Nano or small-scale thermodynamic systems have degree of freedom one higher than the macro thermodynamic systems. Analogous to how the combined expression for first and second law of thermodynamics was altered by Gibbs to include multicomponent systems, the combined expression changes for the small-scale systems and a generalized energy conservation law can be postulated to characterize the behavior of any system regardless of the length scale. Presence of additional degree of freedom in small-scale systems gives them size dependent properties that are not observed in bulk systems. Systems can then be tuned by altering their dimensions and desirable properties can be achieved. INTRODUCTION: In past few decades, technology has turned a major portion of its attention towards materials at smaller length scale to solve intricate problems in the fields of drug and medicine, electronics and communication, structural functional materials etc. This work can be divided into two sections. First section would deal with the mathematical interpretation of presence of size dependent effects in small scale systems; this part would describe how addition of extra degree of freedom in the fundamental law of energy conservation can help in generalizing this law for both types of systems i.e. macro and small-scale system. Whereas other section would deal with tuning of systems to obtain desirable material properties.Generalization of the energy conservation equation to include multicomponent systems was done by Gibbs. In a similar manner, Hill (2001) showed that energy equation for multicomponent macrosystems can be generalized to include small scale systems as well. However, to extend Gibbs’ equation to include small systems, attention would be paid at the ensemble of subsystems rather than the individual subsystems. This is because an individual subsystem is essentially a small-scale system however the ensemble of these subsystems will be a macroscopic system which is a zone of comfort for the Gibbs’ equation (as it is valid only for macrosystems). Analogy between nanothermodynamics and quantum mechanics is further explained by describing how these can act as respective universal sets for the subsets of macro-scale thermodynamics and classical mechanics. GENERALIZATION OF THE LAW OF CONSERVATION OF ENERGY:Nanothermodynamic systems are those systems for which the order of magnitude of the size of system is smaller than the order of magnitude of ranges up to which forces interact with the system (Lucia, 2015). Properties of a material at macro scale tend to differ from properties at smaller length scales because of comparatively higher surface area and surface energy of smaller particles which in turn contribute to the total Gibbs free energy of the system. This results in power law dependence of properties of system on the system size (Delogu, 2005). Thermodynamics, at this tiny length scale where minute details such as interaction between individual particles become significant factors in determining the state of system, takes quantum statistical mechanical approach (Rajagopal et al). Nano thermodynamics proposes inclusion of subdivision potential (e) in the Gibbs equation (Hill, 2001). The parameter e is described on the basis of this understanding that a system containing n particles would not have same energy as total energy of n isolated particles when summed up individually. This happens because of surface effects, thermal fluctuations and size dependent properties of the nano systems. Parameter e is analogous to the chemical potential ? which is the energy required to take one individual particle from a cluster of interacting particles into the system.Mathematically the chemical potential of component I is given by: µi = ( ??E/?N?_i)S,V,NkOn the same note, e is the energy required to take a cluster of a number of interacting particles from a bath of several such clusters into the system (Chamberlin, 2015). Following is the mathematical definition of the parameter E.The combination of first and second laws of thermodynamics for a macroscopic system gives:dE = T dS – P dV (1)For a macroscopic system containing Ni¬ number of components, this equation becomes:dE = T dS – P dV + ?_i?( µi dNi¬ ) (2)Hill in 2001 proposed that this equation was not adequate to account for total energy of small systems such as those at the nanometer scale which should also have a considerable surface energy term in equation (2). Additional terms were therefore needed to account for total energy of such systems. Since equation (2) applies only to macro thermodynamic systems, this addition would need to be done at macro level i.e. collectively for a cluster of X identical and non-interacting small systems (subsystems), rather than adding some term individually for each small system. It is worth attention that a cluster of X small systems would be a macro thermodynamic system in itself.If the system contains cluster of X small systems, then all the variables in (2) should be function of X .i.e.: Et ? X ESt ?XVt ? X VNit ? X Ni(Note: the subscript ‘t’ stands for total, i.e. the total value of extensive variables for the system)Total energy of such system containing X small systems can Therefore be written as:dEt = T dSt – P dVt + ?_i?( µi dNi¬t ) + e dX (4)Note that the term ‘e dX’ has been added to account for the surface effects, thermal fluctuations and size dependent properties of the small systems. Moreover, the parameter e is not roofed under the summation sign implying that e should be a something related to the bulk rather than just one small system. Presence of X small systems within the system can be understood as presence of a subsystem inside the system under consideration. The parameter e is therefore called ‘sub division potential’. e = ( (?E_t)/?X)St,Vt,Nit (5)The above equation gives an appearance that e? 0 as X ? ?. However, we will see that this is not the case in all types of systems. A system may be composed of large number of macroscale subsystems or a large number of small-scale subsystems. Value of e will be altered in different ways for these systems; it is to be assumed that St, Vt, and Nit will be constant under both of these circumstances. When there are too many macro subsystems in the system: If the value of X is large but X signifies the number of macro subsystems rather than number of small subsystems, then the sub division potential can be ignored in (4). This is due to diminutive role of surface effects, thermal fluctuations and size dependent properties at macroscale. When there are too many small-scale subsystems in the system: When X is increased by further division of each small system into two identical smaller-scale systems then e cannot be ignored. As such an increase in X would cause the subsystem size to further reduce and therefore the smaller-scale effects would dominate.Now to obtain the total energy of the system, integrate (4):Et = T St – P Vt + ?_i?( µi Ni¬t ) + e XUsing set of equations mentioned in (3) gives: E = T S – P V + ?_i?( µi Ni¬ ) + e (6)Moreover, (3) gives:dEt = X dE + e dXdSt = X dS + S dXdVt = X dV + V dXdNi¬t = X dNi + Ni DxReplacing the values of dEt , dSt , dVt , and dNi¬t in (4) gives:X dE + E dX = {T X dS – P X dV + X ?_i?( µi dNi¬)} + {T S – P V + ?_i?( Ni¬ µi ) +e} dXNotice that the highlighted part of the above equation is E itself. Using (6) and eliminating E dX from both the sides gives: dE = T dS – P dV + ?_i?( µi dNi¬) (7)Note that Et in (4) was a linear homogeneous function of St , Vt , and Nit however in (7) it can be observed that E is no longer a linear homogeneous function of S, V and Ni and therefore E cannot be integrated to yield (6).Furthermore, differentiation of (6) gives:dE = T dS + S dT – P dV -V dP + ?_i?( µi dNi¬) + ?_i?( Ni dµi¬ ) + deReplacing the value of dE using (7) will cause the highlighted portion to cancel out from the above equation and will yield the expression for de:de = – S dT + V dP – ?_i?( Ni dµi¬ ) (8)Notice that e is an intensive property, which is also evident from (5); moreover, (8) tells that e is a function of T, P and µi which are all intensive thermodynamic properties. However, when e is differentiated with respect to these intensive properties, an extensive quantity is produced:S = – ( ?e/?T)P,µiV = ( ?e/?P)T,µiN_i = ( ?e/?µi)P,T,µk HIGHER DIMENSION OF SMALL-SCALE THERMODYNAMICS:Hill compares generalization of thermodynamic systems (whether small-scale or macro scale) by equations (6), (7) and (8) with generalization of mechanics to include both classical and quantum mechanics. Furthermore, just like classical mechanics can be considered as the subset of quantum mechanics, macroscale thermodynamics can be considered as a subset of small scale thermodynamics. This is because of the fact that macroscale thermodynamic systems have degrees of freedom one less than the small-scale thermodynamic systems. Physically this can be understood by considering that surface effect and other size dependent effects are absent for macroscopic systems.Thermodynamics properties of small-scale thermodynamic systems are dependent on the environmental variables which will best describe the fluctuations in these systems. These environmental variables may be temperature, chemical potential or pressure depending on the type of fluctuations a small-scale system is undergoing (Chamberlin, 2015). A thought experiment is performed to strengthen the argument of dependence of thermodynamic properties of small-scale systems on the environmental variables.Consider two systems, A and B with following specifications: System A is made up of a cluster of NA particles so that the order of interaction between these particles is not of the magnitude as that of the size of whole system. In other words, A is a macro scale system placed in an inert bath at temperature T. In A, the number of particles and volume (and thus pressure) associated with small fluctuations is conserved. Let system A has an entropy SA. System B is an ensemble of several small subsystems or small particles. Being a small-scale system (i.e. made up of several tiny subsystems not clustering together to form a macro body), the particles in B always have a tendency to jump in and out of the system. Due to fluctuations in number of particles present in the system, the environmental variable in this case would be chemical potential ?. We can choose ? such that the average number of particles in the system at equilibrium be NB. We further choose ? such that: NA = NB.The fluctuations in B will be considerably higher than those in System A because there is an interaction between particles in A unlike in B which is just an ensemble of particles. Interaction between the particles prevent them from jumping in and out of the system. Let SB be the entropy of B. Under such condition one may say that:SB > SAHowever, when NB is small then at given temperature, the fluctuations will be comparatively small. Under such situations: SB = SA.This strengthens the argument that the thermodynamic properties of small-scale systems are dependent on environmental variables and selection of these variables is crucial to define a certain small-scale system. This is another fundamental difference between a macro-scale system and a small-scale system as for macro systems, selection of partition functions may be done as per convenience.The dependence of thermodynamics properties of small scale systems on environmental variables and the inclusion of a size dependent term ‘e’ in the energy conservation equation (6) gives small-scale systems a higher dimensionality than their macro counterparts.? APPLICATIONS OF NANO MATERIALS: In this section of the report, a discussion on phenomena occurring at small-scale materials will be done. Furthermore, this section will describe the characteristics of nano systems that allow us to fine tune the material properties.In the previous section of this report, inclusion of subdivision potential in the 1st law of thermodynamics was done. It can be asserted that small-scale systems have this additional potential and therefore the parameter e is characteristic to systems at small length scales and will play a decisive role in defining the properties of small scale systems. Smaller the length scale, more dominant will be the magnitude of this parameter and therefore material properties such as melting temperature, electrical conductivity, chemical reactivity, magnetic permeability, etc. will change accordingly. Therefore, it is often stated that the material properties at this length scale are size dependent. Fig. 1: Electron paths in a nanowire, including imperfections in the wire.(Eric J. Heller, Harvard University) (https://nsf.gov) Gold nanoparticles do not reflect light in the same manner as the bulk gold does and therefore might appear red or purple (https://www.nano.gov). The theory of quantum confinement explains this phenomenon. At such small length scales, the movement of electrons would be spatially confined causing the nano materials to behave completely different than their bulk counterparts. This property of gold nano particles is used to cure tumor as the nano particles accumulate in tumor enabling precise imaging. Imaging is followed by destruction of tumor cells by accurately focused laser beam preventing nearby healthy cells from damage.Another application of quantum confinement is Quantum Dots which are made of semiconductor particles. These have a wide range of applications from optoelectronics to medical devices. The principle behind phenomenon exhibited by quantum dots is that the energy band gap in semiconductors is inversely proportional to size of particles. The energy band gap would therefore increase if the particle size decreases and therefore the particle would emit higher energy or shorter wavelength radiations while falling back from excited state to the low energy state (Fig. 2). This allows researchers to tune the optical properties of quantum dots as needed.Fig. 2: Size dependent optical behavior of quantum dots(Image courtesy: H.S. Mansur, Wiley Interdiscip. Rev. Nanomed. Nanobiotechnol. 2, 113–129 (2010))Another phenomenon that happens at micro/nano scale is Quantum Tunneling which defies the principles of classical mechanics and has remarkable applications (www.eas.asu.edu/~vasilesk). Classical mechanics suggests that for a body to cross any barrier, there is a threshold amount of energy required; however, at smaller length scales, a particle might violate this principle and would penetrate through the barrier that has a potential higher than the kinetic energy of the particle. This characteristic of small particles is used in Resonant Tunneling Diodes (RTDs) and Scanning Tunneling Microscope (STM). RTDs just like other diodes are used as switching devices but usually in fast electronic circuits. STM on the other hand utilize the property of small particles, essentially electrons, to penetrate through the surface of a specimen. As the detector moves away from the specimen the probability of detecting an electron depreciates however if an electric field is induced electric current of these electrons can be detected and therefore producing a magnification of the order of 106 and enabling microscopy at subatomic scale.Nature is known to produce structures and materials using the bottom-up approach. One of the most prominent examples of utilization of bottom-up approach would be self-assembly. Natural structures ranging from planets and galaxies to terrestrial materials are known to be built by self-assembly of nano scale systems. Thermodynamics explains the phenomenon of self-assembly by its fundamental laws i.e. every system would arrange itself spontaneously in a way that would minimize its energy and help it attain stability. An example of this approach in nature would be lotus leaf which is known to have hierarchical structures that give it the multifunctionality of being hydrophobic and self-cleaning (Fig. 2). These hierarchical structures have inspired researchers to build functional materials such as super hydrophobic, self-cleaning surfaces, adhesives that could be reused and in this way opening a whole new avenue of bio-inspired materials. The feet of insects are known to have billions of nano terminal elements. These terminal elements increase the surface area to a level where Van Der Walls forces could defy gravity.Mathematically, Van Der Wall’s force: FW = A/(12 ? D^2 )As the number of terminal elements increase and the size of terminal elements decrease, the surface area (A) increases whereas the distance between the element and surface (D) decreases. Thus increasing FW and creating super adhesive surfaces.Fig. 2: a) SEM image of lotus leaf surface showing hierarchical structures formed by micro papillose epidermal cells covered with epicuticular wax tubules having nanostructures.(Image courtesy: Bharat Bhushan, Yong Chae Jung, Kerstin Koch Phil. Trans. R. Soc. A 2009 367)? CONCLUSION:Nanoscale systems or small systems have additional energy terms which account for surface effects, thermal fluctuations and size dependent properties of these systems. These additional energy terms were included in Gibbs energy equation by defining a sub division potential (e), for the conservation of energy to hold for these systems. Inclusion of sub division potential yields a generalized form of Gibbs energy equation which can be used for thermodynamic system at any length scale. At smaller length scales e would be considerably high because of higher contribution of surface energy towards the total energy of the system, whereas at larger length scales, surface energy and other size dependent properties and thus e would have comparatively little to contribute to the total energy of the system. For this reason, it has been asserted that macroscopic systems have degrees of freedom one lesser than the nanoscale systems and therefore they can be considered as the subset of small scale thermodynamics.Smaller the length scale of a system, more dominant the parameter of sub division potential (e) and greater the deviation of material properties from properties of the bulk. This explains the extraordinary properties of nanomaterials as compared to their macroscopic counterparts making them fit for wide range of applications ranging from material characterization equipment to multifunctional hierarchical structures.