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# Mechanical approach to chemical transport

Edited by R. Stephen Berry, The University of Chicago, Chicago, IL, and approved July 29, 2016 (received for review January 17, 2016)

## Significance

It is useful to have a systematic and simple framework to describe transport and equilibrium in chemical systems. Here, we present a general Lagrangian-based approach to describe transport phenomena, while remaining consistent with chemical thermodynamics. Although we neglect heat flow independent of transport, the ease with which transport and equilibrium relations can be derived for any type of driving factors (electromagnetic, thermal, pressure, surface tension, etc.) should make this approach useful in physics, chemistry, biophysics, engineering, and even economics and social sciences.

## Abstract

Nonequilibrium thermodynamics describes the rates of transport phenomena with the aid of various thermodynamic forces, but often the phenomenological transport coefficients are not known, and the description is not easily connected with equilibrium relations. We present a simple and intuitive model to address these issues. Our model is based on Lagrangian dynamics for chemical systems with dissipation, so one may think of the model as physicochemical mechanics. Using one main equation, the model allows a systematic derivation of all transport and equilibrium equations, subject to the limitation that heat generated or absorbed in the system must be small for the model to be valid. A table with all major examples of transport and equilibrium processes described using physicochemical mechanics is given. In equilibrium, physicochemical mechanics reduces to standard thermodynamics and the Gibbs–Duhem relation, and we show that the First and Second Laws of thermodynamics are satisfied for our system plus bath model. Out of equilibrium, our model provides relationships between transport coefficients and describes system evolution in the presence of several simultaneous external fields. The model also leads to an extension of the Onsager–Casimir reciprocal relations for properties simultaneously transported by many components.

Modern thermodynamics is a mature field whose equilibrium and out-of-equilibrium properties have been investigated thoroughly (1). Thermostatics deals with the initial and final states of systems in contact with a heat bath using the fundamental variables of energy and entropy but does not consider transport processes. Nonequilibrium thermodynamics describes transport rates based on fluctuations, thermodynamic driving factors, and their conjugate variables (2, 3). It is now possible to derive phenomenological transport coefficients from molecular mobilities, thermodynamic quantities of the system, and external Newtonian forces (4).

The importance of building simple models to connect thermodynamics with chemical transport has been clear for a long time and thus has been investigated extensively (3, 5⇓–7). One way to make the connection (8⇓⇓⇓–12) is to start with the Lagrangian *L* of a mechanical system with many particles “*j*” of a chemical component “*i*” at coordinate *L* equals the difference of the kinetic energy and potential energy of the whole system (13). For mechanical systems with dissipation, Lagrange’s equations have been generalized to include Rayleigh’s dissipation function

Here, we seek to develop a model inspired by Lagrangian mechanics, which allows easy derivation chemical transport driven by multiple Newtonian forces, and the associated transport coefficients. The model retains full consistency with equilibrium thermodynamics, without the need to consider thermal fluctuations explicitly. For example, it is worth noting that the Rayleigh dissipation function in Eq. **1** accounts for only half of the dissipated energy, whereas the model we derive below satisfies energy conservation for our open system plus bath (*Proof of Energy Conservation and Alternative Derivation of the Model*). The model can deal with nonhomogeneous distributions of temperature and other variables in the system but does not account for the heat flow associated with transport or chemical reactions in the system. The model is, however, much simpler than full heat-mass transport formalisms and derivations of complete transport equations from statistical mechanics (15⇓–17) and is consistent with classic thermodynamics. We believe that such a model could be useful for describing transport in electrochemistry, spatially heterogeneous kinetics, biophysics, and other areas where position-dependent forces are applied to a system. For example, temperature jumps and photoexcitation patterned with spatial light modulators, chemical and osmotic pressure gradients, or other complex perturbations can nowadays be applied to nonhomogeneous systems ranging from self-assembled monolayers to live cells (18⇓–20) to induce transport phenomena.

## Proof of Energy Conservation and Alternative Derivation of the Model

As mentioned in the main text, our Lagrangian formulation includes the internal energy of the open system under the influence of external fields. Thus, our model can be written so that the whole dissipated energy (not half, as it is in the Rayleigh dissipation function in Eq. **1**) shows up in analysis. The model thus satisfies energy conservation for the system, plus the bath into which the energy dissipated by transport is deposited.

As shown by Biot (10), when internal degrees of freedom of the system are included, the Lagrangian can be written as*i*th component. *E* and entropy *S*, but *T*_{res} is the reservoir temperature, whereas the system can have variable temperature *T*(*q*). It is important to note that Eq. **S1** is valid out of equilibrium. The system temperature *T* can differ from the reservoir temperature and depend on the generalized coordinates *q* (*x* in our 1D simplified notation in the main text). When work done by the system dissipates and equilibrium is restored, the average temperature of the system returns to the temperature of the reservoir. Like Eq. **2**, Eq. **S1** is still restricted to the thermodynamic limit of the system size.

Eq. **S1**, Biot’s final result, can be used as an alternative starting point to derive our transport equations based on the physicochemical potential, leading to its mechanically inspired interpretation. The generalized Lagrange equation contains no velocity or time derivatives on the left. Consider the system without any average acceleration where frictional forces have damped fluxes to constant velocity. Unlike in the main text, where we used Stokes’ law, here, we use a generalized assumption about the frictional force, which follows from a fluctuation–dissipation theorem in a form suggested by Biot:

We now take deviations of species *i* from equilibrium in a coarse-grained volume δ*V* at position *x*, or *x* and *t*. Multiplying Eq. **S1** by ∂*n*_{i}/∂*x*,**S2**, and summing both sides over all species *i*,**S5** and the Lagrangian view with a dissipation function are directly related. Indeed, our derivation shows that when the thermal energy *TS* of a molecular system is taken into account on the left hand side of the Lagrangian, then Raleigh’s dissipation function **1**.

## Results

### Model Assumptions.

We assume that the system is subject to external forces, which can be decomposed into forces acting on each species *i* in the system. We coarse-grain the system into volumes *δV*(*x*) large enough so that *i* with number density *x*. These assumptions correspond to the thermodynamic limit, and the gradients can be described only on a size scale greater than *δV*. For simplicity, we use one coordinate “*x*,” but the generalization to higher-dimensional systems is obvious.

We exclude convective transport and hydrodynamic effects from our model in this paper. Instead we assume Stokes’ law, stating that the frictional force *i*. By considering only drift influenced by a frictional force, we are assuming that the time-average of the leftmost term in Eq. **1** is zero, as it is for highly damped Langevin models. This leftmost term manifests itself only microscopically by the acceleration during collisions, contributing to the thermal energy *δE*_{th}(*x*) = *T*(*x*)*δS*(*x*) in each coarse-grained volume, where *T* and *S* are local temperature and entropy. Thermal energy leads to chemical transport driven by temperature gradients and entropic forces, but not to any average acceleration of the whole system.

We also assume that all free energy dissipated by the system enters the environment as heat, so the temperature pattern *T*(*x*) is not altered by transport or reactions. Thus, our model can explain the effect of temperature gradients on transport and reactions, but both transport and reactions must be small perturbations that do not alter the temperature gradient between the coarse-grained volumes *δV* of the system (5). A full description requires the more complex formalisms of coupled heat-mass transport (15).

A brief rationale for the derivative notation throughout the paper will be useful here because we combine concepts from local equilibrium and coarse graining (*δx*), concepts from total and partial thermodynamic differentials (*dx* and *∂x*), and concepts from transport, where a position differential is usually denoted “*∂x*” instead of “*dx*” when a function also depends on time. Here we will use the symbols in the thermodynamic sense. A total position derivative will be notated *d/dx.* For example, for *dy*(*z*(*x*),*x*), we have *dy*/*dx* = *∂y*/*∂z* ⋅ *∂z*/*∂x* + *∂y*/*∂x*. For non–steady-state processes that depend on time, we will retain the notation ∂/∂*t* for time. Furthermore, the variation *x* to the next at *x* + *δG*/*δx* by a continuous derivative *dG/dx* of the corresponding smoothed free energy as a function of *x*.

### Mechanical Formulation of Chemical Flux.

The Gibbs free energy is the Legendre transform of the system internal energy, a thermodynamic analog of the Legendre transform from Hamiltonian to the Lagrangian (or vice versa) in mechanics. To describe the effect of different fields, including pressure and temperature, we will work in the Lagrangian representation, which is analogous to local thermodynamic equilibrium within a grain but allows out-of-equilibrium transport. In analogy to ref. 10, we define a function, the generalized free energy *x* and external fields:*x* with respect to an adjacent grain at *E*(*x*), *x*). *i*. *i*.

The heat dissipation when components *i* move between adjacent coarse-grained volumes by *n*_{i} particles. For a proof that a full accounting for dissipation and energy conservation leads to internal energy *E* in Eq. **2**, see *Proof of Energy Conservation and Alternative Derivation of the Model*.

If we use Einstein’s mobility *i* as **4** as*δx* by an ordinary derivative *d/dx* of the corresponding smoothed function. Thus, we obtain

For each component, it is convenient to introduce the generalized physicochemical potential **6** must hold for any combination of molar concentrations, so for independent transport of each component *i*, we have**4**). The derivation given here makes a more general connection with equilibrium thermodynamics.

Combining Eq. **8** with the standard continuity relation ∂*C*_{i}/∂*t* = −∂*J*_{i}/∂*x* leads to a description of local concentration as a function of time:

Eq. **9** is similar to the Fokker–Planck equation (17) but also includes entropy and temperature as driving factors.

### Simple Derivation of Chemical Transport and Equilibrium Laws.

The expression *d**dx* on the right of Eq. **9** signifies a total thermodynamic derivative, which can lead to redistribution of *C*_{i}(*x*,*t*) in time and space. This derivative of *x* is (at some instant *t*)*i*, to be used instead of concentration **8** and **10**, we obtain our main result,

Eq. **11** describes the rates of transport driven by several additive forces simultaneously. The forces are given by gradients of driving factors such as temperature, pressure, chemical potential, concentration, etc., as well as gradients of external potentials, such as electric and gravitational potentials. Through its concentration-dependent term, Eq. **11** can describe transport resulting from concentration changes due to chemical reactions. All external and internal forces have the same units of force/mole and may be added directly, in contrast to thermodynamic forces, which must be multiplied by conjugate molar properties to achieve consistent units.

Eq. **11** leads to the classic equations describing the rates of chemical transport driven by each of the molar forces. For example, if external potentials temperature and pressure are independent of *x*, for ideal solutions, Eq. **11** reduces to Fick’s first law of diffusion, **9** then reduces to Fick’s second law **11** gives Faraday’s law for electrophoresis,

The rates of all processes for a given component *i* in our model system are proportional to the same mobility *U*_{i}, and thus transport coefficients are mutually related (4). When the molar properties of transported species are known, this mutual relation allows one to estimate a flux driven by one force, based on the flux driven by another force. For example, the ratio of the diffusion coefficient

Temperature is present in two terms in Eq. **11**, explaining thermoelectric phenomena and thermodiffusion (21). The different signs in the pair *T*(*x*) is treated as an externally imposed field, and heat flow based on the Fourier mechanism inside the system is neglected.

### Connection with Reciprocal Relations.

The chemical flux of each dissolved species leads to a volumetric flux **11**, the transport coefficient for volumetric flux driven by an electric field is therefore **11** for conjugate transport coefficients may be continued. Thus, based on Lagrangian dynamics with dissipation, but without the explicit assumption of microreversibility, we have Onsager’s reciprocal relations when different types of fluxes are carried by the same species. It also becomes clear that if two fluxes are carried simultaneously by several species, each with different concentration, mobility, and molar properties, the traditional reciprocal relations are not valid and need to be generalized (4).

### Velocity-Dependent Forces.

We have discussed the effects of conservative forces, which depend on position, but not on velocity. Velocity-dependent forces, originating from vector potentials, also may be included in our model. This addition could make the model useful for magnetic perturbations of the system, because magnetic manipulation of heterogeneous systems is becoming more commonplace (24). For example, the effect of the Lorentz force per mole *z* is the charge in units of the elementary charge, **v** is the velocity of the charges, ** E** is the electric field, and

**is the magnetic field,**

*B*## Discussion

Our model allows a simple and uniform description of coupled transport of different species in a nonhomogeneous environment subject to external fields. System variables such as *T*(*x*) are treated as position-dependent fields, so heat produced by transport or reaction is assumed to be dissipated without changing the local temperature *T*(*x*). This situation arises in practical applications. Experiments are coming online that can expose nonhomogeneous systems such as live cells or electrochemical interfaces to programmable spatiotemporal patterns of temperature, chemical composition, osmotic pressure, light exposure, and other variables (18⇓–20, 26).

Our model can be used as a starting point for constructing master equations for more general reaction–diffusion models (27) driven by perturbations beyond concentration gradients, what we term a force–reaction–diffusion model. The physicochemical potential includes all driving factors, and it is possible to analyze processes where all fields are simultaneously changing as a function of position. Concentration, molar entropy, temperature, pressure, and external fields are now on the same footing, and independent transport of any component is always directed toward the decrease of its physicochemical potential.

Table 1 lists all major phenomena that may be described using Eq. **11**. These phenomena include nonequilibrium processes driven by a single driving factor (diagonal cells), and processes driven by different pairs of driving factors, or the corresponding equilibrium laws (off-diagonal cells). For example, transport of ions and electric current due to a difference of electric potential are described by Faraday’s and Ohm’s laws. The Boltzmann and Nernst equations describe equilibrium concentrations in the presence of an electric potential difference and vice versa.

Although almost all cells in Table 1 are filled with known laws, our model makes it easy to predict new effects and their corresponding transport coefficients, and fill an “unknown” cell. We predict that a magnetic fluid, moving with velocity **x**-direction, forms a surface tension gradient along **z**. The electromagnetic driving force in the **z**-direction is determined by the vector product of the fluid velocity in the **x**-direction and *q***v**_{x}. A similar analysis may be conducted for several driving factors acting simultaneously, so some electromagnetic field-related phenomena where temperature is involved (27) are also given in Table 1.

The generalized local Gibbs free energy *x*), and the corresponding physicochemical potential for each species *i*, represents the local difference of “internal chemical energy” + “potential energy” − “heat energy” of a system out of equilibrium. In this sense, the model is analogous to the mechanical Lagrangian. Unlike the mechanical formulation in Eq. **1**, which includes the difference of potential energy and kinetic energy of directed movement only, our model includes thermal energy.

As a result, our model is consistent with thermodynamics for the system plus bath as equilibrium is approached. Energy conservation is discussed in *Proof of Energy Conservation and Alternative Derivation of the Model*. To demonstrate consistency with the Second Law, we connect flux with the equilibrium conditions in our model by using the logarithmic relationship between chemical potential and concentration (or activity for nonideal systems). The local chemical potentials in the presence of fields usually are not known, so for our purpose, it is convenient to introduce a “total force” potential *C*_{i}, so Eq. **11** can be rewritten as

At constant temperature, Eq. **13** reduces to

If the coordinate dependence of all potentials *i* in nonhomogeneous media driven by several forces simultaneously, when there is a difference in the physicochemical potential *x* = 0 and position *x* = *l*. According to Eq. **15**, directed transport in the presence of external fields ceases when **15**: if an electric field is the only contribution to **15** with *J*_{i} = 0 becomes the Nernst equation

Eq. **15** makes the connection with the existing thermodynamic transport literature and confirms that in our model, a potential difference induces an opposing flux to restore equilibrium, in accord with Le Chatelier’s principle. The model thus satisfies the Second Law of thermodynamics for our open system plus bath.

Finally, consider the case of equilibrium in isolated system, when all external fields are shut off and the explicit *x*-dependence **10** over the volume of the system (just *x* in one dimension) to obtain*dT* and *dP* become variations of the average temperature and pressure over the whole system. Thus, after summation over all components *i* in equilibrium, Eq. **11** leads to the ordinary Gibbs–Duhem relation (1). An alternate view in terms of entropy generation is presented in *Alternative Proof of Consistency with the Second Law*, confirming yet again that the model satisfies the Second Law for the system plus bath. As in any open system, it is of course possible that *ΔS*_{sys} < 0 even though the system plus bath entropy *ΔS*_{tot} > 0.

Although the Gibbs–Duhem relation in Eq. **16** looks superficially similar to Eq. **10**, the relation has a very different meaning. Eq. **10** is a vector equality between two gradients, and Eq. **16** is a scalar equality after integrating over the whole system volume. In accordance with the Curie–Prigogine principle, scalar processes and vectorial processes are uncoupled. In equilibrium, the left side of Eq. **10** equals 0 and there is no flux. Nonetheless, a nonzero gradient of chemical potential **10** could balance the temperature, pressure, and force gradients.

Also, note the switched roles of a driving force and its conjugate variable in transport processes. For example, mechanical work done on an ideal gas with a piston is *dq* is *n* changes at constant *δV*(*x*) to an adjacent volume *δV*(*x* + *δx*) in diffusion, they are driven by a gradient of chemical potential. Now intensive variables, including chemical potentials, may be functions of both coordinate *x* and time *t*, allowing for reactions to be incorporated into our diffusive transport model.

Summarizing, our approach allows a systematic derivation of nonequilibrium transport rates and equilibrium equations in nonhomogeneous systems with chemical transport driven by position-dependent external forces. The approach does not consider convection or Fourier heat flow in the system, and transport phenomena or chemical reactions should not perturb the externally imposed temperature gradients. Nevertheless, the general model presented here may be used to describe the processes besides traditional physical chemistry applications such as live cell kinetics, electrochemical transport, or spatially patterned perturbation of self-assembled monolayers with simultaneous reactions and diffusion (18⇓–20, 27). Furthermore, the random walk view of consumption (31) and various economic optimization problems can be formulated based on Lagrangians with logarithmic terms. Typically, a utility/consumption term is

## Alternative Proof of Consistency with the Second Law

As seen in the discussion of the Gibbs–Duhem relation (Eq. **16**) (*Discussion*), without external fields, **10** simply becomes the partial molar free energy *G*_{g} of Eqs. **2** and **6** simply becomes the Gibbs free energy. By construction, *ST* term in Eq. **2**. Thus, our model is an extension of the Second Law of thermodynamics to nonisolated and nonisothermal systems consistent with the Gibbs ensemble when equilibrium is reached.

In nonequilibrium thermodynamics, evolution of a system is usually analyzed based on two seemingly contradicting principles, maximum and minimum entropy production per unit time. Now, it becomes clear that the areas of application for these two principles are different. The former principle means that initially the out-of-equilibrium system will move in the direction of maximal forces, which leads to maximum friction and thus to maximum entropy production. The latter principle, known as the Prigogine–Wiame principle, states that the entropy production continuously decreases until it reaches a minimum in the steady state. According to Eq. **9**, the gradient of molar force will lead to accumulation of dissolved species in the area with smaller acting force. Because of this redistribution, the system reaches a steady state. After that, the flux does not depend on time anymore, and for each independent component in homogeneous fluid, total molar force is described by **9**).

Energy dissipation for independent component is always positive,

Note that the rate of energy dissipation here is the same as the energy dissipation in traditional nonequilibrium thermodynamics, where dissipation is the total of the products of all different fluxes in the system by their own conjugated thermodynamic forces. This statement becomes clear if thermodynamic forces are converted into Newtonian forces, and different types of fluxes are converted into the molar fluxes and then added. For an isolated system, positive energy dissipation also means positive entropy production; thus our model is fully consistent with the Second Law.

Here, we look in more detail at the implications of the model for energy conservation and entropy increase, when the full system plus bath is taken into account. Of course, the open system by itself, subject to external fields and transport, may gain or lose energy, and have a negative entropy of reaction, as is also the case for *ΔH*_{sys} and *ΔS*_{sys} in the Gibbs ensemble, when a nonisolated system subject to heat flow and pressure equilibration is analyzed.

## Acknowledgments

We thank Dmitry Babikov and Sergei Khangulov for useful discussions in the early stages of the project. This work was supported by National Science Foundation Grant CHE1012075 (to M.G.) and by Biomime, Inc. (N.K.).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: nikolai.kocherginsky{at}gmail.com or mgruebel{at}illinois.edu.

Author contributions: N.K. and M.G. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1600866113/-/DCSupplemental.

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