**3. Thermoplastic foams: processing and nanocomposites**

A wide variety of thermoplastics such as polypropylene, polyethylene, polystyrene, polycarbonate, polyvinyl chloride, polylactic acid, and polycarbonate have been experienced in foam processing techniques. Depending on their viscosity, melt strength, the formation of cell morphology of the polymer foam changes. Due to the demands of improvements in foam morphology and the mechanical strength of the polymer foams, nanoparticle-reinforced polymer nanocomposites have been developed in last decade. It has known that nanoparticle usage in polymer foam processing improved cell morphology due to the nucleating agent behavior of the nanoparticles in the polymer matrix. The presence of nanoparticles is also effective in improving the mechanical, physical, and chemical properties of polymer foams. In this section, the most experienced thermoplastic foams in industrial applications and their composites are reviewed.

#### **3.1. Polypropylene-based foams**

**Figure 7.** Illustration of the headstream in foam injection molding.

**Criteria Batch foaming Foam extrusion Foam injection molding** Amount of material needed Small amount (in g) Larger amount (in kg) Larger amount (in kg)

in the process itself

–1011 104

sometimes the cells in the core are different in size from those found at the

Composition can be changed at any time. Nucleating agents can be introduced at anytime during processing

Blowing agent is metered but not more than the melt

Solid Melt state Melt state

edges

can take

machine capacity

No needed, molding tool is in the process

It is difficult to obtain foams with uniform

Nucleating agents can be introduced also at anytime during processing

Blowing agent is metered but not more than the melt can absorb at a certain processing condition

Expensive depending on machine capacity and also mold is extra

cost

itself

–108

cell

Pre-molding Necessary No needed, molding tool is

Cell distribution Uniform distribution Generally uniform but

Foaming composition is fixed from the onset. Must be done in the previous processes such as injection molding or extrusion, etc.

the blowing agent until equilibrium is reached

Tooling cost Cheaper than the others Expensive depending on the

**Table 1.** Comparison on batch foaming, foam extrusion, and foam injection molding.

–1016 104

Skin layer thickness (μm) Thin Thin Thick

Surface quality Good Good and glossy Generally poor

Sample state during gas loading/saturation temperature

Cell density range (cells/cm3

124 Recent Research in Polymerization

Addition of nucleating agents/process flexibility ) 106

Blowing agent supply Sample is saturated with

Polypropylene, member of linear polyolefin group, has poor solubility of carbon dioxide and low melt strength. Linear olefins do not show high strain–induced hardening which is the critical requirement in withstanding the stretching force generated in the stages of cell growth. Chien et al. [21] studied on polypropylene foams obtained by conventional injection molded and traditional foaming injection molded using chemical foaming agent content under various molding conditions. They observed the effects of process parameters, part thickness and foaming agent content on the degree foaming. Injection velocity, melt temperature, mold temperature, and back pressure on weight reduction and mechanical properties were investigated. The chemical foaming agent was azodicarbonamide used in their study. It has been reported higher injection velocity induced higher weight reduction due to the reduction in the amount of melt foaming in the screw and provided more melt foaming in the cavity. Higher melt temperature and mold temperature resulted in higher melt foaming in the cavity; consequently, weight reduction was observed. The effect of foaming agent content on weight reduction of thick parts was evaluated, and it was found that concerned the weight decreases with increasing content of foaming agent but less significantly. The mechanical test results of PP foam showed that tensile, flexural strength, stiffness, and part weight decreased with the increasing melt temperature, mold temperature, and injection velocity whereas increased with increasing back-pressure.

Sporrer and Altstadt [19] obtained PP foams by physical foaming, MuCell technique. The effect of process conditions on cell morphology was observed. Two different mold temperatures were studied as 20 and 80°C. When they worked at higher mold temperatures, the thickness of the compact skin layers was reduced by 20% as compared to the part processed using the cold mold. The SEM image is given in **Figure 9**. The mold with 80°C gave 552 μm of layer thickness, and the mold with 20°C gave 442 μm of skin layer. The thinner skin layer is the result of the lower thermal gradient between melt and mold steel and a less rapid heat transfer in hotter mold.

In **Figure 9**, morphologies of polypropylene foams are given that are foam injected molded under 20 and 40°C. The foam of 40°C gave coarser and ruptured cells, while the foam of 20°C gave thicker skin layer. The reason of thicker skin layer is the sudden frozen layer of the material as it is injected into the cold mold wall (20°C).

Xin et al. [22] applied chemical foaming by using azodicarbonamide in order to obtain microcellular polypropylene/waste rubber tire (WGRT). Their aim was to generate "a value added" product by using a waste material. They observed the effects of critical processing parameters on cell morphology and physical properties of the blend foams. They observed that under the

**Figure 9.** Morphology of the PP foams processed with different mold temperatures (a) 40°C (b) 20°C [20].

same molding conditions, the microcellular PP/WGRT blend samples had smaller cell sizes and higher cell densities than the microcellular PP samples. They reported that it was due to the behavior of the waste rubber tire powders as nucleating agent that promoted heterogeneous cell nucleation, resulting in higher cell density. On the other hand, the increase of viscosity in the PP/WGRT blend prevented the growth of the cells, leading to a smaller cell size [23].

foaming agent content on the degree foaming. Injection velocity, melt temperature, mold temperature, and back pressure on weight reduction and mechanical properties were investigated. The chemical foaming agent was azodicarbonamide used in their study. It has been reported higher injection velocity induced higher weight reduction due to the reduction in the amount of melt foaming in the screw and provided more melt foaming in the cavity. Higher melt temperature and mold temperature resulted in higher melt foaming in the cavity; consequently, weight reduction was observed. The effect of foaming agent content on weight reduction of thick parts was evaluated, and it was found that concerned the weight decreases with increasing content of foaming agent but less significantly. The mechanical test results of PP foam showed that tensile, flexural strength, stiffness, and part weight decreased with the increasing melt temperature, mold temperature, and injection velocity whereas increased

Sporrer and Altstadt [19] obtained PP foams by physical foaming, MuCell technique. The effect of process conditions on cell morphology was observed. Two different mold temperatures were studied as 20 and 80°C. When they worked at higher mold temperatures, the thickness of the compact skin layers was reduced by 20% as compared to the part processed using the cold mold. The SEM image is given in **Figure 9**. The mold with 80°C gave 552 μm of layer thickness, and the mold with 20°C gave 442 μm of skin layer. The thinner skin layer is the result of the lower thermal gradient between melt and mold steel and a less rapid heat

In **Figure 9**, morphologies of polypropylene foams are given that are foam injected molded under 20 and 40°C. The foam of 40°C gave coarser and ruptured cells, while the foam of 20°C gave thicker skin layer. The reason of thicker skin layer is the sudden frozen layer of the mate-

Xin et al. [22] applied chemical foaming by using azodicarbonamide in order to obtain microcellular polypropylene/waste rubber tire (WGRT). Their aim was to generate "a value added" product by using a waste material. They observed the effects of critical processing parameters on cell morphology and physical properties of the blend foams. They observed that under the

**Figure 9.** Morphology of the PP foams processed with different mold temperatures (a) 40°C (b) 20°C [20].

with increasing back-pressure.

126 Recent Research in Polymerization

transfer in hotter mold.

rial as it is injected into the cold mold wall (20°C).

Realinho et al. [24], developed flame-retardant polypropylene composite foams by combining a basic hydrated magnesium carbonate (hydromagnesite), an intumescent additive based on ammonium polyphosphate, an organo modified-montmorillonite (MMT), and graphene nanoplatelets with PP. Azodicarbonamide was used in chemical foaming. Addition of hydromagnesite was 60%, while the other nanoparticles were about 1%. They reported that cell size reduced to 100 μm from 900 μm with the addition of hydromagnesite. The presence of nanoparticles enhanced the cell morphology. They also mentioned that solid composites were more successful in improving the flame retardancy than foam composites.

In order to improve mechanical properties of PP foams, Hwang and Hsu [25] used nanosilica particle polypropylene. Physical foaming, MuCell technique, was applied in their study. Particle addition was between 2 and 10%. They observed that when the silica content increased, cell size decreased and the cell density increased. However, a threshold was seen in silica content that the cell size leveled off when the nanosilica loading was greater than 4%. Similarly to the previous studies, dispersion of the nanoparticles in the matrix homogenously improved cell morphology. This is due to the nucleating agent effect of nanoparticles that cells nucleate at the boundary between the polymer matrix and the filler. Hwang and Hsu [25] also experienced the effect of microsilica particles and compare their effect on cell generation. They observed that at the same concentrations of particles, nanoparticles gave denser and smaller cells in size.

Nanoclay is another nanoparticle used in order to improve the properties of polypropylene foams. Nanoclay particles, similar to silicate, act as nucleating agent and lead homogeneity in cell size. Increment in clay content decreased the cell size due to the high viscosity of polymer [16, 26, 27]. Furthermore, authors suggested that clay particles act as secondary layer to protect the cells from being destroyed by external forces. In other words, the biaxial flow of the material during foam processing, the nanoparticles align along the flow direction which is the cell boundary (**Figure 10**). By this way, clay particles help the cells withstand the stretching force. Otherwise, the cell wall will break and weaken the mechanical strength of the polymer foam.

Doruk [28] studied on the effects of the nanocalcite and microcalcite particles on the cell morphology and mechanical strength of the PP foams. Nanoparticles were mixed with polymer in twin screw extruder, and then, foam injection molding was applied by chemical foaming agent (azodicarbonamide). When the fracture surface was observed as given in **Figure 11**, nanoparticle addition improved cell morphology. In **Figure 12**, tensile properties of the PP/ calcite foams are given, and it has been seen that under the same concentration of the particle addition (1 wt.%), tensile strength of PP/microcalcite foam is slightly higher than that of PP/ nanocalcite foam. This is due to the improved cell generation of the PP/nanocalcite foam as given in **Figure 11**. On the other hand, cell generation of PP/microcalcite is very poor, and the ductility of the PP/nanocalcite is apparently higher than that of PP/micro. When the weight

**Figure 10.** Illustration of the alignment of the nanoparticles during foaming process.

**Figure 11.** Cell morphology of PP/calcite foams (a) nanocalcite reinforced (b) microcalcite reinforced [28].

**Figure 12.** Comparison tensile properties of PP foams with microsized and nanosized calcite (1 wt.%) [28].

loss is considered, nanocomposite foam shows the weight loss of 20.7%, while microcomposite foams have 8.3% of weight loss.

The demand of new lightweight materials with improved transport properties for applications in electrostatic discharge, fuel system components, and electromagnetic interference shielding such as fuel cells, gaskets for electronic devices, among others brings the generation of a multifunctional material, carbon-based nanoparticle-reinforced polymer foams. Carbon nanotubes, graphene, have been recently attractive for many applications in electronic industry. Antunes et al. used carbon nanofibers (CFN) with polypropylene in order to improve thermal and electrical properties of the polypropylene composite foams [29, 30]. In their study, they emphasized the importance of particle alignment during cell generation and the importance of this on thermal conductivity of the PP. The foaming of PP with CNF provided a kind of network of the particles through the polymer matrix which increased the thermal conductivity of the polymer. When they compared their results with foamed and unfoamed polymer composites, they observed that the unfoamed composite showed a constant thermal conductivity independently of CFN content, while PP/CFN foams showed an increment in thermal conductivity as the content of CFN increased. This shows that a kind of network of CNFs throughout the polymer was formed formation that makes the material thermally conductive. The formation of this network is similar to the clay alignment as discussed in **Figure 10**. In a different study related to PP/CNF foams [31], electrical conductivity of the polymer composite foams was investigated. When the unfoamed and foamed composite were compared, the lower concentration of CFN gave high electrical conductivity. Also, the cellular structure generated during processing with cells highly elongated along the foam thickness direction increased the through-plane electrical conductivity of the foams with regard to the in-plane one. This indicates the importance of cell morphology on the electrical properties of the polymer foams. An accurately developed cellular structure may help to develop foams for semi-conducting lightweight materials [29–31].

Altan [20] made a research on polypropylene/ nano–zinc oxide (ZnO) foams. Zinc oxide is another alternative material to improve electrical properties of the polymer foams. The concentration of ZnO was 1.5% in weight. When the cell morphologies of PP foam and PP/nano-ZnO foam were compared, it has been seen that the presence of nanoparticles decreased the cell diameter and thickness of skin layer and increased the cell dense (**Figure 13**).

Graphene is the latest nanomaterial applied in polymer foams. Similar to previous nanofillers, in literature, it has been seen that graphene loading to PP between 2.5 and 5 wt.% has great

**Figure 13.** Fracture surfaces of polypropylene foams (a) neat PP (b) PP/ZnO [20].

**Figure 12.** Comparison tensile properties of PP foams with microsized and nanosized calcite (1 wt.%) [28].

**Figure 11.** Cell morphology of PP/calcite foams (a) nanocalcite reinforced (b) microcalcite reinforced [28].

**Figure 10.** Illustration of the alignment of the nanoparticles during foaming process.

128 Recent Research in Polymerization

ite foams have 8.3% of weight loss.

loss is considered, nanocomposite foam shows the weight loss of 20.7%, while microcompos-

The demand of new lightweight materials with improved transport properties for applications in electrostatic discharge, fuel system components, and electromagnetic interference shielding effect in cell nucleation [32]. Besides, the higher expansion of polymer during foaming process induces higher exfoliation of the graphene nanoplatelets in PP matrix and brings higher mechanical strength [32].

### **3.2. Polyethylene-based foams**

Polyethylene (PE) is a member of polyolefin-like polypropylene. High-density polyethylene (HDPE) and low-density polyethylene (LDPE) have been experienced in foam processing. LDPE foams are used as thermoplastic material for applications such as packaging and foamed sheets, sports parts due to its owing low density, high elasticity, water resistance, and low cost. One of the common problems in polymer foams is the loss of toughness and ductility of the material due to the generation of the cells. Sun et al. [33] developed a toughening mechanism for high-density polyethylene/polypropylene blends. They obtained super ductile polymeric blends using microcellular injection molding. They prepared PP/HDPE and PP/LDPE blends which were prepared at weight ratios of 75/25, 50/50, and 25/75 with melt mixing method and then by applying MuCell technique. It has been observed that during tensile test, 75/25 PP/LDPE foamed parts were highly fibrillated along the tensile load direction in the necking region. The researchers reported that the reason of this behavior—the high ductility of polymer foams—was related to two reasons. The first one was due to the microcellular foam structure cell size lower than 100 μm, and the other was an immiscible but compatible submicron-size secondary polymeric phase. During tensile test, the sub-micron phase of the blend debonds from the matrix, and the cavities collapse. Secondly, they interconnect the microscale foam cells along the load direction. This generates many fibrils that make the material highly ductile [33].

Similar to the case of PP-nanocomposites foams, various authors have reported the preparation, characterization, and properties of PE-nanocomposite foams [34–37]. Arroyo et al. [37] developed low-density polyethylene/silica nanocomposite foams by using chemical foaming agent. They applied different concentrations of silica between 1 and 9%, and foaming agent was 5% in weight. The addition of silica particles improved the cellular structure of the LDPE improved with increment of the cell densities and decrease in cell size. However, at silica concentrations over 6%, increase in cell size was reported. There are few reasons about the poor quality of foam cell morphology under higher nanoparticle concentrations. One of them is possible agglomerations of the nanoparticles at higher concentrations that they prevent the formation of the cells. Also, increase in viscosity of the polymer melt because of the higher loadings of the particles makes the cell generation difficult.

Clay is one of the most used inorganic particles in enhancing the properties of PE-based foams. Clay, such as montmorillonite (MMT), is mixed with polymers and the mechanical strength of the polymers increase [36, 38]. In the study of Hwang et al. [38], the effect of MMT on the cell morphology of the low density of polyethylene (LDPE) was observed. First of all, the researchers enhanced the nanoparticle distribution in the polymer matrix by grafting polar maleic anhydride (MA) onto nonpolar LDPE. The concentration of MMT was between 1 and 5%. Their results are similar to the previous studies that the MMT and MA act as nucleating agents that lead to a finer and more uniform cell structure. When the dispersion of the nanoparticles is homogenous, the cell size decreases and the distribution of the cells is homogeneous.

Polyethylene foams, similar to the other thermoplastic foams, can be processed either batch foaming or foam injection molding. Hayashi et al. [39] compared the orgona clay PE-based ionomer composite foams obtained by batch processing and foam injection foaming. The effect of clay on foam morphology of PE is similar to the previous studies that the dispersed nanoclay particles act as nucleating sites for cell formation, and cell growth occurs on the surfaces of the clays. Different from batch processing, in foam injection molding, the moldings have two compact solid skin layers and a foamed core. In both foaming processes, the foam morphology can be improved by setting process conditions correctly depending on the viscosity of the polymer and the temperature and gas pressure limits. Hayashi et al. [39] reported that in batch process, the ionic cross linked structure provided finer cells, and the coalescence of the cells was prevented. On the other hand, by the effect of supercritical nitrogen gas as foaming agent during foam injection molding process, the viscosity of polymer was decreased, and this promoted the nucleation and also coalescence of the cells, especially at high temperatures.

#### **3.3. Polystyrene-based foams**

effect in cell nucleation [32]. Besides, the higher expansion of polymer during foaming process induces higher exfoliation of the graphene nanoplatelets in PP matrix and brings higher

Polyethylene (PE) is a member of polyolefin-like polypropylene. High-density polyethylene (HDPE) and low-density polyethylene (LDPE) have been experienced in foam processing. LDPE foams are used as thermoplastic material for applications such as packaging and foamed sheets, sports parts due to its owing low density, high elasticity, water resistance, and low cost. One of the common problems in polymer foams is the loss of toughness and ductility of the material due to the generation of the cells. Sun et al. [33] developed a toughening mechanism for high-density polyethylene/polypropylene blends. They obtained super ductile polymeric blends using microcellular injection molding. They prepared PP/HDPE and PP/LDPE blends which were prepared at weight ratios of 75/25, 50/50, and 25/75 with melt mixing method and then by applying MuCell technique. It has been observed that during tensile test, 75/25 PP/LDPE foamed parts were highly fibrillated along the tensile load direction in the necking region. The researchers reported that the reason of this behavior—the high ductility of polymer foams—was related to two reasons. The first one was due to the microcellular foam structure cell size lower than 100 μm, and the other was an immiscible but compatible submicron-size secondary polymeric phase. During tensile test, the sub-micron phase of the blend debonds from the matrix, and the cavities collapse. Secondly, they interconnect the microscale foam cells along the load direction. This generates many fibrils that make the material highly ductile [33].

Similar to the case of PP-nanocomposites foams, various authors have reported the preparation, characterization, and properties of PE-nanocomposite foams [34–37]. Arroyo et al. [37] developed low-density polyethylene/silica nanocomposite foams by using chemical foaming agent. They applied different concentrations of silica between 1 and 9%, and foaming agent was 5% in weight. The addition of silica particles improved the cellular structure of the LDPE improved with increment of the cell densities and decrease in cell size. However, at silica concentrations over 6%, increase in cell size was reported. There are few reasons about the poor quality of foam cell morphology under higher nanoparticle concentrations. One of them is possible agglomerations of the nanoparticles at higher concentrations that they prevent the formation of the cells. Also, increase in viscosity of the polymer melt because of the higher

Clay is one of the most used inorganic particles in enhancing the properties of PE-based foams. Clay, such as montmorillonite (MMT), is mixed with polymers and the mechanical strength of the polymers increase [36, 38]. In the study of Hwang et al. [38], the effect of MMT on the cell morphology of the low density of polyethylene (LDPE) was observed. First of all, the researchers enhanced the nanoparticle distribution in the polymer matrix by grafting polar maleic anhydride (MA) onto nonpolar LDPE. The concentration of MMT was between 1 and 5%. Their results are similar to the previous studies that the MMT and MA act as nucleating agents that lead to a finer and more uniform cell structure. When the dispersion of the nanoparticles is homogenous, the cell size decreases and the distribution of the cells is homogeneous.

loadings of the particles makes the cell generation difficult.

mechanical strength [32].

130 Recent Research in Polymerization

**3.2. Polyethylene-based foams**

Polystyrene (PS) is an amorphous polymer, and it has wide application area in polymer foam processing such as thermal insulation, packing material due to its low cost, ease of processing, resistance to moisture, and recyclability. Dow Chemical Company invented PS foams as "Styrofoam" in 1941. Polystyrene foams are basically divided into two; expanded polystyrene (EPS) and extruded polystyrene (XPS). Expanded polystyrene has white color and can be used in cups for hot beverages, insulation material in whitegoods, or in packing industry. EPS consists of 96–98% air and 2–4% polystyrene. Processing method is heating the material with steam and then expanding of the material. Extruded polystyrene (XPS) has smaller air pockets inside and manufactured by extrusion process in the form of boards with different colors to identify the brand type of the product. Zhang et al. [40] produced extruded polystyrene foams (XPS) by using CO2 and water as a co-blowing agent. Okolieocha et al. [41] carried out on a tandem foam extrusion line, similar studies on XPS. They used a slit die (0.5 mm) set at a temperature of 126°C. In order to enhance cell density, they used 1 wt.% of thermally reduced graphite oxide. However, general purpose polystyrene (GPSS) and high-impact polystyrene (HIPS) are suitable for injection molding and structural foaming, and cell generation can be provided similar to the other thermoplastics by chemical or physical foaming agents. Furthermore, PS is not characterized by low melt strength so this makes it suitable for foam injection molding. Hwang et al. [42] applied foam injection molding via MuCell for obtaining clay-reinforced PS foams. Clay was used to improve cell morphology of the polystyrene foams. They obtained PS/clay composite foams with small size cell, which makes the material highly suitable for acoustic and thermal insulation applications. On the other hand, layers of clay-like montmorillonite (MMT) are difficult to fully exfoliate in PS matrix. MMT was modified by stearylbenzyldime-ammonium chloride prior melt mixing with polystyrene, and the concentration of MMT in the matrix was experienced in a narrow range as 0.25-0.5-1-2-3% (wt). It has been seen that organo clay presence of 1% in PS matrix gave the small cell in diameters, leading to maximum tensile strength, thermal stability, and cell density.

#### **3.4. Polylactic acid–based foams**

Poly (lactide acid) or polylactide (PLA) is a biodegradable and biocompatible polymer produced from such renewable sources as cornstarch and sugarcane [1–4]. PLA foam is a competitive material among most of other thermoplastic foams due to its biocompatibility and biodegradability, PLA has been widely used in tissue engineering applications such as skin, bones, blood vessels, due to their highly porous structure as scaffolds in last [4]. The porous surface of the PLA foams enhances the biological activities of both seeded and native cells. High porosity is important for enhancing biological properties of the scaffold such as the adhesion, proliferation, and migration of the cells. However, mechanical properties of foams decrease with the increasing of porosity. Besides, the high strength and brittle properties of PLA make it difficult to use and process it in foaming techniques. Researchers are focusing on generation of PLA with different polymers or PLA matrix composites [4].

Similar to the other thermoplastics, PLA foams with uniform cell morphology are generally obtained by physical foaming agents such as carbon dioxide and nitrogen in foam injection molding and foam extrusion. However, the poor melt strength of PLA brings difficulties in obtaining an enhanced cell morphology. There are several ways to improve PLA's foam morphology by means of improving melt strength of the polymer such as using chain extenders, using polymer blends of PLA, addition of nanoparticles, and improving crystallization kinetics. Low melt strength of PLA induces cell coalescence during cell growth. Addition of chain extenders to PLA increased the rheological properties of PLA, and depending on this, cell morphology is enhanced [43–45].

Crystallization is an important factor in improving melt strength and foaming ability of thermoplastics. The low melt strength of PLA can be promoted by improving crystallization kinetics and the poor viscoelastic behavior of the polymer. However, high crystallinity has negative effect on cell generation by suppressing the foam expansion. On the other hand, during foaming, cell nucleation starts around crystals [46, 47]. Therefore, improving crystallinity can be balanced by some nucleating agents such as additives and nanofillers behave-like nucleating agents. There are several studies on PLA nanocomposite foams that used calcite, sepiolite, and multi-walled carbon nanotube as nanofiller [46–49]. In these studies, nanomaterial addition was found to be nucleating agent for crystallinity and cell generation. There has been great interest to clay-reinforced PLA composite foams due to the enhanced viscoelastic behavior of clay particles in the polymer matrix which improves cell morphology [48, 50]. As the nano clay particles increased, the cell density of the foamed samples increased. It has been reported that even a small amount addition of carbon nanotube (CNT) promoted the cell density due to its effect on cell nucleation [47]. An interesting point about PLA/CNT composite foams that the gas used during foam injection molding behaved like a dispersant for the nanoparticles that homogenous dispersion of CNT could be obtained in the polymer matrix. This is due to the plasticization effect of the supercritical fluid phase of CO<sup>2</sup> [43, 47]. Therefore, in foam extrusion and foam injection molding, the foaming agents do not only provide foaming but also disperse the particles homogenously in the matrix.
