Phase Change Materials with Enhanced Thermal Conductivity and Heat Propagation in Them

Abstract

The review contains information o; n the properties of phase-change materials (PCM) and the possibilities of their use as the basis of thermal energy storage. Special attention is given to PCMs with a phase transition temperature ranging between 20 and 80 °C since such materials can be effectively used to reduce temperature variations in residential and industrial rooms. Thus, the application of PCMs in the construction industry enables one to considerably reduce the power consumption and reduce the negative environmental impact of industrial facilities. Thermophysical characteristics of the main types of PCMs are presented. The heat balance for a room with walls made of PCM-added materials is estimated. The calculations demonstrate that such structures can stabilize the temperature in practical applications as a result of the usage of such materials. The construction of a thermal accumulator on the basis of PCM is proposed and analyzed. This facility uses water as a working fluid and paraffin as a PCM. The thermal accumulator has a modular structure so that the number of similar modules is determined by the quantity of energy to be stored. The potential of wide application of PCMs as a basis for thermal energy storage is rather limited due to a very low conductivity (less than 1 W/(m K)) inherent to these materials. This drawback can be overcome by adding carbon nanoparticles whose thermal conductivity is four to five orders of magnitude greater than that of the matrix material. The problem of fabrication of polymer composites with enhanced thermal conductivity due to nanocarbon particles doping is discussed in detail.

1. Introduction

An intense enhancement of energy production and consumption impacts the environment negatively and requires increased expenditures. Estimations (see, for example, [1]) indicate that the global energy consumption enhances by approximately 30% in a decade. At the same time, about 40% of all energy generated in the world is consumed for heating and cooling residential and industrial buildings. In this situation, the development of new technologies permitting us to decrease or, at least, inhibit the energy consumption growth without decreasing the current living standards and the material production level, appears to be among the top priority problems facing power engineering. One of such technologies is based on the usage of phase-change materials (PCM). These materials can store or release a large amount of energy as a result of a phase transition at a change in the temperature. Building panels containing PCMs enable a considerable decrease of daily temperature variations in residential and industrial buildings without additional expenditures. Thus, the model calculations [2] imply that the utilization of panels with 20% PCM content lowers the amplitude of daily temperature variation in the constructing buildings by as much as 38%. Panels containing PCM microcapsules provide even a much greater decrease in the amplitude of daily temperature fluctuations (up to 80%) without using heat sources or air conditioning [3]. As it follows from the measurements [4], the usage of PCM-based thermal accumulating panels in the construction of residential and industrial buildings can save approximately 15% of the energy consumed for their heating and air conditioning. As a result, approximately 10% of all the energy generated in the world can be saved, which promotes a reduction in the negative environmental impact of the power industry.

Thermal energy accumulators on the basis of PCMs can be utilized not only in the construction industry but also in energy-storage devices of solar and wind power engineering as well as in overheating protection systems of complex electronic systems, supercomputers, or other radio-electronic hardware. The prospects for the application of PCMs in thermal and solar power engineering as the basis for energy accumulators, as well as in electronics and the instrument-making industry, stimulate active research efforts in many laboratories around the world [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The results of these and some other investigations performed in recent years were represented in many reviews and monographs [33,34,35,36,37,38,39,40,41,42].

However, the wide spread of PCM-based thermal energy accumulators is hindered by a set of problems. First of all, the very low thermal conductivity of most PCMs causes high inertia of heat accumulation. The thermal conductivity of most of PCMs accounts for about λ ≈ 0.2–0.5 W/m K, which corresponds to the typical value of the thermal diffusivity of the material a ≈ (2–5) × 10^–7 m²/s. The time τ required for the heat propagation through a material layer of d = 0.1 m in thickness is estimated by the relation τ ~ d²/a and accounts for several hours. For this reason, PCM-based thermal accumulators respond with a large delay to a change in the ambient temperature which makes utilization of such storage hardly workable. Therefore, the problem of the development of PCM-based materials with enhanced heat conductivity becomes the key one. One of the approaches to overcome this problem is in doping a PCM with nanocarbon particles (such as nanotubes, graphene, or soot), whose heat conductivity exceeds that PCM by 4–5 orders of magnitude [43,44]. Another approach to the usage of PCMs in the building design relates to the inclusion of a PCM into thin microcapsules for which the heat exchange time can be shortened down to a second scale by decreasing the size of capsules. This approach has been considered in detail by the authors of [29], who described different forms of PCM capsulation.

The usage of PCM in building construction and in energy storage systems requires the statement and solution of a set of tasks of both technological and theoretical character. First, one should mention the problem of preparation of a spatially homogeneous PCM-based composite material doped with carbon nanoparticles avoiding the known trend of carbon nanoparticles to the aggregation. The next task relates to the determination of the dependence of the thermal conductivity of a PCM based material on the content of doped particles having various geometry and thermophysical properties as well as the heat propagation through a non-homogeneous media depending on the gradients of the doped particle content and phase transition latent heat. All these issues have been discussed in the present article, which also contains a review of investigations addressed to determining physicochemical characteristics of PCMs and their application in thermal energy storage systems. Special attention is paid to the problem of preparing PCMs with enhanced thermal conductivity.

2. Physical, Chemical, and Thermophysical Characteristics of PCMs

We are interested in PCMs having the phase transition temperature in the range between 0 and 100 °C and the phase transition latent heat exceeding notably the product of the heat capacity by the working temperature range (several dozen degrees Celsius). The most common and readily available materials possessing these requirements are paraffin wax, fatty acids, and salt hydrates. Thermophysical characteristics of some of these materials have been presented in

Table 1. Thermophysical properties of commercial paraffin waxes [27] with a specific heat capacity of 2.1 J/g K.

Number of Carbon Atoms in Molecule	Phase Transition Temperature, °C	Melting Heat, J/g	Liquid State Density, kg/m3	Thermal Conductivity, W/m K
9–12	−9 to −53	184	686	0.15
13–16	−6 to 18	196	716	0.19
16–18	18 to 28	212	734	0.21
16–28	42 to 44	214	765	0.21
20–33	48 to 50	218	769	0.21
22–45	58 to 60	221	795	0.21
24–50	66 to 68	221	830	0.21

,

Table 2. Thermophysical properties of fatty acids [27].

Acid	Chemical Formula	Phase Transition Temperature, °C	Melting Heat, J/g	Liquid State Density, g/cm3	Specific Heat Capacity, J/g K	Thermal Conductivity, W/m K
Caprylic	CH3(CH2)6COOH	16	128	0.862 (80 °C)	-	0.148
Capric	CH3(CH2)8COOH	32	136	0.866 (40 °C)	-	0.149
Lauric	CH3(CH2)10COOH	42–44	155	0.870 (50 °C)	1.6	0.147
Myristic	CH3(CH2)12COOH	54	158	0.840 (80 °C)	1.6	-
Palmitic	CH3(CH2)14COOH	63	159	0.847 (80 °C)	-	0.165
Stearic	C17H35COOH	70	191	-	-	0.172

and

Table 3. Thermophysical properties of salt hydrates [27].

Substance	Chemical Formula	Phase Transition Temperature, °C	Melting Heat, J/g	Liquid State Density, g/cm3	Specific Heat Capacity, J/g K	Thermal Conductivity, W/m K
Potassium phosphide hydrate	KP∙4H2O	18.5	231	1.455	1.83	-
Calcium chloride hydrate	CaCl∙6H2O	29.7	171	1.710	-	0.60
Sodium sulphate hydrate	Na2SO4∙10H2O	32.4	254	1.485	1.93	0.54
Sodium hydrogen phosphate hydrate	Na2PO4∙12H2O	35.2	280	1.420	1.55	0.59
Zinc nitrate hydrate	Zn(NO3)2∙6H2O	48.0	147	2.065	1.34	-
Sodium dithionite hydrate	Na2S2O4∙5H2O	78.0	201	1.600	1.46	-
Barium hydroxide hydrate	Ba(OH)2∙8H2O	116.0	267	2.180	1.17	-
Magnesium chloride hydrate	MgCl2∙6H2O		165	1.570	1.72	-

. A diagram showing the interconnection between the phase transition temperature and enthalpy of most known phase change materials is shown in Figure 1. As is seen, all the PCM mentioned in the tables possess rather low thermal conductivity, which makes the problem of the enhancement of this parameter a very topical one.

The most attractive materials among those shown in Figure 1 and Table 1, Table 2 and Table 3 possess the phase transition temperature in the room temperature range from 20 to 30 °C. These materials can be used in the construction of residential or industrial buildings as a basis for heat accumulators. The latent heat of phase change, L, of materials with a room temperature of the phase transition ranges from 100 to 200 J/g, which is approximately two orders of magnitude greater than the corresponding amount of the energy required to heat the material by 1 °C. The thermal conductivity of PCMs is λ ≈ 0.2–0.5 W/(m K), which corresponds to the thermal diffusivity of ≈ (2–5) × 10^–7 m²/s.

The set of PCMs includes organic compounds, inorganic compounds, and eutectic mixtures. The class of organic PCMs is divided into paraffin waxes and nonparaffin compounds, which include fatty acids, esters, alcohols, glycols, etc. The advantages of organic PCMs are a wide phase transition temperature range, chemical stability, no susceptibility to corrosion, and a phase transition temperature proper for room temperature stabilization. Fatty acids, which are generally presented by the formula CH₃(CH₂)_2nCOOH, are stable at cycling. The combination of different fatty acids can yield a material with a melting temperature range of 20–30 °C [38]. However, most of the organic PCMs are not stable at elevated temperatures. The main drawback of the usage of fatty acids as heat storage construction materials is their high price, which exceeds that of paraffin wax by 2.0–2.5 times.

Paraffin waxes present the main material used in thermal energy accumulators. Paraffin waxes with a melting temperature between 20 and 70 °C are used in pilot energy storage systems. These PCMs are considered usually in model calculations addressed to improve the performance of these systems. The intrinsic disadvantage restricting the practical application of thermal accumulating systems on the basis of paraffin waxes is the rather low thermal conductivity of paraffin (approximately 0.2 W/(m K)). Besides that, paraffin waxes experience a large volume change during the phase transition. The cost of paraffin waxes is rather high compared to salt hydrates.

The application of salt hydrates, as well as paraffin waxes, in heat storage materials with phase change also appears promising. The comparison of data presented in Table 1, Table 2 and Table 3 indicates that salt hydrides possess the phase transition enthalpy exceeding that of paraffin waxes by 1.5–2.0 times, while the specific heat capacity is 30–50% lower than that of fatty acids and paraffin waxes. The thermal conductivity of salt hydrides is relatively high, two to four times greater than that of other types of PCMs. Salt hydrates are not expensive and nonflammable. However, their main drawback is a rather bad compatibility with metals because an arrangement of salt close to the metal surface promotes corrosion. In addition, such compounds are hardly useful for impregnation into porous construction materials. Metallic PCMs have a quite high phase transition temperature that makes them unsuitable for construction.

Inorganic PCMs include eutectics, which present mixtures of many materials in different proportions. Eutectics can be divided into three groups according to their consisting materials: organic–organic, inorganic–organic, and inorganic–inorganic eutectics.

The requirements for phase-change materials are presented below [38].

• Thermophysical Requirements:

• Melting point suitable for a specific application of PCMs (between 20 and 80 °C);
• High value of:

  (i)

  Melting heat.

  (ii)

  Specific heat capacity.

  (iii)

  Thermal conductivity of solid and liquid phases.

  (iv)

  Density.

  (v)

  Phase change rate.

• Cyclic stability.
• Low pressure of PCM vapor.
• Small volume change during melting.
• Homogenous structure.

• Chemical requirements

• Stability.
• No degradation during crystallization/melting.
• Reversibility of crystallization/melting.
• Incombustibility.
• Nontoxicity.
• Explosion safety.

• Economic and ecological requirements:

• Low cost.
• High economic efficiency.
• Availability.
• Ecological safety.

3. Parameters of Heat Storage Systems

This section contains estimations of parameters of PCM-based heat storage systems. Firstly, consider the possibility of usage of a PCM panel for decreasing the diary temperature oscillations in a standard living or industrial room. For the sake of definiteness, paraffin wax will be considered as a PCM. Taking into account the data presented in Table 1, let the melting (phase change) temperature of the paraffin wax T_m = 30 °C, specific melting heat H = 200 J/g, density ρ = 0.75 g/cm³, Specific heat capacity c = 2.1 J/g K, thermal conductivity κ = 0.2 W/(m K), thermal diffusivity α = 1.3 × 10⁻³ cm²/s. Let the room be a square of l = 4.7 m per side, S = 20 m² of area, and h = 3 m of height. Estimate the heat balance in this room assuming that the outdoor (night) temperature is 20 °C lower than the inside temperature which is supposed to be T_i = 20 °C. The experience indicates that for fixing the inside temperature in such a room on an acceptable level the thermal power of about W ≈ 2 kW should be provided. If the wall panels contain a PCM this power is released from the panels because of the phase transition. Assuming the duration of the cold period of the day of τ_c = 10 h, one obtains the estimation of the energy ε, which is necessary to store in PCM panels

ε = Wτ_c = 7.2 × 10⁷ J

(1)

This energy can be stored in a PCM having the specific phase transition enthalpy of H ≈ 200 J/g. The minimal mass of PCM M_min, which is necessary to store the above-mentioned energy is expressed through the quantities ε and H as follows:

M = ε/H = Wτ_c/H ≈ 350 kg

(2)

This corresponds to the volume of PCM

V = M/ρ ≈ 4.37 × 10⁵ cm³

(3)

The area of the wall panels S_p is determined as

S_p = 4l²h ≈ 5.6 × 10⁵ cm²

(4)

Therefore, the thickness of the PCM layer d in the building panel is estimated through the obvious relationship

d = V/S_p ≈ 0.8 cm

(5)

The characteristic time τ required for heat propagation through the PCM layer is estimated through the relationship

τ ≈ d²/α ≈ 500 c ≈ 8 min

(6)

Therefore, the time delay in the reply of the PCM panel on temperature oscillations is much shorter than the duration of a night. The estimations performed permit one to believe that the usage of PCM-containing panels allows saving the energy spent for heating and air conditioning residential and industrial rooms. A room whose walls contain PCM panels is well-protected from diary temperature oscillations. This protective action can be illustrated by the results of numerical calculations.

The serious problem arising from the usage of PCM relates to a rather low thermal conductivity of such materials. For this reason, the heat exchange time for elements of constructions containing PCM is usually too long. This time can be shortened either by decreasing the size of the container holding PCM or by the enhancement of the thermal conductivity of the material. One of the ways to decrease the size of PCM containers is the encapsulation of PCM into a set of miniature envelopes [2,3,29,45,46]. In this case, the characteristic heat exchange time can be shortened down to a second level by decreasing the envelop size. The drawbacks of such an approach are the additional expenditures necessary for the preparation of encapsulated material and in decreasing the specific latent energy of the encapsulated PCM with considering the mass of envelopes.

One more version of the heat storage system depleted of the above-indicated drawbacks is the water thermal accumulator (WTA) where the thermal energy is stored in PCM as a result of the passage of hot water. This results in melting PCM, which accumulates the phase transition energy. This energy can be released because of the passage of cold water through a container filled with PCM. The most appropriate configuration of WTA seems to be a set of similar modules. The configuration of the module is represented schematically in Figure 2. Each of the modules contains double concentric cylindrical tubes, where the inner tube is designed for the passage of hot or cold water while the cavity between the outer and inner tubes is filled with PCM. The inner tube of radius R₁ is fabricated from a highly thermally conductive material (for example, copper), which facilitates the thermal exchange between water and PCM. The outer tube of a radius R₂ should be fabricated from a plastic having rather low thermal conductivity, which prevents heat losses through the thermal exchange between PCM and the environment. In such a system, the characteristic heat exchange time can be shortened by increasing the inner tube radius.

The energy E stored in PCM as a result of the phase transition is expressed by the following relation

E = MH

(7)

where M is the mass of PCM and H is the specific phase transition energy. The stored energy can be extracted and used because of passing cold water through the inner tube. Cooling PCM causes the phase transition which results in heating water up to a temperature close to the phase transition point.

The PCM mass is determined by the volume of the space between the outer and inner cylinders which is expressed through the radiuses of those R₁ and R₂:

$$M=\pi (R_2^2-R_1^2)L\rho$$

(8)

where ρ is the density of PCM and L is the tube length. The characteristic heat exchange time τ is expressed by the equation

τ_he = (R₂ − R₁)²/α

(9)

where α = λ/ρc is the thermal diffusivity of PCM, λ is its heat conductivity and c is the specific heat capacity.

Setting for definiteness sake the quantities R₁ = 1 cm, R₂ = 2 cm, L = 1 m and assuming the usage of paraffin wax as a PCM, one obtains in accordance with (2) for the mass of PCM embedded into one module M_PCM = 716 g. This corresponds, according to (1), to the latent energy content of the module ε ≈ 154 kJ. This energy can be stored in PCM as a result of cooling water having the temperature exceeding the melting point by 20 °C. The characteristic heat exchange time estimated by (3) accounts for about 22 min. This time can be shortened considerably as a result of doping PCM with nanocarbon particles. Thus experiment [7] indicates that the thermal conductivity coefficient of paraffin doped with 1% (by weight) thermally reduced graphene oxide enhances by 15 times. Therefore, for such a composite, the heat exchange time is estimated as 90 c. This time determines the minimum duration of the interaction of the water flow with the PCM volume and therefore the maximum water flow velocity: v_max = L/τ ≈ 1 cm/c. The duration of passage of the water flow through the thermal accumulator τ_f is expressed through the mass of water M_w = 1.8 kg used for melting paraffin.

τ_f = M_w/ρvS ≈ 750 s

(10)

This time is of the same order as the characteristic heat exchange time τ_he determined by relation (9). Therefore, hot water has no time to transfer its thermal energy to PCM which requires the usage of a PCM with an enhanced thermal conductivity coefficient.

The character of thermal exchange between the water flow and PCM in the above-described setup was modeled numerically using the code COMSOL considering the phase transition energy sink. The non-stationary heat conduction equation was resolved assuming that the thermal conductivity of the inner copper tube is infinity while the thermal conductivity of the outer polymer tube is zero. The initial water temperature was set to 90 °C while the initial temperature of PCM was set to 20 °C. The results of the simulation are presented in Figure 3 in terms of the dependences of the PCM temperature on the longitudinal and transverse coordinates at various points in time. As is seen, the calculation results are compatible qualitatively with the above-performed estimations. The calculations indicate that at the water flow velocity of 10 cm/s the full melting time of paraffin accounts for about 250 s, which corresponds to the mass of passed water about 1.7 kg. As is seen, the radial dependence of the temperature is relatively slow practically during all the time of the heat exchange.

4. Thermal Conductivity of Carbon Nanoparticles

Carbon nanoparticles such as carbon nanotubes (CNT) and grapheme flakes have very high thermal conduction coefficients and can be considered as a proper dopant to enhance that of PCM. The thermal conduction of CNTs is determined by phonons so that the role of electrons in conduction is negligible [43,47,48]. Short nanotubes (less than ~1 μm) usually do not have defects and both charge and heat transport in them have a ballistic character. In the case of ballistic heat, transport phonons propagate through a medium without scattering, so that the characteristic phonon mean free path relating to the scattering on phonons and structural defects exceeds the length of the nanotube. The simplest description of ballistic phonon thermal conductivity corresponds to a high-temperature limit, which takes place at ħω << T (ω is the characteristic phonon frequency, T is the temperature). In this case, the thermal conductance of each phonon channel is determined by the quantum magnitude G_th, which has the form [48].

$$G_{th}=\frac{{\pi }^2{k}^{2}T}{3h}=9.46\times {10}^{-13}\left(\frac{W}{K^2}\right)T$$

(11)

This corresponds to the quantity G_th = 2.84 × 10⁻¹⁰ W/K at room temperature. The thermal conductance of a CNT is expressed as the product of the quantum conductance G_th and the total number of phonon channels N_p in the nanotube. The latter is a double number of atoms in a unit cell 2N, where N is expressed through the chirality indices (n, m) of the nanotube as [49,50,51].

$$N_p=\frac{2(n^2+m^2+nm)}{d_R}$$

(12)

Here, d_R is the greatest common divisor of (2n + m) and (2m + n). For a CNT having the armchair structure and chirality indices (n, n), d_R = n and N_p = 6n. For example, a single-walled (10, 10) CNT (diameter d = 1.4 nm) has N_p = 120 phonon channels, while a (200, 200) CNT (diameter d = 27.5 nm) has N_p = 2400 phonon channels. Therefore, the ballistic thermal conductance of (10, 10) and (200, 200) CNTs amounts to 120G_th and 2400G_th, respectively. This corresponds to the room temperature thermal conductivity of CNT of L = 1 μm in length λ = G_thN_p(L4/πd²) = 5000 W/m K for (10, 10) CNT and λ = 266 W/m K for (200, 200) CNT.

The thermal conductivity of CNTs decreases abruptly as the nanotube’s length L exceeds the characteristic mean free path l_p of photons due to the scattering of phonons on structural defects and admixture centers. The role of defects can be taken into account through the correcting factor k_d = λ/(L + λ), so the thermal conductivity coefficient of a CNT is approximately expressed by the following relation [43]

$$\lambda =G_{th}{N}_p\frac{L}{\pi d^2}\frac{l_p}{L+l_p}$$

(13)

As is seen, the thermal conductivity of long CNTs does not depend on their length and is proportional to the phonon mean free path. The typical value of the thermal conductivity of short nanotubes is about 5000 W/m K. This conclusion is confirmed by the results of many experiments [48,49].

Similar to CNTs, the thermal conductivity of graphene flakes depends on both the content of structural defects and the geometry of a sample. This parameter was measured using the temperature dependence of grapheme Raman frequencies [52,53]. According to this approach, a laser beam of a specified power is focused onto the middle of a single-layer graphene sheet suspended between two supports. The size of the irradiated spot is 0.5 ± 1 mm, and the temperature increase in the spot amounts to several dozen Kelvins. The temperature of the heated site of graphene is determined by the temperature shift in the position of the G peak in the Raman spectrum. At moderate heating, the increase in temperature depends linearly on the laser power, and the coefficient in this dependence is proportional to the thermal conductivity of graphene.

The experimental setup [52,53] is shown schematically in Figure 4. A set of longitudinal trenches 300 nm in depth and up to 5 mm in width were fabricated by ion etching on the surface of a Si/SiO₂ substrate. Graphene sheets produced by the micromechanical exfoliation of highly oriented pyrolytic graphite were applied onto the substrate in large numbers. Then, elongated graphene samples bridging the two sides of the trench and close in form to a rectangle were selected by means of a Raman microspectrometer. In so doing, the graphene sheets under investigation were put into thermal contact with the graphite particles that also resided on the substrate surface. These particles absorb the heat released on irradiation of the graphene sheet by a focused beam of an Ar-ion laser (λ = 0.48 μm). The size of the focal spot is about 0.5 μm; however, the size of the hot region is increased to 1 μm due to electron diffusion. The measurements yielded magnitudes of the thermal conductivity coefficient ranging between 4840 and 5300 W/m K. The processing of the experimental data also allowed the estimation of the magnitude of the phonon mean free path with respect to the scattering: l_p ≈ 775 nm. Hence, l_p turned out to be much less than the characteristic size (5 ± 10 μm) of the graphene sample, which demonstrates the prevailing role of the diffusion mechanism of heat transport over the ballistic one.

The measurements indicate that the thermal conduction coefficient of an individual graphene layer exceeds by more than two times the appropriate value for crystalline graphite: κ ≈ 2000 W/m K. Therefore, the thermal conductivity of a 2D hexagonal structure (graphene) exceeds notably that of a 3D structure consisting of graphene layers. Transition from a 2D to a 3D structure occurs upon increasing the number of graphene layers, which should be accompanied by a lowering of the thermal conduction coefficient. Such a behavior can be attributed to an additional mechanism of phonon scattering related to the interaction between neighboring layers. This behavior has been studied experimentally by the authors of Ref. [54], who utilized for this aim the Raman spectroscopy method in the manner described above. In this case, Si/SiO₂ wafers with a set of parallel trenches 300 nm in depth and up to 5 mm in width were also used. The heat delivered to the graphene sheet by laser irradiation was removed through thin metal pads deposited near the edges of the trenches. Few-layer graphene samples were produced by micromechanical exfoliation of pyrolytic graphite. The number of layers in the samples under investigation was determined through processing the Raman spectrum [55]. The longitudinal size of the suspended part of the graphene sheet ranged between 5 and 16 mm. The thermal conductivity of the graphene samples was determined by fitting the measured temperature shift of the G Raman peak (≈1579 cm⁻¹) under the action of laser irradiation to the solution of the relevant heat conduction equation for the sample by the finite element method. The dependence of the thermal conduction coefficient of few-layer films on the number of graphene layers measured by the authors of Ref. [54] is presented in Figure 5. These data have been reduced to the common lateral size of 5 mm in order to exclude the dependence of the thermal conduction coefficient on this parameter. The results of simulations performed by various methods are also presented, as well as the result of measuring (for comparison) the thermal conductivity for individual single-layer graphene. As may be seen, the thermal conductivity of a few-layer graphene structure approaches that of crystalline graphite already when the number of layers reaches four.

An alternative approach to measuring the thermal conduction coefficient of graphene was applied in Ref. [60], where the graphene samples were synthesized by the reduction of graphene oxide that was produced using the standard Hummers method [61]. The graphene oxide reduction procedure was performed in flowing nitrogen at a temperature of 450 °C and lasted from 5 to 60 min. The thermal conductivity of the reduced graphite samples was determined by combining the results of measuring the temperature of the sample heated by electrical current with the solution of a 1D heat conduction equation. The temperature of the sample was measured by means of a Pt thermocouple.

The thermal conductivity of samples was measured under vacuum conditions at a residual gas pressure of less than 0.03 Torr. The measurement device included a SiN substrate on which the longitudinal silicon pads used as contacts were placed. The distance between the pads ranged between 0.5 and 3 mm. A graphene sample was applied to the substrate in such a way that the electrical and thermal contacts with the pads were maintained. In one measurement configuration, a graphene sheet was suspended between those pads not having contact with the substrate, while in another one a graphene sheet partially lay on the substrate. The structure of the measuring system permitted taking measurements of both the thermal conductivity and the electrical conductivity of the sample simultaneously. The results of these measurements at room temperature are given in

Table 4. Transport characteristics of thermally reduced graphene samples [60].

Sample	Thermal Conductivity κ, W/m K	Electrical Conductivity σ, S/m	Duration of Annealing, min	Contact Resistance, kΩ
1	2.87	62.2	60	120
2	0.87	6.21	5	2
3	0.14	6.57	5	130
4		19.5	20	300

for four samples. These samples differ in size, contact resistance, and the duration t of thermal treatment at reduction of graphene. Furthermore, samples 2 and 3 were suspended, while samples 1 and 4 were in contact with the substrate surface.

The measurement results presented in Table 4 reveal a notable dependence of the transport coefficients of graphene on the duration of the thermal treatment of samples: the more prolonged the treatment duration, the higher the coefficients of thermal and electrical conductions of the sample. While the mechanisms of the electric conduction and thermal conduction are different (electrons and phonons), the samples having an enhanced electric conductivity also demonstrate enhanced thermal conductivity. This is explained by the removal of oxygen atoms from the graphene surface because of thermal treatment. Oxygen adducts on the graphene surface determine the mechanism of electron and phonon scattering, and their presence lowers the magnitude of the relevant transport coefficients. However, even thermal treatment for one hour does not allow the total removal of oxygen. Therefore, in this case, both the thermal conduction coefficient and the electrical conductivity of graphene remain several orders of magnitude lower than those for graphene samples produced by either the mechanical exfoliation of graphite or the CVD method. This permits the conclusion of high sensitivity of the transport coefficients of graphene samples to both the method used for their production and synthesis conditions.

Recent experiments (see, for example, [62,63,64]) indicate that thermally reduced graphene oxide can possess much higher transport characteristics than those shown in Table 4. Therewith the experiments performed show that reliable, well-reproducible results can be obtained only at a rather low rate of heating. Thus, heating samples with a rate higher than 1 °C/s results in non-controllable explosive-like destruction of the material. For this reason, the rate of heating the furnace from a room temperature up to 200 °C was 2 °C/min, while the rate of the subsequent heating up to the annealing temperature was ~1 °C/s. The duration of the thermal treatment was 10 min at all the temperatures.

The thermal treatment of graphene oxide samples results in the removal of oxygen, which causes a decrease in the density of samples. Figure 6 presents the dependence of the graphite samples density on the annealing temperature. Figure 7 presents the dependence of the electric conductivity of thermally reduced graphene oxide on the annealing temperature measured by the authors of [63,64]. As is seen, the density decreases by about 3.5 times as the annealing temperature enhances from 100 up to 800 °C. The decrease in the density of the samples means that the average distance between graphene oxide flares increases approaching the value of 1.66 nm at the annealing temperature of 800 °C. This value exceeds the inter-layer distance in crystalline graphite by 4.5 times, therefore, one can believe that the thermally treated graphene oxide samples have transport characteristics close to those of graphene. This is confirmed by the results of measurement of the dependence of the electric conductivity of graphene oxide samples on the thermal treatment temperature [63,64] presented in Figure 7.

As is seen, the removal of oxygen through the thermal treatment of graphene oxide samples promotes not only enhancement in the average distance between graphene oxide flakes but also an enhancement of their electrical conductivity. The maximum reached value of the conductivity of the reduced GO (~3500 S/m) is about an order of magnitude lower than the reference value for graphite. However, taking into account that the distance between neighboring graphene oxide flakes is about 4.5 times larger than that of graphite, one can conclude that the conductivity of the material accounting for one graphene layer is only twice lower than that for graphite. Therefore, thermally reduced graphene oxide possesses a layer structure with an average inter-layer distance of about 1.5 nm and the conductivity (accounted for one layer) close to that for graphite. At such a distance, the interaction between the neighboring layers is negligible so it is natural to conclude that the layers involved in such a structure are close to graphene in their characteristics, annealing results in the thermal reduction of GO fragments, which lose added oxygen and transform to a conducting state. The conductivity of such a material has a percolation nature and is determined by the resistance of contacts between neighboring fragments, which decreases as the applied voltage enhances. The data presented in Table 4 demonstrate a rough proportionality between the thermal conduction and electric conduction coefficient. It can be used for the estimation of the thermal conduction coefficient of the thermally reduced graphene oxide samples based on the measured electrical conductivity of those measured in [63,64] and presented in Figure 7. The averaged ratio σ/κ determined on the basis of data of Table 4 accounts ≈ 25.2 S∙K/W. Using this ratio and the data shown in Figure 7, one finds that the thermal conduction coefficient of thermally reduced graphene oxide samples should reach ≈ 140 W/m K.

The thermal conductivity of an individual graphene sample is determined by the character of phonon propagation along the graphene plane. In this case, the phonon mean free path depends on such factors as the graphene size, the type and concentration of defects, and the occurrence of neighboring structures, as is the case in crystalline graphite or in a film consisting of several graphene layers. In the absence of those factors, the characteristic phonon mean free path for elastic scattering is determined by the phonon interaction processes (Umklapp processes), and their inclusion into consideration results in an increasing dependence of the thermal conduction coefficient on the graphene size. The specific shape of this dependence is determined by the Grüneisen parameter γ, which describes the effect of changing the volume of a crystal lattice on its dependence on its vibrational properties. Figure 8 presents the dependences of the thermal conductivity of defectless graphene sheet on its size [65] calculated for different values of the Grüneisen parameter (for longitudinal γ_LA and transverse γ_TA vibrational modes). As is seen, the thermal conductivity is a monotonically increasing function of the graphene sheet size independently of the choice of the Grüneisen parameter. The room-temperature magnitude of the thermal conductivity for L = 10 μm turned out to be close to 4000 W/m K which corresponds approximately to the experimental data (shown by the point).

The heat transport in defectless graphene is hindered by the phonon-phonon scattering (umklapp process). The role of this process increases as the temperature enhances because the number of phonon modes enhances with the rise of the temperature. Therefore, the temperature dependence of the thermal conductivity of defectless graphene is a decreasing function. Results of calculation of this function performed by the authors [65] for various graphene sheet size L are shown in Figure 9. The results of calculations are in satisfactory agreement with experimental data (shown by point).

The structure of real graphene samples can contain both intrinsic defects and, depending on the method of preparation, various surface functional groups. These defects contribute to the probability of acoustic phonon scattering, whereas the thermal conduction coefficient of real graphene samples depends on the number density of the most probable defects. This dependence was evaluated both by utilizing MD simulations and through Boltzmann kinetic equation calculations. Thus, the results of Non-Equilibrium Molecular Dynamic (NEMD) calculations [66,67] indicate an abrupt decrease in the thermal conductivity as the defect number density increases. In this regard, single and multiple carbon vacancies, OH-group adducts, and the roughness of the graphene sheet were considered as defects. Along with the NEMD method, the dependences of the thermal conductivity of graphene on the number density of OH groups and carbon vacancies were calculated by means of the Boltzmann kinetic equation. The results of calculation [66] of the dependence of the thermal conductivity coefficient on the defect number density are presented in Figure 10. The results obtained by the two methods are in qualitative agreement with each other and demonstrate that the defect number density promoting a decrease in the thermal conductivity by a factor of 2 is about 1% at room temperature for OH groups (Figure 10b) and about 0.1% for the case of vacancies (Figure 10a). Notice that the dependences of the thermal conduction coefficient on the defect number density calculated in the above-cited works are in qualitative agreement with the results of simulations for single-walled carbon nanotubes, which also point to a decrease in the thermal conductivity as the defect number density increases [68].

5. Thermal Conduction of Polymer Composites Doped with Carbon Nanoparticles

As it was mentioned above, the effective usage of PCM as a basis of thermal accumulating systems is possible if the thermal conductivity of PCM can be enhanced by several times. This can be reached as a result of doping PCM with nanocarbon particles the thermal conductivity of which exceeds that of PCM by 4–5 orders of magnitudes. The electrical conduction of polymer materials doped with carbon nanotubes has been studied in a lot of publications some of which were reviewed in articles [44,69]. The electric conduction in such composites has a percolation character in accordance with which the charge transport occurs through a limited number of percolation paths formed by the nanotubes contacting with each other. The heat transport in composites doped with carbon nanoparticles proceeds by a similar mechanism, however, this phenomenon has been studied at a rather lower degree.

The thermal conductivity of a polymer material doped with carbon nanoparticles is determined by several factors [70], the most important among which are (1) the characteristics of the particles used (defect content, the geometry, aspect ratio), (2) the dispersion of carbon nanoparticles within the composite, (3) the percent content loading of particles (volume% or weight%), and (4) the interfacial contact between the polymer matrix and the nanocarbon filler. The heat transport in both polymer matrix and nanocarbon particles is provided by phonon propagation. A multitude of interfaces connected doped particles with molecules of the composite hinders the heat propagation through the composite because the interfaces scatter phonons. Mechanisms of heat conduction in polymer-based composites and recent advances in the experimental and calculation research in the field have been reviewed in [71,72].

One distinguishes composites with a random orientation of graphene flakes and that with an ordered orientation. Composites of the first type are comparatively easy in preparation based on standard approaches such as solution mixing, melt mixing, in-situ polymerization, etc. [73,74,75,76,77,78,79,80]. There have been published a lot of experiments indicating a notable enhancement of the thermal conductivity of polymer composites doped with graphene flakes with a random orientation. The results of these experiments are presented in

Table 5. Parameters of polymer composites with a random orientation of graphene [73].

Polymer	Graphene Content, wt%	Thermal Conductivity, W/m K	TCE, % per wt% of Graphene	Preparation Method	Surface Preparation Method	Ref.
Py-PGMA-GNS/epoxy	3.8	1.91	225	In-situ polymerization	Non-covalent modification	[81]
f-GFs/epoxy	10	1.53	66.5	In-situ polymerization	Non-covalent modification	[82]
GnP-C750/epoxy	5	0.45	23.8	In-situ polymerization	no	[83]
DGEBA-f-GO/epoxy	4.64	0.72	52.3	In-situ polymerization	no	[84]
GS@Al2O3/PVDF	40	0.586	4.8	Solution mixing	Coated by alumina nanoparticles	[85]
Al2O3@ GNP/epoxy	12	1.49	56.4	Solution mixing	Coated by alumina	[86]
ApPOSS-graphene/ epoxy	0.5	0.348	115.8	Solution mixing	Covalent modification	[87]
GNP/PBT	20	1.98	61	In-situ polymerization	no	[88]
GNPs/PPS	37.8	4.414	49	Melt mixing	Covalent modification	[89]
PI/SiCNWs-GSs	7	0.577	21	Solution mixing	no	[90]
GP/SR	0.72	0.3	69.4	Mechanical blending	Covalent modification	[91]
PA6/graphene-GO	10	2.14	56.9	In-situ polymerization	Non-covalent modification	[92]
GNP/epoxy	25	2.67	49.4	Solution mixing	No	[93]
PVDF/FGS/ND	45	0.66	3.9	Solution mixing	no	[94]
ApPOSS-graphene/epoxy	0.5	0.348	115.8	Solution mixing	Covalent modification	[95]
IL-G/PU	0.608	0.3012	55.9	In-situ polymerization	Non-covalent modification	[96]
PA/TCA-rGO	5	5.1	357.8	Melt mixing	Covalent modification	[97]
BE/graphene	2.5	0.542	73.7	Solution mixing	Covalent modification	[98]
GNPs/silicone	16	2.6	49.7	In-situ polymerization	no	[99]

Abbreviations: Py-PGMA-GNS/epoxy: Pyrene-end poly(glycidyl methacrylate)-graphene nanosheet/epoxy composite; f-GFs/epoxy: Non-covalently functionalized graphene flakes/epoxy composite; GnP-C750/epoxy: Graphene nanoplatelets (sizes < 1 μm)/epoxy composite; DGEBA-f-GO/epoxy: Diglycidyl ether of bisphenol-A functionalized graphene oxide/epoxy composite; GS@Al₂O₃/PVDF: Alumina-coated graphene sheet/poly(vinylidene fluoride) composite; Al₂O₃@GNP/epoxy: Alumina nanoparticles decorated graphene nanoplatelets/epoxy composite; GNP/PBT: Graphene nanoplatelet/polybutylene terephthalate composite; GNPs/PPS: Graphene nanoplatelets/polyphenylene sulfide composite; PI/SiCNWs-GSs: Polyimide/SiC nanowires grown on graphene sheets composite; GP/SR: Graphene/silicone rubber; PA6/graphene-GO: Polyamide-6/graphene-graphene oxide composite; GNP/epoxy: Graphene nanoplatelets/epoxy composite; PVDF/FGS/ND: Poly(vinylidene fluoride)/functionalized graphene sheets/nanodiamonds composite; ApPOSS-graphene/epoxy: Aminopropylisobutyl polyhedral oligomeric silsesquioxane grafted graphene/epoxy composite; IL-G/PU: 1-allyl-methylimidazolium chloride ionic liquid modified graphene/polyurethane composite; PA/TCA-rGO: Titanate coupling agent modified reduced graphene/polyamide composite; BE/graphene: Bio-based polyester/graphene composite; GNPs/silicone: Graphene nanoplatelets/silicone composite.

[73]. The degree of influence of a dopant on the thermal conductivity of a composite is characterized by the factor Thermal Conductivity Enhancement (TCE) which corresponds to the enhancement of the thermal conductivity per 1% of dopant.

One should note that the preparation of a polymer-based composite doped with carbon nanoparticles presents a great challenge for researchers. An enhancement of the thermal conductivity of a composite because of doping with carbon nanoparticles can be reached under the condition of a homogeneous distribution of a carbon filler over the matrix volume. The attainment of such a distribution is hindered due to a trend of carbon nanoparticles to aggregation. For this reason, various techniques were utilized for the preparation of polymer-based nanocomposites doped with carbon particles. A systematic investigation of the dispersion degree of carbon nanotubes (CNT) in a bisphenol F-based epoxy resin in dependence on the method of composite preparation has been performed by the authors of Ref. [100]. The CNTs used were 1–6 nm in diameter and a few microns in length. Composites were prepared using such dispersion techniques as high-speed dissolver, roll-milling, ultrasonication, etc. The degree of homogeneity of the filler was controlled during the entire processing cycle by means of optical microscopy. There has been shown that the standard roll-mill technique not only enhances the dispersion of CNTs into an epoxy matrix but also that promotes the re-agglomeration of CNTs during the preparation. Polymer-based composites doped with CNT demonstrate a several times enhanced thermal conductivity at a filler concentration of about 10%.

Much higher enhancement in the thermal conductivity has been obtained for composites doped with graphene nanoflakes (GNF). Thus, the authors of [101] studied the thermal conductivity of the composite on the basis of poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF/HFP). The GNF/PVDF-HFP composite films were produced using hot mixing of the components in the presence of a solvent, molding, and subsequent solvent evaporation). GNF were dispersed in N,N-dimethylformamide (DMF) by sonication at 35 °C for 30 min. This dispersion was inserted by parts into the solution of PVDF-HFP in DMF under magnet stirring. Then the mold was thermally treated in an oven at 90 °C for 24 h to remove DMF. The results of measuring the thermal conduction coefficient vs. the filler concentration are presented in Figure 11. As is seen, the thermal conduction enhancement coefficient (TCE) reaches 10,000% at a filler loading 20 weight%.

The filler concentration, filler size, and temperature dependences of the thermal conductivity of graphene nanocomposites were studied theoretically by the authors [102] via an effective-medium approximation based on Maxwell’s far-field matching at a microscopic level. The results are in close agreement with the experimental observations over the average filler size from 200 to 1000 nm and over the temperature from 300 to 360 K, respectively. Figure 12 presents the comparison of the dependences of the thermal conductivity of the polymer-based composite on the volume concentration of the graphene dopant calculated in [102] for various sizes of graphene flakes with the relevant experimental data. These dependences have a form typical for the percolation thermal conductivity.

Figure 13 shows the temperature dependences of the thermal conductivity of polymer-based composite doped with few-layer graphene particles calculated in [92] and measured in [97] for the volume dopant loading 10%. The calculations were performed for the average graphene lateral dimension 2000 nm; the average graphene thickness 1.4 nm; the thickness of interlayer between the neighboring graphene flakes 0.14 nm. As is seen, both calculated and measured thermal conductivity demonstrate a slightly increasing temperature dependence which is typical for percolation phonon thermal conduction.

A careful study of the thermal conductivity of graphene doped polymer-based composites has been performed by the authors of [70] who prepared a set of composite samples based on polymer PVDFHEP. The polymer was doped with 98.5% pure graphene flakes in weight percentages wg varying between 0.1% and 50%. The graphene flakes have dimensions ~(7 × 3.5 × 0.001) μm³. The results of measuring the thermal conductivity coefficient of composites at various graphene content are summarized in

Table 6. Thermal conductivity of polymer-based samples with various graphene loading [70].

wg, %	0	1	6	10	20	25	33	50
Κ, W/m K	0.22	3.24	7.95	16.0	24.5	36.3	38.1	57.51

. As is seen, inserting 1% (weight) graphene results in the enhancement of the thermal conductivity coefficient as much as 15 times.

Along with carbon flakes, carbon nanotubes can be also used as an effective dopant for enhancement of the thermal conductivity of polymer-based composites. In this relation the article [107] should be mentioned, where the effects of doping polymer matrix with expanded graphite particles (EGPs) and multi-walled carbon nanotubes (MWCNTs) on the thermal conductivity of composites are compared. A PCM matrix was used in the mixture of paraffin (melting temperature 20–25 °C; latent heat 122.6 J/g), high-density polyethylene (HDPE), and styrene-butadiene copolymer (SBS). MWCNTs have an average diameter of 10 nm and an average length of 10 μm. The experiments performed have shown that both EGPs and MWCNTs increase the thermal conductivity of PCMs, while EGPs demonstrate a greater thermal conductivity improvement than MWCNTs. The conductivity of EGP-filled PCM reached 0.574 W/mK at 9 wt%, while that of MWCNT was just 0.372 W/mK at the same loading.

Figure 14 presents the dependences of the measured thermal conductivity of a paraffin-based PCM doped with a single filler (MWCNT) and expanded graphite particles (EGP) (a) and hybrid filler (EGP/MWCNT) (b) on the filler content. The measured dependences demonstrate a synergistic effect in the enhancement of the thermal conductivity. Thus, when the EGP/MWCNT ratio was 8:2, the most significant thermal conductivity enhancement to the SSPCM was obtained. The thermal conductivity was 0.674 W/mK, 288% that of the SSPCM and 117% that of 9 wt% EGP-filled SSPCM. The advantage of hybrid filler is in (1D) MWCNT bridges connecting 2D planar EGP. Seemly EGP-MWCNT junctions have a lower thermal resistance compared to that of EGP-EGP and MWCNT-MWCNT junctions.

6. Conclusions

The usage of PCMs in the building trade permits one to limit the level of energy production and consumption, which offers a possibility to lower the negative impact of the industry on the environment and terrestrial climate. The wide spread of PCMs is hindered by a rather low thermal conductivity of these materials. This drawback can be overcome by doping PCMs with nanocarbon particles for which the thermal conductivity coefficient exceeds that for PCMs by 4–5 orders of magnitude. Due to a rather complicated geometry of nanocarbon particles, the theoretical determination of the thermal conductivity coefficient of polymer-based composites doped with them seems to be hardly possible so the main source of reliable data on this subject is experimentation. By now, a large number of experimental works have been published where the thermal conductivity coefficients of polymer composites doped with carbon nanoparticles were measured. The data obtained are characterized by a notable spread but all of them demonstrate a considerable enhancement of the thermal conductivity due to nanocarbon doping. However, the effect of doping polymers with nanocarbon particles on the thermal conductivity of a material depends critically on the method of preparation of composite. This is caused by a trend of carbon nanoparticles toward aggregation, which prevents a homogeneous distribution of the filler over the matrix volume. Besides that, the wide spread of composites in the building industry and other fields is hindered by a rather high production cost of carbon nanoparticles such as carbon nanotubes and graphene. The development of methods of large-scale production of nanocarbon materials with a decreased production cost should stimulate the usage of PCMs with enhanced thermal conductivity in the building industry, in thermal accumulating systems, for temperature stabilization of big computer systems, etc.