Osmosis, diffusion,

convection

SOODAK, H., AND A. IBERALL. Osmosis, diffusion, convection. Am. J. Physiol. 235(l): R3-R17, 1978 or Am. J. Physiol.: Regulatory Integrative Comp. Physiol. 4(l): R3-R17, 1978. -We present a tutorial on the mechanisms of and connections among osmosis, diffusion, and convection. For simplicity, we consider only two-component nonelectrolyte solutions under isothermal conditions, Further, we confine our attention to laminar convection with application to the case of flow through narrow channels, as might occur in membranes containing pores or slits. The application of equilibrium and near-equilibrium thermodynamics to flow processes is just like considerations of mechanics with friction, or hydrodynamics. The description of flow processes of more than two atomistic components, either solutions or suspensions, is identical in the dilute limit to the description we give, except possibly when the curvature of the flow field (at the velocity profile) is significant. Flow fields, therefore, naturally divide into three regimes: 1) “one-dimensional” flow fields, e.g. 7 solutions or suspensions in extended regions, whose velocity profile is macroscopitally flat (compared to the atomistic curvature); 2) flow fields with significant curvature, e.g., Poiseuille or turbulent fields; and 3) high curvature fields, e.g., narrow flow channels.

fluid mechanics; namics; transport;

irreversible thermodynamics; mechanics; two-component flow processes; bulk flow

MANY BOOKS AND PAPERS discuss various experimental and theoretical facets of the problems of mixed hydrodynamic fields involving diffusion, osmosis, chemical reaction, and convection. Rather than offering any methodical review of their content, because our purpose is instead to make clear some of the perplexing connections between osmosis and convection, we would merely like to call the reader’s attention to the wealth of detail that these books provide. The references at the end of this paper give an excellent sample.

thermody-

states that the sum of the ‘“partial volumes” in a unit volume of solution is unity. The variation of Vi with pressure and composition is usually negligible. We may thus regard them as constants under isothermal conditions, and equal to the volumes per molecule or mole of the pure liquids in a solution of nondissociating liquids. Other composition variables are volume fraction +i, mole fraction xi, and mass fraction yi. These are related to the c’s by -

ciJTi;

xi

=

~

l Yi

Cl

THERMODYNAMIC

?TtiCi

Ci 4 i-

=

=

PRELIMINARIES

Equilibrium state functions of a solution may be regarded as functions of pressure p, absolute temperature T, and composition. The composition of a binary solution is characterized by a single composition variable, which may be selected in many ways. We will denote the composition variable by c when we need not specify its choice. When we do specify, we select as the composition variable, the “solute” concentration cz, which is the “particle count” of component 2 in unit volume of solution, either in molecules or moles per unit volume. The “solvent” concentration is then determined from c1q

+ c,v2 = 1

(2)

WC1

c2)

+

m2c2

where mi is the mass of a molecule or mole of i. In the dilute limit where c2 < cl, m2c2K mlcl, and 42K 1 x2 =

c2 --; Cl

m2c2

Y2 =

. 7

m1c1

1 Cl

=

-

V1

Let h denote the chemical potential of component i in a solution at p, T, c, the partial Gibbs free energy per molecule or mole of i. It varies with pressure and composition according to dk = Vidp + pi’dc

(4)

where h’ is defined as the composition derivative

(1)

where Vi is the partial volume of component i (i = 1, 2) per particle count (molecules or moles). Equation 1 0363-6119/78/0000-0000$01.25

Copyright

0 1978 the American

Physiological

Society

R3

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R4 The k’

H.

and the dh satisfy

the related equations (6)

which are isothermal versions of the Gibbs-Duhem equation. In the dilute limit, with cz as composition variable in Eq. 5 R*T ,ul’dC2 = - -0 7 Cl

(8)

c2

where R* is the gas constant per molecule or mole, in terms of which the pressure of an ideal gas (pz) is pz = R*Tc,

(9)

where c, is the gas concentration in molecules or moles per unit volume. The condition for mechanical equilibrium within a solution is the absence of a net external force acting on a unit volume of solution. This takes the form -vp

+ ClfieX + c2fzex = 0

where the negative pressure sure force on unit volume, external force (equivalent to or electrical force) acting per The conditions for transport vidual constituents are -v/J1

+ fi”” = 0 ;

(lo) gradient -VP is the presand hex is the long range or the actual gravity force, molecule or mole of i. equilibrium of the indi-v/J/2 + fz”” = 0

When the fi are field forces with potential molecule or mole, then ex - _ V@ i fi and the conditions

of transport

equilibrium

(11)

energy Qi per (12)

become

V(/Ji + @i) = 0

(13)

or constancy of k + @i (e.g., the “gravitochemical” potential or the electrochemical potential). A simple interpretation of the above results that is consistent with Onsager theory of irreversible thermodynamics is as follows. We regard the negative gradient -Vh of chemical potential as the thermodynamic or nonexternal field force per molecule or mole of i. Differentiation of Eq. 4 with respect to position leads to (14) Vpi = ViVp + /Li’VC which resolves the thermodynamic force

force into a pressure

f i JJ= -ViVp (15) and a “diffusion” force II f i - - j&‘VC (16) These are the forces per particle count (molecule or mole). Equation 11 then states that the net force (17) on each component must vanish at equilibrium fi = 0 ;

f2 = 0

(18)

AND

A. IBERALL

We now consider a unit volume of solution, containing Ci molecules or moles of component i, of component mass density Mi

The total forces (F) acting volume are F1

R*T p2’dc2 = -

SOODAK

=

on the components

Clfl = c,[fi”

in a unit

+ fi” + fi”“]

(20)

F, = c,fi = c2[fzP + fzl) + f2ex] Summing these to find the net force (F”) the matter in a unit volume gives

acting

on all

F n = F, + F, = -VP + clfiex + c&“~~ because of the additivity of partial volumes, Eq. 1, and the cancellation of the diffusion forces acting in a unit volume; ix* qfi” + c2f2n = 0 (22) which follows from the definition of fill, Eq .16, and from the Gibbs-Duhem relation, Eq. 6. Because of their equal and opposite nature, the various diffusion forces for the components may be regarded as purely internal forces, having no effect on the motion of the center of mass of a volume element, and acting only on the internal or relative motion of the constituents. Note that we are defining diffusion as relative motion between the constituents induced by composition gradients. The diffusional forces of Eq. 16 are not the only forces affecting relative motion. The pressure forces and long-range external forces may also act to produce relative motion between the component mass densities M, and M, in a unit volume. A simple discussion of mechanics in APPENDIX I shows that the net forces acting on M2 and MI to produce relative motion Fr are obtained from the Fi’s of Eq. 20 by

--

F lr

Th e mechanical reason for the combination of forces appearing in the parentheses of Eq. 23 is that forces that would result in equal accelerations have no effect on the relative motion. Substituting for the Fi from Eq. 20 results in . F -c,p.gvc 21I=II (24) f y13/2p 2ex _ ‘,,lex - (J!&!5)vpl ’ ’ [ m2 -I where yi is mass fraction, Eq. 2, and p is solution mass density p = mlcl + m2c2

(25)

The equal and opposite forces F2r and Fir act on the comnonent mass densities M9 and Ml in a unit volume of solution to produce relative motion. We can express this as a relative motion force acting per unit particle count (molecule or mole) of solute, component 2, by dividing Eq. 24 by c2. The result may be expressed as

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

R5

OPINION

pi

=

F

PI i

is the “partial density” of component i. Equation 26 expresses the force that acts per molecule or mole of solute, component 2, to produce relative motion between the components. In the dilute solution limit, the solvent mass fraction y1 may be replaced by unity and the diffusion force by VC .-..e.?

f 2 n z -R*T

c2

f 2 r=-(28)

(after use of Eqs. 16 and 8). We find, then, that the net force acting to change the motion of the center of mass of a unit volume element is Fn of Eq. 21 and the force acting to produce relative motion is fzr per molecule or mole of 2. The equilibrium conditions may then be restated as

f2 r--

F n.-- 0 ;

where p is the solution density. Thus, the g field generates the pressure distribution which balances the g forces. In a dilute solution, the density may be regarded as a constant, equal to the solvent density, which we assume hardly changes with pressure. Following the rapid establishment of mechanical equilibrium, the sedimentation rate is controlled by and is proportional to the force fzr that drives relative motion of solute with respect to solvent. We now apply Eq. 26 for f2’ to the dilute case where the solvent mass fraction y1 may be replaced by unity and the diffusion driving force f2[) is given by Eq. 28. The result is

0

(2% We illustrate the use of these notions first with discussion of the flat field forms of sedimentation, osmosis, and diffusion. A flat field is one with no curvature in the velocity profiles.

R*T pvc,-v2

(36)

c2

We note that the g forces have no direct effect on the relative motion, due to the fact that they are proportional to the masses, and thus h/m2 - film, = 0. They do, however, determine the pressure distribution of Eq. 35. The resulting pressure gradient drives relative motion of the solute with a force per solute molecule or mole of - v2 1 - g vp (37) ( Pl,) This force induces composition gradients that in turn generate oppositely directed diffusion forces of

vc

R*T ---J? ILLUSTRATING

FLAT

FLOW

FIELD

Sedimentation We consider a sedimentation process in an artificial centrifugal (e.g., augmented gravitational) field with external forces

f i ex

=

where the augmented gravity

mg

g = w2r

(31) crpis the angular velocity of rotation and r is the distance from the axis of rotation. The effective potential energy in this centrifugal field is c

(32) \ /

and f;:“” may be expressed as d ma2r2 z _____ (33) dr ( 2 )/ When a centrifugal field is imposed on a solution of initial uniform composition, mechanical equilibrium, Eq. 29, is established very rapidly, before any significant relative motion occurs between the components of the solution. The condition for mechanical equilibrium, as obtained from Eqs. 29,21,30, is

f i ex

--

V@i- -_-

-VP + (m,cl + m,c,)g = 0

+ pg = 0

PzVzg

-

(39)

p,$g

which is the effective weight of a molecule or mole of solute, less the effective weight of the displaced solvent. Archimedes’ principle of buoyancy is contained in Eq. 24 1The differential equation for the concentration c2 at equilibrium is obtained by setting the sum of Eqs. 38 and39 equal to zero. After using Vc, = dc,/dr, g = w2r, and m2 = p2V2, the result is 1 dc, --c2 dr

(40)

which integrates to c2 = constant X exp [(I-Z)

s]

(41)

If sedimentation occurs in a normal constant 1.0-g field the result is c2 = constant X exp [-(l&J

E]

(42)

(34)

or -vp

per solute molecule or mole. At equilibrium, the sum of these forces, fzr, is zero. The relative motion pressuref orce of Eq. 37 may be rewritten by using Eq. 35 and replacing p and fil by the solvent density p 1, as is appropriate for a dilute solution. The result is

(30)

potential g is given by

?72iW”P @; = ~ 2

(38)

c2

PROCESSE

(35)

where x is altitude in the g field. In both cases, there appears the “Boltzmann factor,” the negative exponential in

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R6

H. SOODAK

chanical equilibrium

exp

and Diffusion,

A. IBERALL

is again maintained between the

(43) two sides because of the free motion of the membrane

where CD, is the energy per particle count, which in this application is the gravity potential energy including displacement effects, or buoyancy. It is remarkable that the thermodynamics formulated by Gibbs for sohtions of molecules applies just as well to suspensions of Brownian particles, in the dilute limit. This was first pointed out by Einstein (7) in one of his famous papers on Brownian motion and osmotic pressure. Osmosis

AND

Part I

We now apply the same mechanistic interpretation of thermodynamic drives that we used in sedimentation to discuss the mechanistic relationship between diffusion and osmosis. So now consider a horizontal cylinderpiston arrangement in which a piston to the right of a reservoir maintains constant pressure p. The volume enclosed between the piston and the fixed left end is divided in half by an impermeable freely movable partition. In the right half is a reservoir of pure solvent and in the left half, a solution. This is a two-phase system at equilibrium. Each phase is in mechanical and transport equilibrium within itself, and the two are in mechanical equilibrium across the impermeable partition. Now remove the partition, allowing component transfer between the two halves. Mechanical equilibrium is maintained, Vp = 0, and there are no external forces acting on any volume element within the liquid (we ignore gravity here). The components mutually diffuse, solute rightward, solvent leftward until uniform composition is eventually achieved. Since Vp = 0 and @” = 0, the driving forces for the relative motion are only the diffusion forces arising at the locations of composition gradients. From Eqs. 22 and 28, these are

partition. What happens now is that the diffusion force acting on the solvent at the membrane boundary drives solven .t across the membran .e into the solution. Pressure rise in the solution is prevented by membrane movement. Equilibrium is again eventually achieved when the membrane has moved completely to the right, and the solution on the left side is uniformly concentrated. We now consider a third and final arrangement in which the partition is replaced, at time zero, by a semipermeable membrane fixed in position. Again, the diffusion force acting on the solvent at the membrane interface drives solvent across the membrane. As a result, the pressure rises in the solution as mass flows in. The pressure on the solvent side is prevented from dropping (due to exit of solvent) by the piston on the right which maintains fixed pressure p. The buildup of pressure on the solution side of the membrane generates a pressure force, Eq. 15, that acts to drive both components from solution side to solvent side. The force on the solute is balanced by the membrane which prevents its motion. The force on the solvent acts to counter the diffusion force on the solvent. Equilibrium is eventually achieved when the pressure in the solution rises to the equilibrium osmotic pressure (and when diffussion within the solution volume achieves uniform composition). At this point, the solvent is in balance across the membrane, the pressure force balancing the diffusion force, i.e. 9 when fi” + fl*j = 0. Recalling that fi” = v,Vp, we are led to

Vp = R*TVc,

= VII

(481

Integrating this across the membrane interface leads to van’t HofYs law, p (solution) = p + II

(49)

for the equilibrium osmotic pressure difference between the solution and the pure solvent across a semipermeaVC2 D fi” = “*TVs = R*TV,Vc, f 2 = -R*T---; (44) ble membrane. Eq uation 48 could have been -written c2 Cl down immediately by demanding that the chemical potential pl of the solvent component is the same on per molecule or mole of component and both sides of the membrane, but this purely equilibrium F 2 I1 = -R*TVc2 thermodynamic result hides the mechanism. (45) Two points emerge clearly from this analysis. One is =- F 1I1 that diffusion and osmosis are both initiated by unbalon solute and solvent in a unit volume of dilute solution. anced diffusion forces. In the pure diffusion case, with Introducing the osmotic pressure I1 of a dilute solution no partition, both components mutually diffuse in opI-I = R*Tc, (46) posite directions. When the membrane is present that prevents the flow of solute, then only the solvent the equal and opposite diffusion forces Per volume are diffuses across the membrane. The second and related point is that the osmotic pressure di fference tha .t arises F 2I) =- Vi-l (47) across a fixed semipermeable membrane comes from The time course of the approach to equilibrium is the inflow of solvent, as in the third arrangement governed by rate and balance equations to be described discussed above. No pressure difference arises when the membrane is free to move. The view that the osmotic in the next section. We return again to the original cylinder-piston ar- pressure difference is “caused” by the bombardment of solute against the membrane with an ideal gas pressure rangement, but here instead of removing the partition at time zero, we replace it by a freely movable semi- ll = R*Tc2 is not basically correct, even though it gives perm .eable membrane partition that passes only Sol- the right answer. This point is discussed further in vent, an a.rrangement introduced by Einstein (7). Me- APPENDIX v. The detailed mechanism of the osmotic Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

R7

OPINION

flow of solvent across the membrane is discussed in Osmosis, Part III for two Diffusion, Sedimentation, membrane models. Diffusion,

Part II-A

Rate Process

When a net force acts on a free particle, an acceleration results. When a net force is impressed on an element of an ensemble of interacting elements (a thermodynamic system), the relative acceleration is halted by a frictional interaction force. For forces that are not too large, that do not destroy local thermodynamic equilibria, the frictional force is proportional to the flow rate induced by the impressed force. The final result, after a brief acceleration period, is then a steady flow rate that is proportional to the impressed force. This is a mechanical description of the first basic fact of irreversible processes. Flow rates are proportional to driving forces. We apply this idea to the relative motion between solute and solvent in a dilute solution. The net driving force hl’ for this relative motion per molecule or mole of solute is given by Eq. 26, as the sum of a diffusion force, external force, and a pressure “displacement” force (which vanishes when solute and solvent have equal densities, p1 = p2). In dilute solution, y1 = 1, and f211is given by Eq. 28. The net driving force fir per particle (or mole) of component 2 results in a relative velocity uzr that is proportional to fir. The relative velocity can be defined as v2 - ul, the velocity relative to solvent, or as u2 u,, the velocity of solute relative to the center of mass velocity, v,,,, of a volume element of solution. In the dilute limit, there is negligible difference between u1 and urn, and we need not specify the choice. In any case, the two relative velocities are proportional. The proportionality constant between relative velocity and force, u12

compared to a solvent molecule, the mobility may be evaluated from macroscopic hydrodynamics. Regarding the particle as a “Stokes’ ball,” a sphere of radius a, the hydrodynamic friction force exerted on one ball moving with speed u2 through the solvent has a magnitude, given by Stokes, of hrjav,, where r) is the shear viscosity of the solvent. We express this result in the form

f 2f =

6qaN*u2

(55)

where N* = I when &” is the force per molecule or particle, and N* = N,, = Avogadro’s number when hl’ is the friction force per mole. The velocity of the particle is that for which the friction force balances the driving force f& which is vz = fLr/6nqaN*. The mobility is then cc)21 =

1 6rr)aN”

(56)

When the only force driving relative motion is the diffusion force, seeEq. 28, arising purely from a concentration gradient, the resulting current is the diffusion current, with density jn = c2v2= c2cr)21&2rj = -R *Tw,,Vc2

(57)

This is identical in form to Fick’s law

jn = -DVc,

(58)

with a value for the diffusion coefficient D given by D = R*To,,

(59)

which is known as Einstein’s formula. We turn now to a review of diffusion from the kinetic point of view. Solute particles, not too large when compared with a solvent molecule, are considered to undergo a “random walk” process. In APPENDIX II, we treat the simplest one-dimensional random walk in which a particle moves along a line in randomly di(J?J21fI11 v2 r = (50) rected steps, each step covering a fixed distance k in a is called the mobility of solute 2 in solvent 1. fixed time 7, with probability one-half for moving right Let us assume here that the solvent is stationary (as or moving left. We also treat the three-dimensional in the sedimentation process, for example), in which generalization of this process, in which the step direccase v1 = v,,[ = 0. Then u21’is simply the solute velocity tion is with probability one-sixth along the t x, t y, or u2 in the laboratory frame of reference, and t z directions. It is shown that a collection of particles undergoing such a three-dimensional random walk v2 = w2# (51) process diffuses according to Fick’s law, with diffusion The “current” J2 of the solute across an area A of flow coefficient D given by is A” D -(60) 67 J 2 = c2v2A (52) in molecules or moles per unit time, and the “current density” j, , defined by

It is also shown that the mean square, , of the displacement along any coordinate axis that such diffusing particles undergo in a time t, is

J j2 = 2

(54

= 2Dt

(54

Finally, the conservation equation that governs the time course of the concentration distribution is shown to be

is . 52

=

C&

in molecules or moles per unit area and unit time. For a solute molecule or narticle that is large enough,

ik

- = DV2c dt

(61)

(62)

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R8 An

H.

intuitive

view of this equation is presented in The above results are valid for more general random walk processes, as long as 7 is the mean time per diffusion step and h2 is the mean square distance per step. The simple random walk picture is not appropriate for solute particles that are large and massive compared to a solvent molecule. Such Stokes’ balls do not change direction so readily due to their larger momentum. As a result, their motion is more properly described as a hydrodynamic motion with random fluctuations induced by molecular bombardment by solvent molecules. This description of Brownian motion was provided by Einstein (7). It also leads to Fick’s law, and with a D value given by Eqs. 56 and 59. There is evidence (23) that the crossover from simple random walk motion to Brownian motion occurs for molecules of -3-4 A radius diffusing in water. This point is discussed further in APPENDIX IV. We have described two views of diffusion. In the first macroscopic thermodynamics-mechanics view, the diffusion current is considered to arise from a diffusion force that is regarded to be equivalent to an actual force externally applied, or a pressure force. In the second, kinetic-microscopic view, diffusion currents result from isotropically random motion (simple random walk or Brownian motion), and not from any actual force. The bridge between these views, as provided by Einstein, is joined by two arguments. In the first, it is noted that a concentration gradient does generate a current even when there is no real force (from feXt and VP), and that this current is j21’ = -DVc,, arising from random motion. It is further noted that this random contribution to the current can cancel the contribution j21‘eal of the real force, f2rea1. Thus, at equilibrium APPENDIX

III.

-DVc,

+ c2w21f2rea1= 0

where the mechanical relation l

J2

view real

-

(63)

of the real current-force

c26?& fzreal

(64

was used. It is finally noted that the thermodynamic condition for equilibrium (which appears as f2r = 0 in the mechanical view) is -R*TVc,

+ c2hreal =0

(65)

which is the equation f:rr = 0, multiplied by c2, with f21’ replaced by its dilute approximation, Eq. 28. Comparing Eqs. 63 and 65, then results in Einstein’s formula, Eq. 59, relating the random motion diffusion coefficient to the mechanical-hydrodynamic mobility. Thus, although diffusion is a random process, its effect on macroscopic flow rate is identical to that of a real force equal to f2Y The second argument consists of a direct calculation of the diffusion constant by analyzing the details of Brovvnian motion, leading to the identical result. Muerosctopic: j7ours are produced by real forces, and by the corn&nation of isotropic random motion (local equilibrium) plus macroscopic gradients (global disequilibrium).

Diffusion,

Sedimentation,

SOODAK

AND

Osmosis,

A. IBERALL

Part III

In the last section we introduced the mechanical view of diffusion rates, in which a diffusion driving force is balanced by a friction force proportional to relative velocity. In this section, we generalize this approach to situations with external forces, such as gravity in sedimentation and membrane forces in osmotic processes. We still consider only flat flow fields, in which boundary friction is absent. Further, the treatment does not apply during transient periods of accelerated flow, Also, we do not distinguish here between solvent velocity ul and center of mass velocity u,,,, and thus we restrict the discussion to dilute solutions wh.ere this difference is negligible. Boundary friction does not arise in flat flow fields, in fields with no velocity curvature. Yet friction may arise from the interpenetrating relative motion between the components, and from relative motion between any one of the components and an interposed “flat field” membrane. These friction forces are proportional to the relative velocities and are directed on each component so as to reduce the relative velocity. Thus

f ;1 = - k 21( v2 - 4); f 12 = - k 12( Vl v2>;

flv = - v,) (66) f ;cL= -&-Lh - q-J 42Jv2

u, is the where vl, u2 are the component velocities, membrane velocity (generally taken to be zero), f ij for i, j = 1 2 is the friction force (appropriately directed) exerted on one molecule (or mole) of i due to its motion relative to component j, and fiP is the friction force exerted on one molecule (or mole) of i by the membrane. The total friction force on one molecule or mole of i is f {

fi = fil + f&b;

fl = fin

+

(67)

fLL

The friction forces mutually exerted between all the molecules of solute and solvent in a unit volume must be equal and opposite. Thus

c,fi2 + c,f& = 0 which tells us that the friction are related by elkI

(68)

coefficients

k2, and h,,

+ c2k21 = 0

(69)

Finally, the friction coefficient k21 for the friction force exerted on one molecule (or mole) of solute 2 due to its relative motion through the solvent 1, is the reciprocal of the mobility CC)~~ introduced in Eq. 50 k 21 =

1

(70)

921

The component flow velocities ul, u2 arise in response to the driving forces and rapidly build up to the values at which friction balances the driving forces. The equations describing this balance for each component, are

fi” + fi” + fi”” + f ;z + f ;, = 0 jy + h” + fi”” + f& + f ;, = 0 where we recall hex is the external particle or mole of i

force acting

(71) va per

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

R9

OPINION

f i 1, =

-viJ7p

(73

and VC f 2I) = -R+T-?

(74

c2

f 1)- R*T 1-

2 = R*T

vc

v,VcZ

(75

Cl

Equations Multiplying

74 and 75 are valid in dilute solutions Eq. 71 by cl, Eq. 72 by cz, and adding gives

-vp + ClfieX+ czfzex+ c,fip + czf;p = 0

(76)

because the diffusion forces f;:” and mutual friction forces f ij add to zero on all the component contents in a unit volume of solution. Equation 76 states that the net force, driving (due to pressure and external field) plus friction (with the membrane), on a unit volume of solution is zero. As a first application, we consider a pure diffusion case, one with no external force and no membrane but with a concentration gradient. The solution is stationary, Vl = v,, = 0, and only diffusion takes place. The condition for mechanical equilibrium, Eq. 76, reduces to vp = 0, a uniform pressure. Equation 72, the balancing of the net force on solute, reduces to

f2/’ + f& = 0

(77)

which leads to the correct results

and j, = c2v2 = -DVc,;

D = R*Tw,,

(79)

We apply the equations to the sedimentation process considered earlier, where

f ex = 1

m,g ;

fLex = m,g

(80)

and there is no membrane. The condition for mechanical equilibrium (in the rotating centrifuge frame), Eq. 76, reduces to the correct hydrostatic pressure distribution -vp+pg=o;

p = mlcl + m2c2= p1

w

where p is solution density andp, is density of pure solvent. The flow of solute through the stationary solvent (v, = v,,, = 0 in the rotating frame) is obtained from Eq. 72 which reduces to

f I) 2

Substituting

v2Vp + m,g - kz1v2 = 0

for the pressure gradient, then gives v2

=

~2lW

+

cm2

-

Plvz)gl

(83)

after using Eq. 70. Solute moves in response to a driving force that is the sum of the diffusion force and the solute weight corrected for buoyancy. The quantity p1v2g is precisely the weight of displaced solvent. As a final application of the flat field force equations, we consider a variant of the osmotic flows described

earlier. Here, a fixed semipermeable membrane across a cylinder separates a solution compartment on the left from a pure solvent compartment on the right. Furthermore, both compartments are maintained at constant and equal pressures p by the action of pistons. We use the simplest model for the membrane that is consistent with flat field conditions. The membrane is homogeneous and of thickness Z, extending from x = 0 on the left to x = Zon the right. The membrane effect on solute is represented by a force field fzexdescribed by a 2, which is zero to the left of potential energy barrier CD the membrane (x < 0)and rises steeply to a large value some small distance E K Z into the membrane. This potential energy value is assumed to be much larger than the thermal energy R*T. Thus, solute molecules are strongly blocked by the membrane. If czo is their concentration in the solution and at x = 0, then the concentration c2 in the membrane drops to zero well inside E. At the same time we assume that the membrane presents no potential barrier to solvent molecules, which thus can move freely into and out of the membrane. However, their passage through the membrane is impeded by a friction force fiP proportional to relative velocity. Let us postulate an initial condition of uniform pressure in the membrane, equal to the fixed pressure p on both sides of the membrane. Our picture of what follows is this: in the very thin region between x = 0 and x = E, there is a concentration gradient. (It is the only region where Vc2 # 0 initially. To the left ofx = 0, c2 = czo, and to the right of x = E, c2 = 0.) This generates diffusion forces fi”, fzlj, that act to drive solvent leftward and solute rightward. The solute flow is blocked. Solvent diffuses leftward out of the thin region, and across the membrane interface. As a result, the pressure in this thin region decreases to a value below p, and a pressure gradient is developed across the thickness Z of the membrane that drives solvent through at a steady rate (assuming that the concentration czo at the membrane interface remains constant). The time required for this pressure gradient to develop is very short because the distance E:is assumed very small. The force equations are not adequate to deal with the rapid transient stage, but will tell us the following: starting from the left solution compartment, and moving rightward, the pressure is constant at p till the membrane interface at x = 0 is reached. It then drops suddenly to p - IYI in the short distance less than E, and rises at constant gradient Il/Z to the pressure value at the right boundary of the membrane, x = Z, which is the same p value in the situation being considered. This pressure gradient then drives an ohmic flow of solvent through the membrane at a rate determined by the pressure gradient and the frictional resistance presented by the membrane to the flow of solvent. The pressure “discontinuity” that arises is equal to the osmotic pressure of the solution lYI = R*Tc,”

(84)

To demonstrate this, we consider the force balance Eq. 71 for solvent, written out as

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

RlO R*T ~-Cl

H.

de, dx

d v, - p - k 12 ( 4 dx

-

v2>

-

klCLCV1

-

v,)

=

0

(85)

where we have used flex = 0. We apply this equation to the membrane region between x = 0 and x = E. We set v, = 0 because the membrane is fixed. Multiply through by cl, use c& = 1 (dilute), and Eq. 69, to replace elk,, by - c2k21. The result is R*T

dc, --dx

dp + c2kgl(vl - v,) - clklPvl dx

= 0

(86)

It is immaterial whether or not we set v2 equal to zero, because the term it appears in is of negligible consequence. Multiplying Eq. 86 by dx gives R*T

de, - dp + Q dx = 0

(87)

where Q denotes the two velocity terms in Eq. 86. We now integrate Eq. 87, starting at x = 0 and going to x = E. The result is R *m2k)

-

c2”] - [P(E) - p] + E

= 0

(88)

where c2(e) is the c2 value at x = e, p(e) is the pressure atx = E, and is an average value of the sum of the two velocity terms. But c2(e) can be set equal to zero for our membrane, and thus p(e) = p - R*Tc20 + The term E is negligible compared due to the smallness of E, and thus

E with

the others

PW = P - Jl

w9

We now consider Eq. 86 in the remaining part of the membrane, the region from x = e to x = Z, inside which c2 can be set equal to zero. Thus dP - = clklpvl dx Neglecting the small compressibility of solvent current density j, = clvl (particles unit area and unit time) must be constant of constant cross section and also cl must Thus, cl, vl, and dpldx are constant in the the right of E. The pressure gradient dpldx is dP - PM - P(E) = p - P(E) I-I -dx 1 -E 1 -c =I-E=i leading to a current

density

solvent, the or moles per along a flow be constant. membrane to

n

(92)

of solvent

The net force exerted on the membrane by solute and solvent can be demonstrated to be zero. The solvent exerts a leftward friction force equal to the pressure drop IYl per unit area of membrane, and the solute exerts the opposite force of lYl per unit area against the membrane barrier. We recall that the solvent cannot exert other than a friction force on this membrane model under consideration.

SOODAK

AND

A. IBERALL

The above discussion demonstrates that the flow induced by the osmotic difference I-l, with no pressure difference; is the same as the flow induced by an equalsized pressure difference, with no osmotic gradient. The reason for this is clear. The osmotic gradient is transformed into an equal-sized pressure gradient within the membrane, caused by an initial equilibration of solvent forces at the solvent-solution i nterface. Thus the flow through the membrane, except for the E region near the interface, is driven by pressure gradients. When both compartments linked by the membrane are solutions with arbitrary p values and lI values, equilibrating action and pressure j umps take place at both solventsolution interfaces to convert the osmotic difference Al-l into an equal-sized pressure difference which is impressed within the membrane, in addition to the exterially imposed pressure difference. The flow is then equivalent to that produced by a pure pressure difference of Ap - AIL We note that this result is also true for a homogeneous membrane that does impede entry of solvent. This blocking effect can be modeled by a potential energy barrier QI, which is zero outside the membrane and rises to a maximum value Qlmax a short distance into the membrane. In this case equilibrium of solvent requires constancy of pl + Ql across the membrane interface. Thus Apl = -A@, = -Qlmax. Using Apl = vlAp, we arrive at a pressure jump of Ap = -@lmax/~l. Thus the pressure drops as we cross the interface from liquid region to membrane region. But this drop is the same at both membrane faces, and does not, therefore, contribute the pressure difference within the membrane. The characteristic of the membrane model that is responsible for the equivalence is the high degree of impermeability to solute, as measured by the small size of the effective barrier distance E compared to the membrane thickness. The same processes occur in porous semipermeable membranes that operate by allowing solvent to enter pores freely while blocking completely entry of solute. Equilibration of solvent forces at the pore-solution interfaces results in pressure jumps which transform the osmotic pressure difference between the solutions linked by the membrane into equal-sized mechanical pressure differences across the pore. Such pressure jumps in narrow channels have been discussed and observed by Mauro (19; also see Ref. 21), who measured negative pressures (tensions) in water channels linking osmotic solutions; p - lI easily goes negative for p = 1 atm and Il = several atmospheres. One difference between the porous membrane and the homogeneous model lies in the mechanism of the resistante offered to solvent flow. The homogeneous model provides a flat field local friction, and the pore resistante initiated at the internal pore surface results in a curved flow field in the pore. For pore sizes large enough compared to solvent molecules, this flow resistance is controlled completely by solvent shear viscosity and is Poiseillian. We cornme& on this flow in the subsequent section on flow fields with curvature. As a final comment we note that the flow rate

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

Rll

OPINION

through the membrane is given in terms of the instantaneous value of solute concentration, cZo, at the solution-membrane interface, x = 0. The time course of the osmotic flow is then controlled by the time course of cZo. In the cylindrical geometry considered above, the flow of solvent leftward across the interface displaces solution leftward, and would result in c20 decreasing to zero were it not for diffusion. If the solution compartment was well stirred, then c20 would remain constant (assuming a large volume of solution compartment). If there were no stirring at all, the value of czo would drop, decreasing the driving force and rate of osmotic flow through the membrane. The time course of the entire process could be described by combining Eqs. 84 and 93 with the equations describing diffusion in the solution region. We have neglected the effects of gravity in the above discussion. The vertical pressure differences resulting from gravity action are not constant throughout, bottom pressures being greater underneath higher density places. The resulting pressure gradients drive convection currents and provide unforced mixing. Without forced stirring and without gravity activated “natural” convection, an osmotic flow can be as slow as an unstirred diffusion flow, because the osmotic flow would then be rate limited by diffusion in the solution side of the membrane (cf. Ref. 8). A differing view on osmotic mechanisms has been introduced to the physiological community by Hammel and Scholander (see for example Ref. 11). In APPENDIX v, we offer some critical commentary on their view. FLOW

FIELDS

BOUNDARY

Poiseuille

WITH

CURVATURE

(VISCOUS

CONDITIONS)

called the kinematic viscosity Y (momentum ity). The rate at which momentum is convected field is the momentum density p u multiplied velocity u to give p u2 as momentum per unit unit time. The ratio of incoming momentum momentum outgoing by diffusing toward the walls is called the Reynolds number. V2

Re=---

diffisivinto the by flow area per pu2 to channel

r0u

(95) r) e-0

-

lJ

When the ratio is small enough, e.g., Re < 1,000 in channels, molecular diffusion controls the friction process, by diffusing momentum toward the wall, and the flow is laminar. Laminar flow with no-slip boundary conditions at stationary channel walls is called Poiseuille flow. The relation between longitudinal pressure gradient dp/dz and the steady-state velocity profile across the channel is obtained by balancing the driving (pressure) force on a unit volume, -VP, and the viscous friction force. Thus -VP

+ F’ = 0

WI

where F’ is the friction force (with its appropriate direction) exerted on a unit volume by neighboring volume elements, through the mechanism of momentum transfer by molecular diffusion. This mechanism is identical to that of particle or mass transfer by molecular diffusion, described in APPENDIX II. For F’ we take the net momentum diffusing into a unit volume in unit time, and this is given by the product, momentum diffusion coefficient U multiplied by the Laplacian of the momentum density, V2p u, as discussed in APPENDIX III for particle or mass diffusion. Thus

Flow

F’ = 7j-v2v

The resistance to liquid flow through a channel is initiated at the walls of the channel; otherwise, the fluid would move as a solid plug. Except for kinetic considerations at molecular size scale, the boundary acts to prevent fluid slip motion, and the macroscopic wall boundary condition is zero fluid velocity with respect to the wall. This necessarily results in a transverse velocity variation with curvature (maximum at the channel center). Molecules diffusing in the transverse direction carry their longitudinal velocity u along with them, thus mixing u values and causing viscous friction. Another view is that there is a net diffusion of axial momentum toward the walls of the channel, with a rate measured by

for an incompressible

fluid with

(97)

constant

density,

vp = 7)vzv

(98)

or g=?(g+g)

(99)

where x, y are transverse channel coordinates. In a circular tube of radius ro, the solution that satisfies u = 0 at r = r. is u=uo

and

I---(

X2

Y”

ro2

ro2,)

of Eq. 99

or ,v=?E r0

P ro

(94

where r. is a measure of transverse channel size, u of central flow speed, p is the liquid density, and q is the liquid shear viscosity. This expression gives the shear stress, force per area, or equivalently, the momentum flow per area and time (due to molecular diffusion). The ratio u/r0 is a measure of the transverse velocity gradient, and pu/r, measures the gradient of momentum density p v (per volume). The quantity q/p is thus the diffusion coefficient for momentum diffusion, and is

V =

u.

I--;

r2

uo=---

ro2 dP ww

47j-dx > where r = dm is the distance from axis, and v. is the maximum, central velocity. Averaging over the cross section area gives the mean flow velocity (v) as (

f-O2

(v) = 1 - u. = - -To2 -dP (100 2 87 dz Finally, multiplying by the area n r-2 gives the volume current JV (in cm3/s) as

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R12

H. SOODAK

AND

A.

IBERALL

In a deep topological sense, quasi-static stability

in

(102) the distributed elastic system and dynamic instability

in the fluid mechanical field are connected, but we do not stop to explore this issue (see Ref. 10). Suffice it to say that there are many competing force systems in (103) fluid mechanics that produce transitional forms, that are responsible for many examples of complex form and where y measures transverse distance from slit center, function. In hydrodynamics the transitions are generally identified by a dimensionless parameter that gives and the ratio of the competing force systems. We will name Yo2 dP (104) a few. The first such process variable -illustrating the comu0 = - G z petition between momentum diffusion and momentum 2 convection of kinetic energy-was the Reynolds num(4 = - uo uw 3 ber, which characterizes turbulent flow in a channel, or around a compact object. Competition between thermal and diffusion and momentum convection in Benard cells J \ = (v) A = $ u,,A (106) (Rayleigh number), or competition between vertical buoyancy and momentum convection (Richardson number) governs instabilities in the atmosphere. On water for a slit of area A. With regard to solution flows in channels, where the surfaces there is a competition between momentum channel size is large compared to the particle sizes, yet convection and the gravity potential associated with viscous shear develops flow field curvature, the convec- surface waves, measured by the Froude number which tive and diffusive (relative) motion of the components determines both surface boundary resistance and also are independent. However, the combination of the shear cross-channel transfer, e.g. 9 aeration rate. Such procand diffusive flows, introducing competing flow proc- essesillustrate “facilitated” diffusions in which a macroscopic scale “eddy” diffusivity acts to augment the esses (see the next section), results in convective diffuslower molecular diffusivity, either axially or transsion, an enhanced or “facilitated” axial diffusion (see versely. In fractionating columns, with a continuous APPENDIXVI). With regard to pore flow, the theory developed in the plate (as compared to a discrete plate column), the previous section for osmotic flow through semipermea- difference between the radial flow process and the axial flow process is exploited to produce fractionating sepable homogeneous membranes also applies, with slight amendment, to porous membranes which allow free ration. A very rich literature, ably exploited by chemientry of solvent into the pores but block entry of solute. cal, civil, aeronautic and naval engineers, investigates The equilibration discussed there, occurs here at the many of the mechanisms that are of significance for facilitated transport in biology. We have selected one interface between solvent in the pore and solution adjacent to the pore. The integration of the solvent force clear mechanical problem, involving one of the many balance equation across the pore interface can be car- contributions of G. I. Taylor (l), for exposition in ried through to lead to the same pressure jump, or APPENDIXVI. perhaps slightly altered (69 21), depending on the nature (NEAR-CONTINUUM FLOW) of the barrier preventing solute from entering the pore. HIGHLY~URVEDFIELDS Further, the flow rate through the membrane is then Flow in Narrow Channels and Boundary Effects the sum of the rates through all the pores. New effects become significant for flow of solutions Competing Flow Processes and suspensions through channels in which the ratio cy That aspect of flow fields which has provoked the of particle size a to channel size r. greatest excitement physically has been the resultant a a=---of competing force fields. There is a richness of interac(107) r0 tion which is responsible for most, if not all, levels of emergent evolution including life. This result might is not negligible. Effects appear at the channel wall and perhaps be viewed as somewhat surprising, because a in the core of the channel. The simplest wall effect has casual view of elementary statics is that a vector force been called steric hindrance, and is accounted for in the system simply produces a static resultant. Yet as soon flow field equations by merely restricting the particle as we address any field continuum which permits non- centers to the allowed core region r < r. - a. The rigid movement, even elastic deformation issues of radius a is here regarded to be that of the solute stability emerge. This fact was first demonstrated by particles, and we assume for simplicity that solvent Euler in the theory of the buckling column. It is molecules are small. In the allowed core region particle interesting to note that Poincare introduced the prob- size affects the friction force fil exerted on a solute lem of dynamic stability of orbits, not with a timeparticle moving relative to the solvent, resulting in dependent problem, but with the static problem of the solute-solvent slip flow, and in a reduced diffusion snap diaphragm (edge buckling in a plate). coefficient. For small enough particle sizes or wide For a slit of half thickness,

y.

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

R13

OPINION

enough channels (cx < l), the variation of solvent velocity (due to field curvature) across a solute particle is negligible and the friction force reduces to the Stokes drag flat field result, Eq. 55. An approximate representation of the altered drag force .S

- k21 i u2 f ;I ==+

-

h)(7

(108)

where v2 is the velocity of the center of the solute particle, imagined to be a Stokes’ ball, and ( vJn is the value of the Poiseuille flow field velocity averaged over the surface of the particle of radius a. This equation describes a nonlocal friction force which depends on velocities some distance removed from the particle center. This leads to slip flow, as we now demonstrate. Consider a Poiseuille flow field (Eqs. 100 and 101) ilW where (~1,) is the average value of v1 across the pore area n ro2, with a solute particle suspended in the field at a particular radial position r. We consider a case where the force t2” driving relative motion is zero, which occurs when no external forces act and when the solute particle has the same density as solvent (see Eq. 26). The particle moves along a streamline with velocity v2 which is such that the friction force balances the driving force. But the driving force is zero for the case under consideration. Thus, from Eq. 108, we see that the particle velocity is given by v2 = (v,)” According

to

APPENDIX

III,

(110

Eq. A3.2 L

( Vl > 0

E

v1

+

a-

v2v1

6

(111

The study of further aspects can be begun by examining Refs. 3, 5, and 14. More elaborate commentary and kinetic studies on narrow channel effects are not warranted in this review, except to note that the discussion presented above does not apply to channels less than approximately 10 solvent molecules wide, in which kinetic effects at the channel walls play a dominating role. APPENDIX

I

Internal

Force

We consider two masses M1, M,, acted on by forces F1, F, which may both be purely external forces. We here regard the two masses as a system even though they may not exert forces on each other. The motion may be resolved into external and internal parts. The external motion moves with the center of mass and has acceleration a,,, given by a,,, =

F, + F, Ml

+

(Al .l)

M2

The internal motion may be characterized by the relative motion between the masses, with acceleration a2 a,, or the motion relative to the center of mass, with accelerations ai - a,,, . The forces Fi can also be resolved into parts F.

1

=

F.ext I

+

(Al .2)

F.i”t I

where FFxt, Fiint are the parts of Fi that contribute separately to the external and internal motions. The external part of the force is that which would not result in any internal motion and is thus that part of Fi which would produce an acceleration of particle i equal to a,,,. Thus Feext 1 = Mia,,,

and from Eq. 109

(Al .3)

and V$

( > = -8 ?!!ro”

Fi int

(112) leading

=

Fi

-

Fiext

(Al .2)

to

There results 4 74 = Vl - - a2(v,) 3

F2int=

($-$I

M*

(113) =

The particle is not dragged along with the flow field velocity vl, but slips relative to it by the slip speed (413) a”(~,). Th us, due both to steric hindrance and slip, a hydraulic flow of a uniformly concentrated solution through a narrow channel does not transport solvent and solute in the same proportions as they exist in the solution. The friction coefficient in Eq. 108 is given approximately (for a 5 0.4) by

(114 which reduces to the Stokes result for cy = 0. The (axial) diffusion coefficient is inversely proportional to this friction coefficient, according to Eqs . 59 and 70, and is thus reduced by the factor 1 - 2 a.

(Al .4)

-Flint

and a2

where

- a, =

F 2 int . 7 M*

M* is the “reduced M* =

&.int

CXi- arr, = M

(Al .5) i

mass” MlM2 M, + M,

(Al .6)

The internal forces so defined sum vectorially to zero, and may or may not be due to internal interaction. In this derivation, it has been a matter of indifference whether the masses were totally divorced from a fluid field (i.e., isolated masses) or in fact interpenetrating masses of the two components in a unit volume of solution or suspension.

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R14

H.

The derivation is meant to emphasize most transparently the centrality of the Newtonian mechanical character of thermodynamic-hydrodynamic fields. APPENDIX

and thus the specific current

II

Basic Mechanics

SOODAK

AND

A. IBERALL

j is

(A2.6)

= -DVc for Diffusion

as a Random

Walk

We review the simplest one-dimensional random walk, in w ,hich a parti cle moves wi th equal probabilities rightward or le ftward along a line for a fixed distance h, and then repeats this diffusion step. The time 7 for each step is equal. This motion can be regarded as that of a particle that moves with fixed speed uL) = h/7, that collides every h, and comes out of the collision by going forward or backward with equal probabilities. The net displacement x in a time t is a statistical quantity with mean value Neither absolute constancy of the diffusion the restriction to one dimension is intrinsic

(A2.14)

step h , nor , and if in

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

R15

OPINION

fact the diffusion find

steps- were

inhomogeneous -

g = V*(DVc)

we would

(A2.15)

should the diffusion step vary in different regions of the field (e.g., with locally patchy regions that are more or less facilitated). APPENDIX

III

Diffusion

and the Laplacian

Imagine a region in which diffusing particles are making their random walks. Consider a tiny volume element of this field. Particles in it move out, and particles from neighboring elements move in, both processes in stepwise fashion. If the particle conbentration were uniform, there would be no change in concentration, even though there was an exchange of particles. The concentration in the element of consideration would increase with time, if the concentration in its surrounding neighbors exceeded its own. The Laplacian VQ=-+-+-a2c ax2

a2c ay2

a2c

-“-c=Fv2c

= co + kl-2 = co + k(x” + y2 + z2)

(A3.2)

= co + hii2

(A3.4)

and the excess is kX2, the left side of Eq. A3.2. Differentiating Eq. A3.3 gives V2c = 6k

(~3 .5)

Thus, the right side of Eq. A3.2 is also kX2. Equation A3.5 may be used to derive the balance equation for a diffusion process. -

at

= DV2c

(A3.6)

-

theorem

W-7)

is

A2 AN = -V2N 6 Dividing

(A3.8)

this by the time 7 for one diffusion

the rate Of change Of N as aN A2 -- V2N at - 67

step gives

(A3.9)

that N = cx3, there results ac -=DV”c; at

APPENDIX

IV

Modeling

of Liquid

D=G

A2

(A3 .lO)

Diffusion

The following numerical consequences of the last section may be of interest. We attempt to compare the coefficient of diffusion of a molecule in solution with the diffusion of a “particle” in suspension. For a molecule in solution. We have X2 67

Di A molecular diffusive step at of a molecular diameter, e.g., step for the diffusion 7 = Q time Tjit which is of the order

(A3.3)

where co is the concentration at the element of interest, r is distance from this point, and k is a constant. The average value of c at distance h from the central point is here

dC

by our previous

(A3 .l)

In this equation, c is the concentration in the element of interest, and ”

AN = ”

Recalling

is the measure of the extent to which the average concentration of the surrounding elements exceeds the concentration c of the central element. More precisely

C

We divide the region into volume elements that are cubes h on a side, where h is the random walk diffusion step length. Then in one time step 7, all the contents of the control cube leave through all six faces. Concurrently, one-sixth of the contents of each of the six adjacent cubes enter the central element. Let N = cx3 denote the number of particles in any element where c is the concentration. In one time step, the contents of the central element changes from its original value N to the new value h, the average N value in the surrounding elements whose centers are a distance h from that of the element of interest. Thus

Tjit

ZZ

(A3 .lO)

equilibrium is of the order h = 2 x 10ms cm. The time is not the molecular jitter of A

(A4.1)

Uth

where

z,+his the thermal

velocity, 711=

NjitTjit

but (A4.3)

where Njit is the number of jittering fluctuations before a diffusive step is made. Iberall and Schindler (14) have shown that a fair number for Njit is of the order of 10. The root-mean-square fluctuational (thermal) velocity is not the propagation velocity in the liquid medium, but the translational jitter available by motion in a local liquid cell. 3R (A4.4) m This results from equating the fluctuational kinetic energy, muth 212, to 3RT/2, where R and m are given per mole. Thus Uth

=

J

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

R16

H. SOODAK

6 x lO-5

DE This may be compared (23) of

(A4.5)

?7,-p2

with

D=

the result

8 x 1o-5 m1’2

given

by Stein

(A4 3)

for small molecules, e.g., a coefficient of diffusion or self-diffusion in water of about 2 x lo+ cm2/s. For a particle in suspension. The Stokes-Einstein relation gives kT DE-------6qa



R k XZ___ N0

W-7)

where N, is Avogadro’s number. Assuming water density ()oO= 1 g/cm”) and viscosity (rl = 0.01 poise), we obtain 4 na”pJV,, = m 3

(A4.8)

3 x lo-” D=-----ml 13

(A4.9)

Stein (23) also gives D=

3 x 1o--5 ml/3

(A4.10)

These two relations cross over, per Stein’s data, at a molecular weight of about 250, corresponding to a molecular diameter of about 8 A., The result indicates that for larger molecules there is essentially no difference between “solution” and %uspension” as far as transport is concerned. The “colloidal” nature of larger molecules, e.g., protein, is further indication of that thesis. Large molecules are partly in solution. They may be hydrophilic (“wetted”) in patches, or they may be hydrophobic. The separation of chemical bonding energies and potentials from the physical interaction forces makes for scaled hydrodynamic transitions that are extremely interesting for life. In this section, we have pointed up an underlying difference between little and big processes at molecular scale. APPENDIX

V

Tensile Water Tensile water exists. Briggs (4) has put water under a negative pressure of 260 atm in a centrifuge apparatus. We expect water in membrane pores near the interface with osmotic solutions to be under negative pressure, p - Kt, where p = 1 atm is the solution (hydrostatic) pressure and ll is its osmotic pressure, R*Tc, for a dilute solution. Mauro (19) has observed these tensions. In these examples, the water is under tension and is actually elastically stretched just as water under pressure is elastically compressed. The stretching by Briggs was done by a centrifugal force field driving water away from the central region of the centrifuge tube, in both directions from the center. The

AND A. IBERALL

water in the semipermeable membranes is stretched by the outward diffusive drive acting on the membrane water at the solution interface. This is our mechanism for how an osmotic pressure difference is converted into a hydrostatic pressure difference, inside the membrane, which then drives osmotic flow hydraulically through the membrane. On the other hand, Hammel and Scholander (11) propose that the water in solution is under tension. If solution pressure is p and osmotic pressure is ll (-R*Tc,), then the water in that solution, they say, is under an effective pressure p’ = p - II, which can easily go negative. As a mechanism for generating this negative pressure, they propose the bombardment, from the inside, of the free surface of the solution by the solute particles in solution. The implication is that the tension of the water component of the osmotic solution pulls water through the membrane, and they state that without their suggested mechanism, there is no way of converting an osmotic difference into a hydrodynamic flow. It is clear that we disagree on a number of counts. 1) We have proposed (in the section, Diffusion, Sedimentation, Osmosis, Part 111) a different mechanism of converting from osmotic difference to hydrostatic mechanical pressure difference that drives hydrodynamic flow. We believe our analysis to be correct. 2) We do not believe that it makes sense to regard the water in solution to be under tension p’ = p - ll . First, there is no clear-cut method of partitioning the total pressure of a solution (p) into a sum of pressure contributions (in this case with one of them negative) from the individual components: p’ from water and Il from solute. Only in ideal gases is such a partitioning straightforward. 3) Their suggested partitioning of pressure cannot apply to a solution of liquids (water and alcohol) in relatively equal proportions. Which component is solvent, which solute? They cannot both be under tension if the partitioning of pressure is supposed to be valid. 4) In a solution of solvent and solute molecules that are similar in size and in structure, it is clear that the statistical distribution of positions and motions of the solvent particles is much the same as in pure solvent. A decrease in pressure, by ll, does not make sense here. 5) The solute particles, even in dilute solution, do not form an ideal gas. They are in close interaction with the solvent, even though they do not interact with each other. The solute vapor pressure may be far smaller than the osmotic pressure TII. 6) The proposed mechanism for generating the tension, that of surface bombardment by solute molecules, is far-fetched. APPENDIX

VI

Convective Diffusion Consider a convective laminar flow along the x direction through a circular tube of radius a, with a velocity v(r) that varies with distance r from the central axis. In the absence of diffusion, solute particles are carried along by the flow, and thus the average velocity of

Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

INVITED

Rl7

OPINION

solute that is uniformly distributed in the radial direction is equal to the mean convective flow . The random diffusive motion of the particles has two different consequences. Axial diffusion leads to normal axial diffusion effects. Transverse or radial random motion of a particle carries it from regions of higher than average axial v to regions of lower than average axial v and vice versa, effectively resulting in an axial convective random walk around the mean flow. Thus, the interaction between transverse random motion and axial flow leads to enhancement of the effective diffusion coefficient for axial diffusion. We estimate the convective diffusion coefficient D,. Let D be the diffusion coefficient of the particles in the fluid. Divide the cross section of the tube into equal areas consisting of a central circle and an outer annulus. The flow velocity in the inner region is higher than the average by vl, and in the outer region, lower by ul. Let Q measure the time required for a particle to d&se transversely between the inner and outer regions. We may regard Q as the time between random changes in velocity. We thus arrive at the following picture: in the reference frame of a tube section moving with average flow velocity , transverse diffusion results in an axial random walk with time step Q, and step length h = By use of Eq. A2.3, the convective diffusion v 1% coeficient is then l

D,,

= u1 ” d

2

wander

halfway

across the channel.

To find Q, we use Eq. AZ.2 in the form t = .y2/2D where y is a transverse axis . If we use the value-(a/2)2 for y2, the resulting t value is an estimate for the time Q to

a” 7 -I) - 80

(A6.2)

D,, = a2v1” 16D

(A6.3)

and then

the variThe quantity v12 that appears here represents ante of the flow around the mean value. For Poiseuille flow in a tube, v12 = ~/3 and thus our estimate agrees with the result calculated by G. I. Taylor (1) a2” D, = ~ 480 This result form

may be expressed

(A6.4)

in terms

of ratios,

in the

(A6.5) The ratio of convective diffusion to molecular diffusion is given in terms of the rato a/D, which is called the Peclet number, measuring the ratio of convective flow a to diffusive flow D. Taylor’s calculation involved the solution of the differential equation for convective diffusion, which is dC

(A6.1)

This gives

d”C

z + vVc = D i)x”

(A6.6)

and is the balance equation, as is Eq. A2.14, for the solute contents in a volume element moving with the fluid at convective velocity V.

REFERENCES 1. BATCHELOR, G. (editor). Scientific Papers, Sir G. I. TayZor. New York: Cambridge, 1950-1971, vols. l-4. 2. BIRD, R., W. STEWART, AND E. LIGHTFOOT. Transport Phenomena. New York: Wiley, 1960. 3. BRENNER, H., AND L. GAYDOS. The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius. J. CoZZoid Interface Sci. 58: 312-356, 1977. 4. BRIGGS, L. Limiting negative pressure of water. J. AppZ. Phys. 21: 721-722, 1950 5. CURRY, F. A hydrodynamic description of the osmotic reflection coefficient with application to the pore theory of transcapillary exchange. Microvascular Res. 8: 236-252, 1974. 6. DAINTY, J. Osmotic flow. Federation Proc. 19: 75-85, 1965. 7. EINSTEIN, A. Investigations on the Theory of the Brownian Movement. New York: Dover, 1956. 8. EVERITT, C., AND D. HAYDEN. Influence of diffusion layers during osmotic flow across bimolecular lipid membranes. J. TheoreticaL BioZ. 22: 9-19, 1969. 9. GOLDSTEIN, S. Modern Developments in Fluid Mechanics. London: Oxford, ~01s. 1 and 2, 1938. 10. GUREL, O., AND 0. R&SLER (editors). Bifurcation theory and applications in scientific disciplines. NY Acad. Sci. In press. 11. HAMMEL, H., AND P. SCHOLANDER. Osmosis and Tensile SoZvent. New York: Springer-Verlag, 1976. y 12. HAPPEL, J., AND H. BRENNER. Low Re;moZds Number Hydrody-

namics. Leiden, Nordhoff, 1973. 13. HOUSE, C. Water Transport in Cells and Tissues. London: Arnold, 1974. Transport. 14. IBERALL, A., AND A. SCHINDLER. Physics of Membrane Upper Darby, Pa.: Gen. Tech. Services, 1973. 15. LANDAU, L., AND E. LIFSHITZ. Fluid Mechanics. London: Pergamon, 1959. 16. LEVICH, V. Physicochemical Hydrodynamics. Englewood Cliffs, N. J.: Prentice-Hall, 1962. 17. LIGHTFOOT, E. Transport Phenomena and Living Systems. New York: Interscience, 1974. 18. LIGHTFOOT, E., J. BASSINGTHWAIGHTE, AND E. GRABOWSKI. Hydrodynamic models for diffusion in microporous membranes. Ann. Biomed. Eng. 4: 78-90, 1976. 19. MAURO, A. Osmotic flow in a rigid porous membrane. Science 149: 867-869, 1965. 20. MOELWYN-HUGHES, E. PhysicaZ Chemistry. New York: Macmillan, 1966. 21. RAY, P. On the theory of osmotic water movement. Plant Physiol. 35: 783-795, 1960 22. SCHEIDEGGER, A. The Physics of Flow Through Porous Media. London: Univ. Of Toronto Press, 1960. 23. STEIN, W. The Movement of MoZecuZes Across Cell Membranes. New York: Academic, 1967.

H. Soodak Physics Department, City College of New York, New York City 10031 A. Iberall General Technical Services, Inc., Upper Darby, Pennsylvania 19082 Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (196.011.235.237) on August 17, 2018. Copyright © 1978 American Physiological Society. All rights reserved.

Osmosis, diffusion, convection.

Osmosis, diffusion, convection SOODAK, H., AND A. IBERALL. Osmosis, diffusion, convection. Am. J. Physiol. 235(l): R3-R17, 1978 or Am. J. Physiol.:...
3MB Sizes 0 Downloads 0 Views