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USDA-ARS, National Soil Tilth Lab., Ames, IA 50011
* Corresponding author (logsdon{at}nstl.gov)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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Uniformity of chemical potential, evaluated for systems with inert constituents, is properly regarded as a necessary, but not a sufficient, requirement for thermodynamic equilibrium. Thermodynamic equilibrium requires each of the variables appearing in the chemical potential function to be equal independently at all points in a fluid phase (Sposito, 1981). Since the gradients of each of the variables are zero, it is clear that a function of the coordinates exists that will be uniform at equilibrium.
Unfortunately, it is frequently assumed that the negative gradient of chemical potential indicates the direction of transport by diffusion. A typical example of this concept is illustrated by a quote from Salisbury and Ross (1992). They state, "A diffusing solute tends to move from regions at high chemical potential to regions at low chemical potential." The latter concept persists in current literature of soil and plant science despite a number of papers that have attempted to point out the fallacy of the concept (Corey and Kemper, 1961; Corey and Klute, 1985; Corey and Auvermann, 2003). Evidently, the papers cited are not clear enough in their explanation to convince many authors of current textbooks dealing with mass transport in porous media.
An understanding of how chemicals are transported across porous membranes is essential to an understanding of the physiology of plants (Jungk and Classen, 1997). Clays can also function as semipermeable membranes in soils and geological formations (Fritz, 1986). Marine and Fritz (1981) describe a system in which a clay layer acts as a semipermeable membrane between overlying fresh water sediments and underlying saline rocks. Clay skins on aggregates affect solute diffusion in and out of the aggregates, which influences the availability of solutes for leaching (Kohne et al., 2002). Chemical engineers are interested in the diffusion of chemicals into and out of aggregates in chemical reactors. Civil engineers (Malusis et al., 2001) are interested in transport through clay barriers for retaining chemicals in holding ponds, and in the performance of installations for removing salt from brine by reverse osmosis. The above are a few examples illustrating the need for the scientific community to understand principles governing mass transport across porous barriers.
Error in predicting the direction and rate of diffusion (based on the chemical potential gradient) is large wherever a significant gradient of pressure exists. The gradient of chemical potential correctly evaluates the driving force for diffusion only where pressure is constant along a flow path. False assumptions regarding the application of chemical potential, commonly presented in current literature, often predict movement of a constituent in the opposite direction to that actually observed. This fact has been demonstrated experimentally by Corey and Kemper (1961), and by many investigators of osmotic efficiency, such as Malusis et al. (2001).
Homaee et al. (2002) examined several equations to combine matric and osmotic potential effects on transpiration and plant growth, including additive, multiplicative, and combined thresholds. The equation that provided the best correlation was with individual thresholds for each potential. They measured the rates of production of dry matter and transpiration of alfalfa (Medicago sativa L.) treated with combinations of matric and osmotic stress. Both matric suction and osmotic potential independently affected growth rates, but the additive equation did not correlate with growth or transpiration.
This paper attempts to clarify the principles of diffusion by approaching the subject from a new perspective. In particular, we have derived Fick's law of diffusion from Newton's second law of motion, rather than accepting the law by analogy with heat diffusion, as originally suggested by Fick (1855) and by authors of a recent paper dealing with diffusion and advection (Corey and Auvermann, 2003). Our objective is to show that chemical potential is useful for describing equilibrium, but not for evaluating transport.
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BACKGROUND |
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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We regard force resulting from a concentration gradient as an internal force because it derives from gradients of kinetic energy (of molecular translation) within reference volumes of fluid. Force associated with diffusion is not given by a pressure gradient acting externally on reference volumes of fluid. Corey and Auvermann (2003) have explained this fact under the heading of "Pressure and Normal Surface Stress." However, both the pressure gradient and internal driving force may result in motion of the center of mass of reference volumes of fluid.
The resultant driving force (acting on a reference volume of a fluid as a whole) is not given by the vector sum of a pressure gradient and body forces, if a density gradient exists. A differential reference volume may be treated as a single "fluid particle," in applying Newton's second law of motion, only for homogeneous fluids. The fluid dynamics approach to mass transport does not apply where density or thermal gradients exist. Nonuniform kinetic energy within each reference volume is responsible for diffusion of particular molecular species if a thermal gradient exists (Corey and Kemper, 1961).
Interdiffusion of particle species having identical masses results in no net mass transport. Forces associated with such transport within reference volumes of fluid have no resultant work relative to an external frame of reference. Work done is exclusively internal. No work is done against the surroundings of reference volumes. However, when particles of different mass interdiffuse, there is a displacement of the center of mass of reference volumes relative to an external frame of reference (Farr, 1993). Work is done that is not exclusively internal, and a net mass transport results that is not in response to a pressure gradient and body force on reference volumes of fluid.
Diffusion and Advection
Diffusion is defined as transport of a fluid constituent induced by a concentration gradient. It is widely believed that average velocity of a constituent, induced by a concentration gradient, is a velocity relative to the velocity of the centers of mass of reference volumes of a fluid as a whole (Bird et al., 2002; de Groot and Mazur, 1984; Haase, 1969). However, Corey and Auvermann (2003) have challenged the Bird et al. (2002) concept. A concentration gradient induces an increment of constituent velocity relative to the fluid boundaries, not relative to the centers of mass of reference volumes of fluid.
Advection is usually defined as transport induced by forces acting externally on reference volumes of the whole fluid, including all constituents composing the fluid. However, the analysis presented in the following sections does not support the concept that advective velocity is synonymous with velocity of the center of mass of reference volumes of fluid, as often assumed. Our analysis indicates that net mass transport can be evaluated only by including diffusion as an independent mechanism of transport in cases where diffusion represents a significant contribution to net transport.
Direction and magnitude of net transport of a constituent cannot be evaluated by the gradient of chemical potential, because this potential does not evaluate resistances associated with advection and diffusion. It cannot predict direction of diffusion, because chemical potential includes pressure as a variable. The gradient of pressure induces velocity gradients normal to solid boundaries and a resistance proportional to fluid viscosity. A gradient of concentration or temperature does not induce velocity gradients normal to the direction of flow, so that resistance is a different function of boundary geometry. Consequently, pressure should not appear as a variable, along with concentration, in a potential gradient indicating force associated with diffusion.
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ANALYSIS |
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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Driving Force for Advection
A pressure gradient is the only driving force for advection with the simple fluid system assumed. Force per unit volume resulting from a pressure gradient is given by:
 | [1] |
Resistance Force for Advection
Resistance to motion induced by a pressure gradient is of two types, depending on whether viscous flow or slip flow is considered. Slip flux refers to the increment of flux associated with a nonzero local velocity at solid boundaries. For most cases of liquid flow, slip flux is insignificant. Viscous resistance is proportional to rate of angular deformation of reference volumes of fluid. The coefficient of proportionality is called viscosity. Force per unit volume, resulting from the rate of deformation of reference volumes, during viscous flow is given by:
 | [2] |
Setting driving force equal to the resistance force gives a version of Stokes' law without the gravity term. Poiseuille's equation can be derived from Stokes' equation for flow in small tubes. By induction, we can also derive Darcy's equation for flow through porous media (Bear, 1972, Chapter 5; Sposito, 1978; Corey, 1994, Chapter 3). For systems in which the only driving force is the gradient of pressure, Darcy's equation (in terms of local velocity) is given by:
 | [3] |
is porosity of a porous medium and k is the permeability coefficient.
Kozeny (1927), and later Carman (1937), derived an expression for k as a function of channel geometry in granular porous materials:
 | [4] |
2 is the mean value of the hydraulic radius squared, ks is a shape factor having a value of about 2.5, and the ratio Le/L represents tortuosity of the flow path (length of flow path, divided by the direct distance between points where the pressure is measured). Tortuosity squared has a value of about 2 for fully saturated granular media. The Kozeny–Carman equation is presented here to illustrate the sensitivity of permeability to viscous flow on channel geometry, particularly the hydraulic radius. However, the Kozeny–Carman equation is based on the assumption that local velocity approaches zero at fluid boundaries, that is, there is "no slip" at solid surfaces. Experimental evidence indicates the no-slip assumption is valid for most cases of viscous flow of liquid solutions, but it is inadequate for advection of gases (Klinkenberg, 1941).
Driving Force for Diffusion
Diffusion of a constituent is in response to a concentration gradient. Concentration gradients are often regarded as the driving force for diffusion, although a concentration gradient does not have the dimensions of force. Fick's law relating diffusion to a concentration gradient apparently was originally derived by analogy with heat or electrical conduction (Bird et al., 2002). However, a better understanding of diffusion can be obtained from the derivation of Fick's law from Newton's second law of motion.
Diffusion involves mass transport of a fluid constituent against a resistance force equal in magnitude to the driving force. Newton's second law of motion defines force as the rate of change of momentum. In this case the change of momentum involved is the rate of change of momentum of a particular constituent as the constituent moves relative to the fluid boundaries in response to a concentration gradient. We define diffusion velocity as:
 | [5] |
The mass per unit volume transported by diffusion is:
 | [6] |
 | [7] |
Components of vt exist in all directions; however, vt has a resultant velocity in the direction of maximum decrease in concentration equal to jici–1. Equation [7] is consistent with Graham's law of diffusion (Graham, 1833); that is, diffusion velocity is inversely proportional to the square root of particle mass. By the same reasoning, Eq. [7] indicates that diffusion velocity is directly proportional to the square root of Kelvin temperature.
An expression for force resulting from a concentration gradient can be derived by writing an expression for the rate of change of momentum of particles moving in response to a concentration gradient. Momentum per unit volume associated with a diffusing constituent is micivdi, where vd is an increment of local velocity, vi – va, associated with diffusion. Diffusion velocity corresponds to the resultant average particle velocity of translation relative to fluid boundaries, a function of temperature and particle mass.
Force per unit volume (associated with diffusion) is the rate of change of momentum given by:
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For an isothermal system, we assume vd, as well as mi, is a constant. Applying the chain rule to the time derivative of concentration gives:
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Equation [9] is rewritten and generalized for three dimensions as:
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Assuming diffusion flux is proportional to the driving force:
 | [11] |
 | [12] |
Theory leading to Eq. [11] indicates that the coefficient in Fick's law, DF, is a function of concentration, molecular mass, and velocity of translation. Dependence on velocity of translation indicates a dependence on temperature. Dependence on concentration indicates that, for gases, DF also depends on pressure. This means that treating Fick's coefficient as a constant along a flow path should be regarded as an approximation only.
Corey and Auvermann (2003) presented a constitutive equation analogous to Eq. [11] for concentration dependent diffusion, as well as equations for temperature dependent diffusion. However, they forced their diffusion coefficients to assume the dimensions of the coefficient in Fick's law so that the true effect of concentration, temperature, and molecular mass are not properly accounted for in their equations.
It is important to observe that fluxes (evaluated by both Eq. [11] and Eq. [12]) are fluxes relative to fluid boundaries, not fluxes relative to mean velocity or the center of mass of reference volumes. Diffusion flux represents an increment of velocity resulting from a concentration gradient that must be added to the advective velocity to obtain the net transport of a constituent.
Resistance Force for Diffusion
A coefficient for diffusion through a porous medium, such as a granular bed, reflects the resistance opposing motion. Consequently, the diffusion coefficient is also a function of channel geometry. However, the diffusion coefficient does not include mean hydraulic radius as a variable. A concentration gradient induces transport of individual molecules, and does not induce angular deformation of reference volumes of fluid. Consequently, fluid viscosity is not a direct factor in resistance to motion induced by a concentration gradient, so resistance is not sensitive to dimensions of channel cross-sections.
Investigators of diffusion in porous media assume a diffusion conductivity function, analogous to Eq. [4], by modifying Eq. [12] to account for medium geometry (Bear, 1972, Chapter 4):
 | [13] |
Equation [13] applies for diffusion in porous media and accounts for the fraction of the medium cross-section available for fluid transport, and for tortuosity of the flow path.
Advection and Diffusion Combined
Advective driving force given by Eq. [3] includes the sum of partial pressure gradients of each constituent composing a fluid mixture. Advective driving force on a particular constituent per unit volume of fluid is the negative of the partial pressure gradient. An expression for the total driving force per unit volume on a particular constituent is given by:
 | [14] |
Unfortunately, Eq. [14] is not useful for evaluating net flux by advection and diffusion because the coefficients applicable for the two driving forces are dissimilar. Any attempt to combine the two coefficients encounters a problem because the combined coefficient will be a function of the ratio 
pi/ci. In any dynamic process this ratio will vary in time and space.
Equation [4] reflects resistance to advection and includes the mean hydraulic radius, but Eq. [13] (for diffusion) does not include the hydraulic radius. This is the reason chemical potential cannot be used to predict the direction of net transport of a constituent resulting from a pressure gradient and a concentration gradient. A chemical potential includes no terms referring to channel geometry.
Transport across Membranes
Porous membranes are of several types. One type is a semipermeable membrane that excludes all solutes and is permeable to a solvent only. Another type allows the passage of some solutes but not all. A third type allows the transport of all solutes, but some solutes are more restricted than others, and to a greater extent than the solvent.
Transport of a solvent through a membrane that excludes all solutes is easy to analyze, because transport of the solvent responds exclusively to the partial pressure gradient of the solvent (Meyer et al., 1973). A concentration gradient has no effect on transport through such a membrane, because there is no concentration gradient within the membrane. All solutes are excluded. In this case, transport of the solvent is by viscous flow and/or slip flux in response to a pressure gradient in the solvent.
Membranes that allow transport of solutes but restrict their passage to a greater or lesser extent require a more complex analysis. In the latter case, transport is affected by the total pressure gradient and also concentration gradients of all individual constituents. Without more detailed information about a particular membrane than is likely to be available, there is no way to predict direction of net transport of any solute or solvent. There is no function, referring to characteristics of the fluid only, that would permit such a prediction (Corey and Kemper, 1961).
Transport of water and solutes from soil–water solutions into plant cells through root hairs and across other cell membranes within living organisms is a very important process. Cell membranes in living organisms never totally exclude all solutes, or even most solutes. Consequently, transport is by advection as well as diffusion.
The following thought experiments illustrate the fact that a function (referring to fluid variables and fluid constituents alone) cannot be a predictor of the direction of net transport across a membrane of unknown properties. Figure 1 depicts a membrane separating a concentrated solute solution left of the membrane from a less concentrated solute solution on the right. However, the pressure in the less concentrated solution is greater than pressure in the more concentrated solution.
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The membrane illustrated in Fig. 1 is permeable in some degree to both solute and solvent, but the degree of permeability to each constituent is unknown. Coefficients of advective permeability and diffusion have not been determined. We ask the question, Will the solute move "from points of high chemical potential to points of lower chemical potential"? The answer is perhaps, and perhaps not.
If the difference in concentration of solute across the membrane is sufficiently great, the difference in pressure relatively small, and the permeability to advection is also relatively small, the solute may, in fact, move from a position of high chemical potential to a position of lower chemical potential. However, if the channel sizes through the membrane are sufficiently large, the concentration difference will be of insignificant consequence. In the latter case, the net flux of solute will be from high pressure to lower pressure.
We also conclude that the direction of net transport of solvent (in this case water) cannot be predicted from the gradient of the chemical potential. Figure 2 is presented to illustrate this conclusion.
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If permeability of the membrane in Fig. 2 to the liquid solution is very small, and the pressure gradient in the vapor phase is negligible, diffusion of water through the vapor phase can be the dominant mechanism of transport. As the liquid level in the reservoirs in Fig. 1 and Fig. 2 moves, water in the vapor phase is transported by both advection and diffusion. For the case illustrated in Fig. 1, advection moves water vapor counter to a chemical potential gradient.
An analogous condition exists in soils where liquid water content is a fraction of the total pore space. For example, when liquid water replaces the vapor phase in a soil during a rain or irrigation, the vapor phase is driven ahead of a wetting front by advection, as well as by diffusion. Similarly, when a wet soil drains, air (including water vapor) replaces liquid water by advection counter to a chemical potential gradient. Fluctuations of barometric pressure also promote pressure gradients within the vapor phase in soils. The role of advection in aeration of soil profiles is often overlooked, and an unjustified assumption is commonly made that the process is entirely a result of diffusion.
Where both pressure and concentration gradients exist in the vapor phase, it is not correct to state that water necessarily will move in the direction of decreasing chemical potential. Chemical potential governs direction of diffusion only in cases where pressure is constant. In the latter case, gradients of concentration and chemical potential are equal.
Direction of transport of the solute or solvent across a membrane at a particular time cannot be determined without an evaluation of membrane properties. It is possible to predict only that equilibrium will eventually be reached when both pressure and concentration are equal on both sides of the membrane. In the latter case, the chemical potential is also equal on both sides of the membrane.
Direction of Transport and the Second Law of Thermodynamics
It is sometimes argued that the second law of thermodynamics requires a constituent to move always in the direction of decreasing chemical potential, otherwise the Gibbs function for the system would increase, thus contradicting the second law of thermodynamics. However, one should observe that the second law applies to a system as a whole, not to a particular constituent at an instant of time.
Referring to the case illustrated in Fig. 1, we assume that the chemical potential of the solute is less in the more dilute solution on the right of the membrane, despite the slightly greater pressure in the more dilute solution. However, if the membrane is sufficiently permeable to advection, the net flux of solute will be toward the solution where the chemical potential is initially higher. Despite this, the entropy of the system increases and the Gibbs function of the system decreases. There is no contradiction with the second law of thermodynamics. The concept that a solute, or a solvent, must move in a direction of decreasing chemical potential is not a valid corollary of the second law.
We also note that regardless of the initial direction of net flux of the solute for the case illustrated in Fig. 1, equilibrium will occur when the pressure is equal on both sides of the membrane (at equal elevations). The concentration of both solute and solvent will also be equal. Chemical potential will be equal at all points in the system when equilibrium is established. We conclude that chemical potential is useful for describing conditions of equilibrium, not for predicting direction of transport.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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APPENDIX |
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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v, local velocity (LT–1)
va, advective velocity (LT–1)
vt, velocity of molecular translation (LT–1)
vd, diffusion velocity (LT–1)
j, molar diffusion flux (L–2T–1)
State Variables
T, Kelvin temperature
V, volume (L3)
C, concentration (L–3)
Coefficients
k, advective permeability (L2)
D, coefficient of diffusion (L3TM–1)
DF, coefficient in Fick's law (L2T–1)
Other
R2, hydraulic radius squared (L2)
ks, shape factor (dimensionless)
m, mass of a number of particles equal to Avogadro's number (M)
t, time (T)
, porosity (dimensionless)
L, direct distance between two points (L)
Le, length of flow path (L)
Subscripts
i, a particular constituent
r, resistance force
s, shape factor
a, advective velocity
t, velocity of translation
d, diffusion
Received for publication September 13, 2004.
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TOPABSTRACTINTRODUCTIONBACKGROUNDANALYSISCONCLUSIONSAPPENDIXREFERENCES |
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