are produced in the form of slabs (less often in the form of half-cylinders or segments). Polyethylene foam is produced in the form of mats or rolls.

A special feature of products made of polyethylene foam is the possibility of manufacturing seamless insulating shells [1–3]. The second group includes mineral fiber and cellular mineral materials: foam glass, cold-curing cellular glass, as well as foam gypsum and heat-insulating cellular concrete. Foam glass undergoes a pyroplastic state during the formation of its structure and is used to manufacture either individual products (blocks, slabs, shaped pieces) or crushed stone [4–6]. Materials based on mineral binders harden under natural conditions, and therefore the manufacture of products based on them is a less energy-intensive process [7–9].

In general, the basis of the properties of lightweight materials are their structure, determined by porosity,

CONSTRUCTION MATERIALS SCIENCE which can reach 98% for foamed plastics and 82–86% for mineral-based materials. Cellular structure materials are of the greatest scientific and practical interest. Firstly, these are foamed plastics (polystyrene foam and polyolefin foam) obtained by extrusion; secondly, their noncombustible analogues: foam glass, cellular glass, foam gypsum, and heat-insulating cellular concrete.

There are five main types of structure (four of which are shown in Fig. 1, and a mixed one in Fig. 3): closedcell or closed; open-cell or open; mixed; integral; and syntactic. Closed porosity is characteristic of extruded foam plastics; mixed porosity is characteristic of mineral lightweight materials [2, 5].

The common characteristics of these materials are a high degree of cellular porosity and, consequently, low average density. The average density of lightweight materials should not exceed 400 kg/m³. Such properties of LWM as thermal conductivity, water absorption, sound absorption and sound insulation are associated with low average density and structural features. The main and fundamental difference is the flammability of materials based on foamed polymers (with the release of toxic substances in some cases) and the non-flammability of LWM based on a formed mineral matrix.

Digital methods based on digital modelling and the use of statistical methods to plan experiments and process their results are useful for both studying technological processes and solving prescription problems. It is worth paying attention to the appeal of digital methods for optimizing and interpreting the results of experiments. In particular, methods of analytical optimization and the formation of models using methods of differential analysis and vector algebra [10–12].

The aim of the research presented in the article was to study the influence of technological factors on the properties of foam gypsum, as well as on the features of the formation of its structure and optimization of its compositions. The basis of the methodology was digital meth-

Fig. 1. Porosity structure of lightweight materials: а – closed-cell; b – open-pore; c – integral; d – syntactic

CONSTRUCTION MATERIALS SCIENCE ods of planning, processing results, and their analytical optimization.

METHODS AND MATERIALS

Short setting times and low water resistance significantly limit the possibilities of using gypsum-containing materials. The directions for optimizing the properties of gypsum binders are the introduction of mineral additives containing components that exhibit pozzolanic activity, as well as the modification of gypsum stones with polymerizing substances. To make LWM from gypsum or modified gypsum binders, one must mix the mineral matrix with stable foams or fill it with gas [13, 14]. This procedure opens up the pores in the material. The experiment was designed based on statistical methods for planning and processing its results. The following were adopted as variable factors: the consumption of dihydrate gypsum (X1); the polymerizing component (X2); and the consumption of an aqueous solution of the foaming agent (X3). The consumption of semi-hydrate gypsum, the main component of the gypsum binder, remained constant in the experiments. The conditions for preparing foam and foamed gypsum were equal in all cases: foam multiplicity, stability, and viability of the freshly prepared foamed mass. In particular, the foam multiplicity, stability and viability of the freshly prepared foam mass were constant. Part of the research was carried out on the equipment of the Center for Collective Use (CCU) NRU MSUCE.

• У ₁, as a response function, provides the strength of the foam gypsum samples after 7 days of hardening. The average density of the foam gypsum is given by У₂. The softening coefficient of the foam gypsum samples obtained from climate tests is given by У₃. The experimental conditions are given in Table 1.

The Statistika program was used to process the results of the experiments. Statistical hypotheses were also tested, and confidence intervals were found to see how important the regression equation coefficients were. In the end, regression equations were made for the foam gypsum’s compressive strength (У1), its average density (У2), and its softening coefficient (У3). Confidence intervals were de- termined for strength ∆b1 = 0.02 MPa, for average density ∆b2 = 4 kg/m³, and for softening coefficient ∆b3 = 0.008. The obtained models were tested for adequacy according to the Fisher criterion. They found that the Fisher criteria values for the compressive strength model F1 = 16.2, the average density model F2 = 15.9, and the softening coefficient model F3 = 15.1 are the same as the values in the table. This means that the models are correct.

The following mathematical models (polynomials) were obtained:

– for compressive strength

• У ₁= 0,69 + 0,05Х₁+ 0,04Х₂– 0,08Х₃– 0,03Х₁²; (1)

  – for medium density:

• У ₂= 282 + 28Х₁+ 11Х₂– 81Х₃+ 9Х₁Х₂;         (2)

  – for the softening coefficient:

• У ₃= 0,34 + 0,01Х₁+ 0,04Х₂– 0,02Х₃+ 0,01Х₁Х₂. (3)

RESULTS AND DISCUSSION

We looked at the equation coefficients for a constant amount of semi-aqueous gypsum use and found that the amount of foaming agent used (coefficient at X3 is minus 0.08) has the most significant effect on the compression strength of foam gypsum (У1), making it weaker. The effect of the consumption of dihydrate gypsum is not unambiguous, which is emphasized by the coefficients at factor X1 = 0.05 and at factor X12 is minus 0.03. This result enables further use of the analytical optimization method. Increasing the consumption of the polymerizing component in the intervals provided by the experimental conditions contributes to an increase in strength (coefficient at X2 equal to 0.04).

One of the most important factors that affects the average density of foam gypsum (У2) is how much of the foaming agent is used (the coefficient at X3 is minus 81); other factors and how they interact with each other have a smaller effect. The increase in the softening coefficient (У3) is largely determined by the increase in the consumption of the polymerizing component (the coefficient at X2 equal to 0.04); the influence of the consumption of

Table 1. Experimental conditions

Factor	Symbol Хi	Average value of the factor, Xi	Variation interval, ΔХi	Factor values at levels
				–1	+1
Consumption of dihydrate gypsum Cg, kg/m³	Х1	60	20	40	80
Consumption of polymerizing component Cp, kg/m³	Х2	8	2	6	10
Consumption of aqueous solution of foaming agent Cfa, kg/m³	Х3	80	20	60	100

CONSTRUCTION MATERIALS SCIENCE dihydrate gypsum and the aqueous consumption of the foaming agent is manifested to a lesser extent.

In the context of further implementation of the digital methodology, the analytical optimization method is used and tested in the study of various processes for building materials production and solving prescription problems [1, 2, 9, 11]. This method consists of the sequential implementation of the following stages: determination of the optimal value of the factor X1 in formalized and natural form; obtaining optimized functions (1–3), implementation of interpretative solutions, and verification of the reliability of the obtained results.

• 1)    Determining the optimal value of factor X₁

∂У1/Х1 = 0.05 – 0.06Х1 = 0 → Х1 = 0.05/0.06 = 0.83.

By using the information in Table 1, we can figure out the best way to use dihydrate gypsum. Pg = 60 + 20 × 0.83 = 76.6 kg/m³.

• 2)    We perform optimization of functions (1-3) under their conditions X₁ = 0.83:

• – For compressive strength

• У ₁ = 0,69 + 0,05×(0,83) + 0,04Х₂ –

  – 0,08Х₃– 0,03×(0,83)².

  – For medium density:

У2 = 282 + 28×(0,83)+ 11Х2– 81Х3+ 9×(0,83)Х2.

– For the softening coefficient:

• У ₃= 0,35 + 0,01×(0,83)+ 0,04Х₂ – 0,02Х₃ +

+ 0,01×(0,83)Х2.

As a result, we obtained the following optimized dependencies:

– for compressive strength

• У 1= 0,71 + 0,04Х2– 0,08Х3.(4)

  – For medium density:

• У 2= 305 + 18Х2– 81Х3.(5)

  – For the softening coefficient:

• У 3= 0,35 + 0,05Х2– 0,02Х3.(6)

• 3)    We implement interpretive solutions

We interpret the results by constructing graphical dependencies based on optimized models (4–6) and then combining these graphs into a nomogram.

The nomogram (Fig. 2) includes three sectors. In sector I, the relationship between the average density of foam gypsum and the consumption of the polymerizing component, as well as the aqueous solution of the foaming agent, is established with an optimized consumption of dihydrate gypsum equal to 76.6 kg/m³. In the second sector, the dependence of the compressive strength on the same factors is established. In the third sector, the dependence of the softening coefficient on the same factors is established.

Fig. 2 shows the implementation of the direct problem of digital modelling, which involves predicting a material’s properties based on specified values of variable factors, with red lines. The order of implementing the direct problem is as follows. We set (at the optimal value of the consumption of dihydrate gypsum equal to 76.6 kg/m³) the consumption of the polymerizing component and the aqueous solution of the foaming agent; for example, 8.5 and 7.5, respectively. In the corresponding sectors on the coordinate axes, we mark the specified values of the consumptions and draw straight lines parallel to the coordinate axes. In sector I, at the intersection of the lines, we determine the average density (in this case, it is equal to 308 kg/m³). In sector II, we similarly determine the compressive strength (equal to 0.76 MPa). In sector III, we determine the softening coefficient. We find the intersection point between the straight line of the consumption of the aqueous solution and the polymer consumption line (Pc). From the intersection point, we drop a perpendicular to the ordinate axis and determine the softening coefficient (equal to 0.348).

The solution to the inverse problem of choosing the optimal value of factors according to the optimization parameter is implemented in the process of checking the reliability of the results obtained from the optimized functions (4–6).

• 4)    Verification of the reliability of the obtained results

The reliability of the data obtained as a result of working with the models is checked using the optimization parameter, which is the softening coefficient of foam gypsum. The calculated values are obtained from the polynomial (6) and are entered into column 5 of Table 2. Next, mixtures of the designed compositions are prepared, foam gypsum samples are made, and after 7 days of hardening, their compressive strength is determined. Then they are kept in a humid environment, the strength of the wet samples is determined, and the experimental values of the softening coefficient are calculated. The results are entered in Column 6. The relative difference between the calculated and experimental values is determined (column 7 is formed), and the compositions that correspond to the best indicators for the softening coefficient are selected.

In this case, compositions No. 4 and 5 have the highest softening coefficient. Using polynomials (4 and 5), we determine the strength and average density of foam gypsum. Based on the strength and average density indicators, we select composition No. 5 (compressive strength 0.75 MPa, average density 320 kg/m³). Consumption of dihydrate gypsum is 76.6 kg/m³; consumption of the polymerising component is 9.8 kg/m³; consumption of an aqueous solution of foaming agent is 80 kg/m³.

The macro- and microstructure of ultra-light mineral materials largely determines their properties. Firstly, this

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Consumption of aqueous solution of foaming agent, kg/m

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Fig. 2. Nomogram for predicting the properties of foam gypsum with an optimized consumption of dihydrate gypsum equal to 76.6 kg/m³.

Table 2. Verification of the reliability of calculated values with an optimized consumption of dihydrate gypsum equal to 76.6 kg/m³

No.	Consumption, kg/m³		Softening coefficient values
	Polymerizing component	Aqueous solution of foaming agent	Calculated	Experimental	Δ, %
1	2	3	5	6	7
1	8.2	62	0,37	0.36	2.7
2	8.2	80	0.35	0.36	2.7
3	9.8	98	0.38	0.36	5.3
4	9.8	62	0.42	0.43	2, 4
5	9.8	80	0.40	0.42	5.0
6	6.2	98	0.28	0,30	7.1
7	6.2	80	0.30	0.32	6, 7
8	6.2	62	0.38	0.36	5.3

CONSTRUCTION MATERIALS SCIENCE is the average pore size (cells) and the distribution of pores by size. Secondly, the nature of porosity: communicating or closed. Thirdly, the density of intercellular partitions, as well as the density and topography of their surface. Based on the location and structure of the pores of foam gypsum, one can conclude that the porosity is predominantly communicating. The distribution of pores by size groups is not traced in the studied fragment of the sample. In Fig. 3, closed macropores are marked in yellow; the average size varied from 150 to 400 µm.

Orange color indicates through pores forming a system of open porosity; the average size on the studied fragment was 40–170 µm. Through pores are located in the body of macropores. The surface of the pores is predominantly torn and loose.

In general, the structure of foam gypsum is partially closed macropores and macropores with through porosity, consisting of intergrowths of gypsum crystals (Fig. 4). The average sizes of typical gypsum crystals were (13–18) × (1.1–2.0) µm (average sizes of the maximum and minimum sides of the crystal). The needle-like shape of individual gypsum crystals is clearly visible.

Gypsum-containing materials are used mainly in dry rooms. The reason for this is that the strength of gypsumbased products in general and foam gypsum in particular depends a lot on how wet the material is [13–15]. In this regard, it is important to study the drying mechanism of foam gypsum samples at different temperatures. The experiment was carried out on foam gypsum samples measuring 50×50×50 mm.

The experiment indicated that the samples absorb moisture during the first 30 minutes, and when they are kept in water for up to 2 hours, their moisture content increases slightly. The test results are shown in Fig. 5.

Analyzing the results, we can conclude that the drying of foam gypsum occurs most intensely at positive temperatures. At negative temperatures, drying takes place, but the process slows down significantly. This becomes especially relevant given the possible areas of application of foam gypsum as a filling material in building structures (Fig. 6).

The use of modified foam gypsum in the construction of low-rise buildings has a number of technological advantages. Firstly, the construction of enclosing structures allows one to do without lifting mechanisms (Fig. 5). Secondly, various methods of sheathing frame walls become possible. In particular, it becomes possible to use chrysotile cement sheets, chipboard, or orientated strand boards and their analogues as permanent formwork.

The use of lightweight materials based on modified foam gypsum as thermal insulation allows for an increase in the comfort of the indoor microclimate. Given the low thermal conductivity of foam gypsum and the monolithic nature of the fill, it becomes possible to form seamless

Fig. 3. General view of the pore structure of foam gypsum

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Fig. 4. Structure of the interpore partition

Fig. 5. Dynamics of drying of foam gypsum samples with an average density of 320 kg/m³ at temperatures, °C: 1 – minus 11; 2 – minus 5; 3 – 20; 4 – 50

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Fig. 6. Using foam gypsum as a filling material: a – filling the ceiling with foam gypsum mixture; b – foam gypsum mixture placed in the voids of frame-sheathing walls [8]

insulating shells, which to some extent allows to solve problems related to energy consumption.

CONCLUSION

Lightweight materials are a large group of materials based on foamed polymers or based on mineral composites, which include foam glass, cold-curing foam glass, aerated concrete (D100-300), foam gypsum, and foam gypsum based on a modified binder.

Mineral lightweight building materials, and in particular foam gypsum, are classified as non-flammable and environmentally friendly materials, which significantly increases their attractiveness both from the point of view of fire safety and from the point of view of increasing comfortable conditions in the premises, as well as reducing the negative impact on the environment.

The properties of foam gypsum are directly determined by its structure both at the macro level and at the micro level. Self-organization of the microstructure of gypsum, forming intercellular partitions, allows obtaining crystalline systems that have partial permeability for vapor-air mixtures, which has a positive effect on the sorption characteristics of the material and allows recommending it for interior decoration not only from the standpoint of its acoustic characteristics but also as products that provide regulation of the level of air pollution in rooms.

We can recommend modified foam gypsum for making one-of-a-kind items (for indoor use) and for using as a heat-insulating material in low-rise homes made of frame-sheathing structures, bricks, and small blocks because its performance characteristics have already been measured.