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

From

Figure 2, it is evident that for a specific range of gap thickness (

h/a = 10.5 to 1.5 i.e., a gap thickness varying from 6.14 nm to 0.88 nm) and a fixed temperature (

T = 100 K for this case), the

C_{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].

The enhancement in heat capacity of the confined liquid was attributed to an increase in non-configurational and configurational contributions [

17]. The non-configurational contribution can be viewed as a combination of two effects–firstly, the vibrational (phonon) contribution and secondly, the anharmonic modes contribution that increases in proportion with the temperature [

17].

The configurational contribution arises preliminarily due to the changes in both the average coordination (size effect) and short-range order in liquid’s structure (depends on its temperature and pressure) [

17]. It can be viewed as the sum of three effects: non-uniform density distribution, energy transfer due to molecular motion and guided molecular mobility of the confined liquid. The non-configurational and configurational contributions are interdependent. They are a direct function of the volume and structure of the constituent liquid since the higher energetic configurational states can endure lower phonon frequency with enhanced anharmonic phonons [

12].

Although currently there is no straight-forward mathematical formulation of the heat capacity of liquid, phonon theory has been devised recently to bypass the problem of system-specific interactions of liquid molecules [

11,

29]. Usually, when the liquid film thickness is large enough compared to the phonon mean free path (MFP), phonon travels diffusively which is driven by frequent phonon collisions [

9,

30]. Heat capacity of the confined liquid in such cases resembles the bulk value as shown in

Figure 2 for

h/a > 10.5. However, as soon as the liquid film thickness becomes commensurate with the phonon MFP, ballistic transport starts to dominate the thermal energy transportation process [

9]. In such cases, the heat retaining capability of a nanodevice strongly depends on the boundary scattering dynamics. Consequently, the effect of its geometric size and shape comes into play [

9]. Hence, the ability of a nanodevice to retain thermal energy is strongly influenced by the scattering dynamics at the boundary and its geometrical size and shape [

9]. Phonon MFP was calculated [

28] using Equation (8):

where, ‘

d’ denotes the hypothetical collision diameter whose value is 0.33 nm for Ar [

28], ‘

σ’ is the collision cross section which is 0.342 nm for Ar [

28], ‘z’ presents the frequency of collision and ‘

v’ is the magnitude of average velocity of thermal phonon.

This results in a phonon MFP of 6 nm when the liquid Ar is at 100 K whereas the anomaly in heat capacity starts to be pronounced below h/a = 10.5 i.e., below a liquid Ar film thickness of 6.14 nm. This explains why C_{v} of nanoconfined liquid deviates from C_{v} of the bulk liquid after a certain gap thickness.

As soon as

h/a becomes closer to a certain critical value at which the maximum heat capacity occurs (denoted by (

h/a)

^{*}; (

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.

When the wall separation distance is further lowered beyond the critical gap thickness, some long phonon wavelength propagation modes are suppressed. Joshi and Majumder [

34] introduced transient ballistic and diffusive phonon heat transfer for this kind of thin film earlier. The reduced frequency of thermal phonons produces a larger phonon MFP leading to a greater phonon transmission rate [

30]. Eventually,

C_{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.

However,

Figure 2 indicates that when

h/a < 1.5 i.e., when there is only one nanolayer, heat capacity of the entrapped liquid is lower than its bulk counterpart. Wang et al. [

23] have explained it satisfactorily. They reported that for a confined monolayer of liquid, the thermal boundary resistance dropped by one order of magnitude compared to larger gap thickness [

23], which was also supported by the earlier works of Liang and Tsai [

22] and Cui et al. [

35]. A smaller thermal boundary resistance perpetuates easy flow of thermal energy, decreasing its heat capacity compared to the bulk counterpart. Israelachvili and Gourdon [

36] explained it in a different fashion. According to their research, when there is only one nanolayer of liquid in the entrapped volume, a structural change of the liquid layer occurs. Depending on the mismatch of the walls and the confined liquid molecules, the liquid film rearranges itself to a ‘structure’ that might be more solid-like, ordered or crystalline in nature. As the heat capacity of the solid is considerably less than its liquid counterpart, heat capacity is observed to drop beyond

h/a = 1.5 (gap thickness,

h = 0.88 nm).

Nonuniform density distribution primarily contributes to the configurational change of a nanoconfined liquid. The number density profile of liquid Ar as presented in

Figure 3b exhibits a strong oscillation of liquid density near the wall extending about 2 nm from the surface of the wall after which nearly constant density of liquid Ar results. This is attributed to the force-field effects of the wall and the local interactions among the liquid atoms [

24]. To calculate the number density of liquid Ar, the values have been binned spatially depending on its present coordinates and averaged over longer timescales. Severe density fluctuations result in almost no molecules within a few bins near the confinement.

Many previous MD studies have reported similar behavior where the wall adjacent liquid atoms are absorbed and form an atomic scale nanolayer [

22,

23,

24]. This absorbed liquid layer, with much lower mobility compared to the rest of the liquid molecules, is termed as a ‘Solid-like-liquid’ region as indicated in

Figure 3a. Cui et al. [

35] have reported an entirely different microstructure of such layers absorbed onto the solid walls compared to the bulk liquid. They showed that these layers were more ordered, and close enough to that of the solid material. The different behaviors of the ‘Solid-like-liquid’ region adjacent to two solid walls (the leftmost and rightmost portions of

Figure 3b) can be attributed to the randomly vibrating surrounding liquid molecules, which act as a vibration dampener of these solid-like-liquid molecules.

At lower temperatures, the effect of solid-liquid wall interactions on the dynamics of liquid molecules are non-negligible [

31]. The solid-to-liquid phase transition, which takes place within the confined liquid leads to a more intricate phenomena involving the dislocation induced forcing out of the layers [

36]. These things play as precursors for the unusual behavior of the heat retaining capability of the confined liquid.

A change in the mode of energy transfer influences the configurational contribution of nanoconfined liquid as all the possible modes of energy transfer (Equation (9)) governs the mobility of its constituent molecules [

24].

where, ‘

J_{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}’.

The equation has two parts. The first part represents the energy transport as a result of molecular motion. The second portion of the equation denotes energy transfer because of the molecular interaction. For a bulk liquid under equilibrium, a molecular interaction governs the energy transfer process, while for a confined liquid the molecular motion dominates the whole process. This was observed in

Figure 4a. In stochastic thermodynamics, one approach to quantify the random walks of the molecules is to calculate the self-diffusion coefficient (

D). In this work, Einstein’s formula [

25] was employed to calculate

D using Equation (10).

The entrapped liquid in nanoscale confinement has a lesser value of this self-diffusion coefficient. With the decrease in wall separation distance, its value reduces further as shown in

Figure 4b. This is due to the presence of an immobile liquid layer adjacent to the wall surface and strong interaction between the solid and the liquid molecules. Reduction in heat diffusion capability and number of molecules transporting the thermal energy indicates an increase in heat capacity as

h/a approaches (

h/a)

^{*}.

Depending on the size of the confinement, the direction of molecules’ motion changes. Confinement reduces the random walk of the liquid molecules to a guided well-defined path when the size effect comes into being as shown in

Figure 5b. Due to a more ordered and structured movement, transportation of heat in confined liquid is lesser than the bulk resulting in an increase in heat capacity of the confined liquid [

25].

On a free-standing slab or in a confinement with a larger gap thickness, a guided well-defined path i.e., an increase in

C_{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.

The simultaneous contribution of all these parameters plays the key role for increasing the heat capacity at a nanometer length scale.

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

The variation of heat capacity (

C_{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.

Figure 7 illustrates that heat capacity of a nanoconfined liquid that shows distinct peaks at different temperatures that move to lower liquid film thickness at a higher temperature. Analogous behavior of nanoconfined water was reported earlier [

38] where the distinct peaks of

${C}_{p}=\frac{dH}{dT}$ of the confined water was found to move to a lower temperature on compression which was attributed to the structural transformation into a four-coordinated liquid [

38]. For instance,

Figure 7a shows that

C_{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.

Another significant finding is that although the range of gap thickness where

C_{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.

The simultaneous effects of factors like—higher phonon transport, premature damping of the density oscillation, increased heat carrying molecules in the liquid region, enhanced molecular mobility, change in thermal resistance, etc. govern this temperature-dependent heat capacity of the nanoconfined liquid.

A higher temperature leads to a higher rate of transport of thermal phonons [

32]. These phonons are associated with lesser frequencies i.e., the transport moves to the ballistic regime with higher wavelengths [

39]. This results in an easier thermal transport and consequently a lower heat capacity at higher temperatures.

Density oscillation is affected by the variation of liquid temperature [

24].

Figure 8a shows that the oscillation in liquid’s density relaxes quickly when the temperature is higher, which eventually increases the number of liquid molecules in the liquid region. As a result, at higher temperatures, the liquid has a lower than usual heat capacity [

24].

Heat, temperature and the motion of molecules, are all inter-related. The addition of thermal energy causes faster vibration of the atoms and molecules in a substance. Mean squared displacement (MSD) is a measure of the spatial extent of this stochastic random movement [

24] that is usually expressed in the generic formula as in Equation (11),

where, 〈

r(

t)〉 is the ensemble average of the position at time

t and

$\tau $ is the lag time.

The variation of MSD per unit time with temperature of the confined liquid is depicted in

Figure 9. It is evident that as the temperature increases, the slope becomes steeper. We relate the increase in MSD to the enhanced transportation of thermal energy of the confined liquid [

25] resulting in a decrease in heat capacity with the increasing temperature.

During the flow of thermal energy from a solid to liquid or vice versa, at the junction, a thermal resistance is generated owing to interfacial phonon scattering. This interfacial thermal resistance (ITR) is known as Kapitza resistance [

24]. ITR primarily depends on factors like the solid–liquid interaction strength, mismatch of thermal phonons’ vibration of the two sides, wettability of the mating surface and temperature at the interface [

24,

40]. It is observed in

Figure 10a that ITR has a negative power-law relationship with temperature,

R_{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.