Introduction

Deep eutectic solvents (DESs) [1] are formed by the eutectic mixture of two compounds, a hydrogen bond donor (HBA) and hydrogen bond acceptor (HBA). The HBA–HBD mixing at a suitable ratio lead to a system with melting point lower than those for the corresponding components [2]. Therefore, forming liquid systems at close to ambient temperature conditions [3], which can be used as solvents or heat transfer materials. The decrease in the melting point upon HBA/HBD mixing is produced by the development of strong hydrogen bonds [4]. Although there is some controversy in the literature for the definition of what a DESs is, we follow in this work the definition of DES based on Abbot’s work [1], considering classical DES for which the HBA–HBD interaction through hydrogen bonding is a necessary condition.

The number of possible HBA/HBD combinations is extremely large, thus forming a new group of solvents (materials) with properties tuned by the suitable combination of HBA–HBD compounds to be applied in many technological challenges [5]. Likewise, there is a large group of natural HBA–HBD compounds, which may form DESs, leading to the so-called natural DES (NADES) [6], thus forming a suitable platform for developing environmentally friendly DESs [7].

The large attention attracted by DES is justified considering their promising physicochemical properties [8], which can be fine-tuned through suitable HBA–HBD combinations [9] to obtain environmentally friendly solvent profiles with low toxicity [10, 11] and biodegradability [12]. DES have been proposed to be used in different applications as solvents for solvent replacement studies [13]. Likewise, applications in pharmacology [14], as drug delivery vehicles [15], for gas capture operations [16] with the emphasis on CO2 [17], bio-refining [18], fuels treatment [19], extraction operations [20], water treatment [21], or nanomaterials development [22]. Biodegradability and toxicity of DES–NADES [23, 24] is good enough to consider them as vehicles for pharmacological applications [25], and their effect on environment may be considered as minor [26]. Likewise, the possibility of developing DES from different HBA–HBD combinations, many of them with natural origin, is a competitive advantage for developing these fluids at low to moderate cost [27], in comparison with the closely related counterparts like ionic liquids [28]. Moreover, DES, and specially NADES, have proved suitable physicochemical properties [29], with densities, usually larger than water [30], moderate to large viscosity [31], which may hinder some of their possible applications but can be fine-tuned through composition effects or by mixing with organic solvents or water [32]. Likewise, DES have almost negligible vapour pressure [33], which consider them as suitable replacements of volatile organic compounds, a.k.a. VOCs [34]. Regarding the thermal properties [35], although the available information is still scarce and no systematic studies have been developed, the heat capacities of these fluids are large enough [36] for considering them as suitable fluids in many heat transfer applications [37]. The state-of-the art of DES physicochemical properties has been recently reviewed in different publications [38,39,40,41] showing their relevance for the application and scaling up of DESs for several types of technologies. These studies have probed that among the most relevant thermal properties of DES, thermal conductivity (λ) is crucial for their application in heat transfer operations [42], considering that these fluids have been proposed for several applications including their use as heat transfer fluids [43, 44]. Therefore, the development of systematic knowledge of thermal conductivity for DESs as a function of HBA–HBD types, temperature, and eutectic composition is pivotal for the advance in the possible industrial applications of DESs as alternative compounds to traditional heat transfer fluids. The available literature on thermal conductivity for DES is reviewed in this work with the objective of analysing understanding the state-of-the-art as well as the suitability of using these materials in thermal-related operations. The relationships between HBA–HBD composition and thermal conductivity is also analysed as well as the nanoscopic origin of this property with the objective of developing suitable micro- to macroscopic relationships for designing suitable DESs for thermal industrial operations. The novelty and need of this work may be summarized as follows:

  1. (i)

    The critical review and analysis of thermal conductivity of DESs is reported for the first time in terms of physical conditions (temperature), composition (HBA–HBD components), and intermolecular forces.

  2. (ii)

    The molecular level basis of this property is analysed for the first time.

  3. (iii)

    The available predictive models are critically discussed in terms of their accuracy and performance.

  4. (iv)

    A strengths and weaknesses analysis of the state-of-the-art for this property is reported for the first time with reported recommendations for future research directions and applications.

  5. (v)

    The technological relevance of and accurate and systematic knowledge of thermal properties for DES require the reported analysis and discussion to show the current problems hindering scaling up as well as the possible solutions.

Experimental thermal conductivity database

Pure DESs

The available literature reporting experimental measurements for the thermal conductivity of DESs is summarized in Table 1. In spite of the relevance of thermal conductivity for the aforementioned applications, the reported data are still very scarce and mostly limited to choline chloride (ChCl)-containing DESs, with only 14 relevant publications in the 2016–2020 period. Most of the reported data were obtained using the transient hot wire method, thus leading to 3–5% accuracy for this type of fluids, as indicated in Table 1. The analysis of the literature shows the following needs: (i) extension of measurements to non-ChCl DESs, (ii) validation of transient hot wire method in comparison with other methods, and (iii) effect of purity on DESs thermal conductivity including absorbed water effect.

Table 1 The literature sources reporting experimental measurements of thermal conductivity for DES. For each DES, the involved HBA:HBD components are reported as well as the corresponding molar ratio (i:j)
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The analysis of the literature shows some of the most relevant DESs being reported in different literature sources showing different values, Fig. 1. The values for ChCl:urea or :glycerol or :ethylene glycol, all of them at 1:2 molar ratio, show differences across the literature but being in the 5% range, which may be considered reasonable in terms of fluids with different purities, i.e. water content, and experimental methodologies.

Fig. 1
figure 1

Experimental thermal conductivity, λ, at 298 K for the reported DESs from different literature sources

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The data reported in Fig. 2 show the HBD effect on the thermal conductivity for ChCl-containing DES. The reported results indicate that although including different HBDs into the DES formulation lead to different thermal conductivity values, this effect is very minor considering that an average thermal conductivity of 0.208 W m−1 K−1 is inferred for all the considered ChCl-based DES, with just 0.017 W m−1 K−1 standard deviation, considering HBDs with very different molecular structures. Thus, HBA type determines the value of thermal conductivity with a minor effect considering the HBD type. Nevertheless, some exceptions were found in the literature, Gautam et al. [42] reported λ = 0.245 W m−1 K−1 for ChCl:urea (1:2) at 298.15 K whereas 0.182 W m−1 K−1 were reported for ChCl:thiourea (1:2) at the same temperature. Thus, the replacement of an oxygen atom for a sulphur one decreases the λ values, which may be justified considering the inverse relationship between λ and molar mass. Likewise, all the HBDs considered in Fig. 2 contain the same type of atoms (C,H,O) with the exception of thiourea, thus leading to lower λ for ChCl-based DESs.

Fig. 2
figure 2

Experimental thermal conductivity, λ, for ChCl:HBD at 298.15 K from Ibrahim et al. [50]. Grey dashed line indicates average value (standard deviation) for the reported DESs

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Regarding the effect of HBA type, the scarcity of the literature does not allow to infer systematic conclusions. For closely related HBAs, the literature shows very minor effects. Gautam et al. [42] reported λ = 0.245 W m−1 K−1 for ChCl:urea (1:2) and λ = 0.242 W m−1 K−1 for diethylethanol ammonium:urea (1:2), both at 298 K. Likewise, Chen et al. [57] studied alkylammonium bromide:glycerol (1:2) reporting 0.217, 0.209, and 0.209 W m−1 K−1 at 293.15 K on going from ethyl to propyl to butyl, respectively, in the HBA. On the contrary, Chen et al. [57] studied ChCl:ethylene glycol and methyl-triphenylphosphonium bromide:ethylene glycol (1:3) reporting λ = 0.195 W m−1 K−1 and 0.168 W m−1 K−1 at 298.15 K, respectively, thus confirming that large changes in thermal conductivity are only obtained when involving structurally very different HBAs.

An additional factor to be considered for analysing DESs thermal conductivity is the HBA:HBD molar ratio. Although in a strictly thermodynamic sense, the eutectic composition is a single value for each HBA:HBD combination, different compositions around the true eutectic ones are considered for DESs development in the literature. Therefore, selected DES with different HBA:HBD molar ratios were considered and reported in Fig. 3. These results show almost negligible effect of HBA:HBD i:j molar ratios on thermal conductivity, with increasing HBD content in the DESs leading to almost the same thermal conductivity data for a fixed HBA.

Fig. 3
figure 3

Experimental thermal conductivity, λ, for ChCl:HBD (1:j) at 305 K (for ChCl:glycerol [44]) or 298.15 K (remaining DESs [57]) as a function of j. MTPB stands for methyl-triphenylphosphonium bromide

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The temperature effect on DESs thermal conductivity was also analysed. Results in Fig. 4 show temperature evolution of thermal conductivity for ChCl-based DESs. According to the theory of molecular liquids [50], the thermal conductivity should decrease when intermolecular distance increases, i.e. with increasing temperature. Nevertheless, results in Fig. 4 show more complex behaviour. The available literature shows severe discrepancies between the available sources even for the most common DESs included in Fig. 4. In the case of ChCl:urea (1:2), decreasing thermal conductivity with increasing temperature is inferred, following a linear behaviour. Nevertheless, the (dλ/dT) slope is very different considering the literature sources, Fig. 4a. Values from Ibrahim et al. [50] show negligible variations, below the measurement uncertainties, with temperature whereas Gautam et al. [42] reported a large decrease with increasing temperature. The cause of these large discrepancies may be on the purity of the samples, including water content, and on the experimental methods. Negative slope is also inferred for ChCl:ethylene glycol (1:2), Fig. 4c. Likewise, positive (dλ/dT) slopes are inferred for some DESs as for ChCl:glycerol (1:2) reported in Fig. 4b but with large discrepancies considering the different literature sources. Positive (dλ/dT) slopes have been reported in the literature for other types of DESs such as ChCl:lactic acid [51], or methyl-triphenylphosphonium bromide:ethylene glycol [60]. For non-ChCl-based DES, studies on temperature effect are very scarce, and in some cases like alkylammonium bromide:glycerol [56], thermal conductivity remains constant for the studied temperature ranges (30 °C). Therefore, the temperature effect on thermal conductivity for DESs is still not well understood, and there is a certain need of additional studies for (i) inferring the sign of (dλ/dT) slopes in terms of the involved HBA:HBD compounds and (ii) determining the values of (dλ/dT) slopes, which, in most of the reported literature studies, are below the experimental uncertainty.

Fig. 4
figure 4

Temperature, T, effect on thermal conductivity, λ, for selected DESs. Continuous lines show linear fits, with the slope indicated inside each plot. Dashed lines show 99% confidence interval of the linear fits

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Modelling and prediction of thermal conductivity

The development of theoretical models for predicting the thermal conductivity of DESs is required considering the large number of possible HBA:HBD combinations as well as for screening purposes to find the most suitable molecular combinations for DESs thermal applications. The Bridgman equation [61] is a semiempirical model which allows to calculate thermal conductivity, from molar volume (i.e. density) and speed of sound, Eq. [1]:

$$\lambda =a{\left(\frac{{N}_{\mathrm{A}}}{V}\right)}^{2/3}{k}_{\mathrm{B}}u$$
(1)

where a stands for an empirical parameter, NA stands for the Avogadro number, V stands for the molar volume, kB stands for the Boltzmann’s constant, and u stands for the speed of sound. Thus, this model allows to predict λ, which is a property difficult to be measured and usually reported with large uncertainties, with two easily available properties such as density and speed of sound. Nevertheless, the available literature information on density and speed of sound also shows non-negligible discrepancies. Results in Fig. 5a show experimental density data for ChCl:urea (1:2) with some literature sources showing relevant differences with other ones. Likewise, the literature on DESs speed of sound is also scarce and large differences between the available sources are inferred, Fig. 5b. Therefore, uncertainties in the use of Bridgman equation may rise from the differences in the available density and speed of sound data.

Fig. 5
figure 5

Experimental density, ρ, speed of sound, u, and dynamic viscosity, η, for ChCl:urea (1:2) at 298.15 K from different literature sources. Lapeña et al. [62], Haghbakhsh et al. [63], Yadab et al. [64], Agieienko et al. [65], Jabbar et al. [66], and Gautam et al. [67]

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A relevant issue in the application of Eq. (1) for predictive purposes stands on the value of a factor, which is frequently considered as a = 3.0. The results of the application of Bridgman model for some selected ChCl-based DESs are reported in Fig. 6. Two different a values were considered (a = 3 or 4), with the model leading to poor predictions for a = 3 and excellent results for a = 4. Therefore, the model works properly for the selection of a suitable a factor.

Fig. 6
figure 6

Thermal conductivity, λ, for ChCl:HBD (1:2) at 298 K from experimental data (Ibrahim et al. [50]) and those calculated from Bridgman equation, with two different a values (Eq. 1), from experimental density and speed of sound obtained in the literature (Ref. [66])

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Despite the relevance of developing predictive models for thermal conductivity, the available modelling studies are very scarce in the literature. Considering that thermal conductivity is a dynamic property, the possible relationship with viscosity may be considered, as previously done for ionic liquids [68], according to Eq. (2) based on the Mohanty theory [69]:

$$\frac{M\lambda }{\eta }=\mathrm{constant}$$
(2)

where M stands for the molar mass, and η stands for the dynamic viscosity. The development of λ versus η relationships has a relevant problem on the available η data from the literature. Results in Fig. 5c show literature η data for ChCl:urea (1:2) DES at 298.15 K from different sources showing large differences. Although the main reason behind the fluctuation on η data is not certain, it is most likely due to different DES purities, experimental methodologies or unknown factors not properly reported in the corresponding literature sources (e.g. poor calibration). This situation has been previously reported for many different types of DESs [8] and currently hinders a systematic evaluation of λ versus η relationships. Nevertheless, the possibility of DESs fulfilling Eq. (2) was analysed for several selected ChCl-based DESs, Fig. 7a. These results discard a suitable fulfilling of Eq. (2) even for the most studied DESs, with the linear relationship mostly discarded. Likewise, the application of Eq. (2) for ionic liquids [68] showed a quasilinear evolution with negative slope, whereas results in Fig. 7a show positive slope for ChCl-based DESs. The analysis of the direct relationship between dynamic viscosity and thermal conductivity is reported in Fig. 7b, showing a nonlinear evolution but inferring that increasing viscosity leads to an increase in thermal conductivity. This behaviour can be attributed to an increase in intermolecular forces (hydrogen bonding) as the origin of increasing viscosity, which maintain the molecules closely packed together, thus leading to a larger thermal conductivity. Nevertheless, considering the complexity of intermolecular forces in DESs, in terms of type and extension [70], a complex η versus λ relationship may be expected as reported in Fig. 7b. The development of these η versus λ models would require the availability of accurate and reliable viscosity data, which, at the current state, are not available in the literature, Fig. 5c.

Fig. 7
figure 7

a Results for ChCl-based DESs for testing the Mohanty model and b viscosity versus thermal conductivity for the same DESs. All values at 298.15 K. Thermal conductivity data from Ref. [50] and viscosity data from the reported sources (Refs. [65 [71, 72]]). The considered DESs are ChCl:urea (URE) (1:2), ChCl:glycerol (GLY) (1:2), ChCl:ethylene glycol (EG) (1:2), and ChCl:triethylene glycol (TEG) (1:3)

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Regarding the modelling of the temperature evolution of thermal conductivity and its possible relationship with viscosity evolution, the available results show large decrease in viscosity upon increasing temperature whereas thermal conductivity suffers minor changes upon heating, Fig. 4, i.e. (dλ/dT) <  < (dη/dT). Thus, the η versus λ representation in Fig. 8 for ChCl:urea (1:2) does not show any relationship, and although both are dynamic properties, the decrease in viscosity upon heating, as a consequence of the weakening of intermolecular forces and increase in the average intermolecular distance is not accompanied by a large decrease in thermal conductivity, Figs. 4 and 8. It should be remarked that some authors have reported thermal conductivity increasing with temperature, Fig. 4b, whereas DESs viscosity always decreases with increasing temperature [73].

Fig. 8
figure 8

Relationship between thermal conductivity, λ, and dynamic viscosity, η, for ChCl:urea (1:2) as a function of temperature indicated inside the plot. Experimental values from [50, 65]

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Additional attempts to develop predictive models for DESs thermal conductivity are almost absent in the literature. Ibrahim et al. [50] reported a model, so-called extended Wu model [EWM], which was used for the estimation of ChCl-based DESs thermal conductivity. This model used the Wu approach [74], and it is based on a group contribution approach, which requires parameters for (i) the determination of boiling and critical temperatures of DESs and (ii) for the determination of λ0 parameters of the model. Likewise, this model considers a DESs as a mixture of two components, and thus, the thermal conductivity of the mixture (DESs) is obtained from those of the components (HBAs and HBDs) using suitable mixing rules and correlative parameters, which need to be adjusted to experimental data. Therefore, although the model led to deviations with experimental data lower than 2%, it requires a large number of group contribution parameters, which are available only for selected compounds, as well as experimental data for the mixing rules, i.e. it is not a purely predictive model, and thus, it may have limited practical application. A recent review on the modelling of thermodynamic and transport properties of DESs did not include any relevant effort for thermal conductivity prediction [75]. From the molecular level viewpoint, molecular simulation studies may provide information on the nanoscopic roots of thermal conductivity in DESs. Nevertheless, these modelling studies are very scarce in the literature. In a recent work, Celebi et al. [76] reported a molecular dynamics (MD) study on thermal conductivity for aqueous solutions of DES. These authors probed the role of the nature and strength of HBA–HBD hydrogen bonding on the values of thermal conductivity. Likewise, the negligible effect of temperature was confirmed as well as the large effect of water content as it affects the extension of DESs hydrogen bonding. Although molecular modelling studies on the relationships between thermal conductivity and intermolecular forces (hydrogen bonding) are scarce in the literature, the previous studies have showed the relevance of hydrogen bonding for dynamic properties such as viscosity or self-diffusion coefficients. The studies by Perkins et al. [77, 78] and Celebi et al. [79] probed that the nature, extension, and strength of the HBA–HBD hydrogen bonding determine the values of dynamic properties, with a common trend to maximize the hydrogen bonding network between the different molecular moieties. Therefore, there is a crucial need for developing accurate predictive approaches, based on the reliable understanding of molecular level effects controlling the property, considering the large number of HBA:HBD combinations leading to DESs as well as the validity of DESs for thermal engineering applications.

Conclusions

The analysis of the available literature on thermal conductivity for deep eutectic solvents had led to several conclusions:

  1. (i)

    The available data are mostly limited to choline chloride-based solvents with scarce studies on other types of systems.

  2. (ii)

    The temperature effect on thermal conductivity should be clarified, considering that minor changes are inferred for most of the systems and increasing or decreasing values are obtained upon heating.

  3. (iii)

    The main effect on the property is caused by the type on considered hydrogen bond acceptor. In order to infer on the role of the hydrogen bond donor, more data are required.

  4. (iv)

    The relationship with other relevant properties, especially viscosity, is not clarified considering the large differences among the values reported from different sources.

  5. (v)

    Predictive models have not been reported in the literature, thus limiting the applicability of the considered solvents.

Therefore, the following recommendations have been made for developing a detailed analysis on the thermal properties of DESs:

  1. (i)

    Systematic studies on selected groups of deep eutectic solvents, considering different groups of hydrogen bond donors and acceptors are required.

  2. (ii)

    The purity and water content of the DES samples should be fully clarified, and their effect on thermal conductivity should be reported.

  3. (iii)

    The studies should be carried out in wider temperature ranges using highly accurate experimental methods with reduced experimental uncertainties.

  4. (iv)

    Predictive models should be developed, for which group contribution approaches following a parallel effort to ionic liquids could be considered as a suitable starting point.

  5. (v)

    Molecular modelling studies, mainly molecular dynamics, should be developed for analysing the relationship between molecular structures of hydrogen bond donors–acceptors, nature and extension of hydrogen bonding, and thermal conductivity, thus allowing to infer the molecular level and nanoscopic roots of this property.

A strengths, weaknesses, opportunities, and threats (SWOT) analysis is included in Fig. 9 to summarize the conclusions obtained in this work.

Fig. 9
figure 9

SWOT analysis of thermal conductivity for DESs

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