# Enhanced Specific Heat Capacity of Liquid Entrapped between Two Solid Walls Separated by a Nanogap

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{v}) on a nanometer scale have been investigated by controlling the temperature and density of the liquid domain using equilibrium molecular dynamics (EMD) simulations. Lennard-Jones (LJ) type molecular model with confinement gap thickness (h) 0.585 nm to 27.8 nm has been used with the temperature (T) ranging from 100 K to 140 K. The simulation results revealed that the heat capacity of the nanoconfined liquid surpasses that of the bulk liquid within a defined interval of gap thickness; that the temperature at which maximum heat capacity occurs for a nanoconfined liquid vary with gap thickness following a power law, T

_{Cv,max}= 193.4 × (h/a)

^{−0.3431}, ‘a’ being the lattice constant of Argon (solid) at 300 K; and that for a specified gap thickness and temperature, the confined liquid can exhibit a heat capacity that can be more than twice the heat capacity of the bulk liquid. The increase in heat capacity is underpinned by an increase in non-configurational (phonon and anharmonic modes of vibration) and configurational (non-uniform density distribution, enhanced thermal resistance, guided molecular mobility, etc.) contributions.

## 1. Introduction

_{2}with diameters 1.3, 2.8 and 5.1 nm. Recently, Khler et al. [4] have shown that for wider nanopores, the size dependence is less relevant once the fluid structure is bulk-like. Leng [14] reported a 5 to 16-fold viscosity of water confined in 0.7–1.35 nm nanopores compared to its bulk counterpart. Heat Capacity is related to the viscosity (µ) via Prandtl Number (Pr) and Prandtl Number can be correlated with the reciprocal of the reduced temperature (T

_{r}= T/T

_{c}) [15]. Hence a qualitative enhancement of heat capacity of liquid due to confinement is expected [16] although the magnitude may vary quantitatively.

_{2}within polymeric norbornene aerogel and reported its freezing behavior. They have reported a greater effective heat capacity of H

_{2}inside a norbornene-based aerogel than its bulk counterpart up to a certain range of temperature (0−300 K). They attributed it to a thermodynamic effect instead of the kinetic behavior of the entrapped H

_{2}molecules [7]. Earlier, Cleve et al. [21] reported temperature dependencies of heat capacity of Hydrogen and Deuterium within the silica aerogel for a variety of filling fractions which actually indicates size- and temperature-dependent heat capacity of nanoconfined Hydrogen and Deuterium.

_{SF}) between the solid–liquid thin film drops by one order of magnitude compared to its bulk counterpart when the liquid consists of only one layer. However, as soon as the film thickness increases, R

_{SF}increases rapidly towards R

_{SL}. They explained it with the difference in vibrational density of states (VDOS) of the nanolayer and the solid walls. However, they failed to provide a concrete explanation regarding why the solid-film thermal boundary resistance increases insignificantly after a few layers and finally approaches a constant value [23].

_{v}) of liquid Ar at different liquid film thickness (h) and temperature (T) has been studied. Finally, non-configurational and configurational contributions of the entrapped liquid have been analyzed to explain this behavioral anomaly of nanoconfined liquid.

## 2. Modeling and Simulation Procedures

^{3}) bounded by two solid Copper (Cu) (8960 kg/m

^{3}) plates as can be seen from Figure 1. There are three monolayers of solid Cu at each plate. The simulation box is 6.14 × h × 6.14 nm

^{3}. Each monolayer consists of 578 molecules of Cu and is in FCC (100) crystal structure. A cutoff distance (r

_{c}) of 2.5σ is used considering the computation cost which is also supported by previous studies carried on similar systems [23,24]. (For r

_{c}= 2.5σ, the resulting error in the number density profile of liquid Ar is ≤2%). The truncated Lennard-Jones (LJ) potential as shown in Equation (1) is used to mimic the Van der Waals interactions between liquid–liquid and solid–solid system. The interaction parameters for solid-liquid interface has been determined using the Lorentz–Berthelot mixing rule [25].

_{overall}) and liquid region’s thermal resistance, (R

_{liquid}) were calculated using Equations (4) and (5) respectively [24].

_{T}’ denotes the total energy of liquid Ar, ‘K

_{B}’ is the Boltzmann constant and ‘T’ is the absolute temperature of liquid Ar.

## 3. Results and Discussions

#### 3.1. Effect of Gap Thickness on Heat Capacity

_{v}of the nanoconfined liquid significantly enhanced, beyond which, this enhancement suddenly vanished approaching C

_{v}to its bulk value. For instance, the C

_{v}of the confined liquid Ar started to increase from its bulk value (19.95 J/mol.K) when h/a decreased beyond 10.5 (gap thickness 6.14 nm). At h/a = 7.0 (gap thickness 4.1 nm), the C

_{v}became the maximum with a value of 44.36 J/mol.K. Quantitatively, this value is about 2.3 times greater in magnitude than its bulk equivalence at 100 K. If the wall separation distance is lowered further, C

_{v}descends from its apex. Interestingly, when h/a = 1.5 (gap thickness 0.88 nm), the C

_{v}of the nanoconfined liquid again followed the value of its bulk counterpart. The increased heat capacity of liquid Ar was observed to be consistent with the experimental results reported earlier for water under molecular scale confinement [12,13,14,16,17].

_{v}of nanoconfined liquid deviates from C

_{v}of the bulk liquid after a certain gap thickness.

^{*}; (h/a) = 7 for Figure 2), ballistic transport starts to dominate across the thin film [31,32]. The Umklapp scattering starts to take the lead over normal scattering of thermal phonons leading to a lower than usual transmission of thermal phonon [33]. Consequently, an increase in C

_{v}of the nanoconfined liquid results in when h/a approaches (h/a)

^{*}as depicted in Figure 2.

_{v}of the nanoconfined liquid drops when the film thickness goes beyond the critical gap thickness. Liang and Tsai [22] observed a similar incident in a nanogap confined liquid consisting of a variety of crystalline and liquid thin film.

_{y}’ denotes the thermal energy flux of ‘V’ volume of atoms each having velocity ‘v

_{i}’ and potential energy ‘φ

_{i}’, ‘F

_{ij}’ represents the force of interactions between atom ‘i’ and ‘j’ separated by a distance ‘y

_{ij}’.

^{*}.

_{v}could also be expected. However, actually in these cases, C

_{v}remains the same as that of the bulk liquid because the effect of the slab or confinement is actually nullified by the natural tendency of the liquid molecules to follow a disordered path within their unconstrained larger portions. Figure 5a,c supports this hypothesis.

#### 3.2. Effect of Temperature on Heat Capacity

_{v}) with the confinement gap thickness (h) has been extended from 100 K to 140 K. This range is selected to ensure a subcooled liquid phase considering the critical temperature of Ar to be 150.687 K at a density of 1320 kg/m

^{3}[37]. The observed results represent the effect of nanogap confinement on heat capacity, which is valid up to a specific range of gap thickness that is strongly a function of temperature of the entrapped liquid beyond which C

_{v}of the nanogap confined liquid approximates to C

_{v}of the bulk liquid. As an example, the unusual behavior of the heat capacity occurs up to h/a ∼ 10.5, 8.0 and 6.0 for T = 100 K, 110 K and 120 K respectively as observed in Figure 2 and Figure 6. For T = 130 K and 140 K, this variation of C

_{v}is observed up to h/a ∼ 5.5 as can be seen from Figure 6. This leads to the conclusion that as the temperature increases, the range of gap thickness where the unusual behavior of heat capacity takes place gets narrower. Additionally, from Figure 6, it is evident that for all these cases, when h/a < 1.5, lower than usual value of the heat capacity occurs which was explained earlier.

_{v,max}occurs at h/a ∼ 7.0, 5.5, 4.0, 3.5 and 3.0 for T = 100 K to 140 K indicating that for a higher temperature, C

_{v,max}occurs at a lower film thickness. From Figure 2 and Figure 6, it is evident that the maximum value of heat capacity is 44.36, 33.4387, 29.71, 27.68 and 26.57 J/mol.K for T = 100 K to 140 K. This leads to the hypothesis that the higher the temperature of the confined liquid, the lower the value of its maximum heat capacity.

_{v}of nanoconfined liquid is greater than its bulk counterpart and the maximum heat capacity (C

_{v,max}) is a function of temperature, in all the cases, the positive and negative slopes were almost equal in magnitude as illustrated in Figure 7. This actually proposes that up to a critical value of the h/a ratio, denoted as (h/a)

^{*}, the change in heat capacity is independent of the gap thickness.

_{ITR}= 133.7 T

^{−0.77}. Such a phenomenon is supported by numerous earlier studies [24,41,42]. Previously Song and Min [41] showed that the ITR reduces with temperature in the form of R

_{ITR}∝ T

^{−α}irrespective of the interface where ‘α’ is any nonzero positive number. This points out that with the increase in temperature, ITR decreases. This in turns allows easy transportation of thermal energy. As a result, the heat carrying capability of the medium decreases. It is evident from Figure 10b. However, this overall reduction of ITR is a function of the gap thickness. For example, when h/a = 4.0 i.e., h = 2.34 nm, ITR varies with T as a power of −0.90. However, when h/a = 25.0 i.e., h = 14.63 nm, ITR decreases as a power of −0.38. When the overall thermal resistance becomes lower, thermal energy transportation becomes easier. As a result, when the gap thickness is small, a lower temperature produces an enhanced heat capacity of the liquid.

#### 3.3. Relation among Gap Thickness, Temperature and Heat Capacity

#### 3.4. Maximum Heat Capacity of the Confined Liquid

#### Relation between Gap Thickness and Temperature for Maximum Heat Capacity

_{v,max}) is occasionally of great interest in many physical processes, it is investigated separately. Actually, maximum heat capacity, the temperature and gap thickness at which it occurs–all are correlated. Maximum heat capacity occurs only at some specific gap thickness and temperature of the confined liquid as is presented in Figure 12a beyond which heat capacity deviates from its maximum value. This one leads to an anticipation that the temperature and gap thickness at which maximum heat capacity occurs for a nanogap confined liquid are correlated. As observed from Figure 12b, the temperature at which C

_{v,max}occurs varies with gap thickness following a power law, T

_{Cv,max}= 193.4(h/a)

^{−0.3431}. However, for the bulk liquid, it is only the temperature that influences its heat capacity and there is no influence of liquid film thickness on heat capacity of the bulk liquid.

## 4. Conclusions

_{v}) of nanogap confined liquid using equilibrium molecular dynamics simulation and a specified solid–liquid model. Key findings of the molecular simulations are:

- Nanoconfinement increases the heat capacity of the liquid within a specific range of gap thickness and temperature beyond which heat capacity of the confined liquid approaches to that of its bulk counterpart. Dependence of heat capacity on pore size and temperature was reported earlier for different fluids including water, hydrogen, deuterium, etc. confined in cylindrical nanopores [12,16,20,38].
- For a fixed temperature, as the gap thickness increases beyond one atomic layer, the heat capacity increases dramatically at first; reaches a maximum value at some specified gap thickness and eventually decreases to align to that of the bulk liquid;
- This specific range of gap thickness depends directly on temperature. This range gets narrower with the increase in temperature;
- Maximum heat capacity, gap thickness and temperature all are correlated. The temperature at which maximum heat capacity occurs for a nanoconfined liquid is observed to vary with gap thickness following the power law, ${T}_{Cv,max}=190.7{\left(\frac{h}{a}\right)}^{-0.3345}$;
- The broad maximum in C
_{v}against h/a plot for nanoconfined liquid shifts to a lower value at higher temperature i.e., the more the temperature of the confined liquid, the lower the value of its maximum heat capacity; - A heat capacity, almost two times larger than its bulk counterpart, is achievable for a predefined gap thickness and temperature (C
_{v,nc}≈ 2.3 × C_{v,bulk}for h = 4.1 nm and T = 100 K); - The enhanced heat capacity is attributed to the simultaneous effects of configurational and non-configurational contributions of the confined liquid.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**(

**a**) The ’Solid-like-liquid’ region (Rendered from xz plane for gap thickness, h = 2 nm) (

**b**) Number density distribution of the nanoconfined liquid.

**Figure 4.**(

**a**) Contribution of molecular motion in total energy transfer, and (

**b**) diffusion coefficient of entrapped liquid under varying thickness and constant temperature.

**Figure 5.**Trajectory of molecules’ displacement (T = 100 K) at a predefined time interval (Frame interval 0 to 1000) (

**a**) Nanogap confined liquid with h/a = 12.5 i.e., h = 7.3 nm. (

**b**) Nanogap confined liquid with h/a = 3.0 i.e., h = 1.8 nm. (

**c**) Bulk liquid.

**Figure 6.**Heat capacity of nanoconfined liquid as a function of gap thickness at various temperatures (110 K to 140 K).

**Figure 7.**Size and temperature effect on heat capacity of nanogap confined liquid. (

**a**) Actual, (

**b**) normalized.

**Figure 8.**For different system temperatures (

**a**) density profile of the liquid Ar near the Cu wall, (

**b**) density profile of the liquid Ar far from the Cu wall.

**Figure 10.**Temperature dependence of (

**a**) interface thermal resistance, (

**b**) overall thermal resistance.

**Figure 12.**(

**a**) Combined effect of gap thickness and temperature for maximum heat capacity of nanoconfined liquid. (

**b**) Relation between temperature and gap thickness for maximum heat capacity.

Pair Interaction | ${\mathit{\u03f5}}_{\mathit{i}\mathit{j}}\left(\mathbf{J}\right)$ | ${\mathit{\sigma}}_{\mathit{i}\mathit{j}}\left(\mathrm{nm}\right)$ |
---|---|---|

Liquid-Liquid | 1.67 × 10^{−21} | 0.3405 |

Solid-Solid | 6.59 × 10^{−20} | 0.2340 |

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**MDPI and ACS Style**

Mahmud, R.; Morshed, A.K.M.M.; Paul, T.C.
Enhanced Specific Heat Capacity of Liquid Entrapped between Two Solid Walls Separated by a Nanogap. *Processes* **2020**, *8*, 459.
https://doi.org/10.3390/pr8040459

**AMA Style**

Mahmud R, Morshed AKMM, Paul TC.
Enhanced Specific Heat Capacity of Liquid Entrapped between Two Solid Walls Separated by a Nanogap. *Processes*. 2020; 8(4):459.
https://doi.org/10.3390/pr8040459

**Chicago/Turabian Style**

Mahmud, Rifat, A.K.M. Monjur Morshed, and Titan C. Paul.
2020. "Enhanced Specific Heat Capacity of Liquid Entrapped between Two Solid Walls Separated by a Nanogap" *Processes* 8, no. 4: 459.
https://doi.org/10.3390/pr8040459