1002 



HANDBOOK OF I'HYSIOI.OUY 



CIRCULATION II 



P 



Ax 



n 

 D~Ac 



(7.6) 



Once the pore area per unit path length, A p Ax, has 

 been established from equation 7.6 for a given large- 

 pored membrane, the membrane may be used to 

 determine free diffusion coefficients of test molecules 

 (233, 264, 316). Membranes employed for this 

 purpose generally have pores which are at least 

 100-fold larger than the diffusing molecules. 



In the case of diffusion through membranes having 

 pores of molecular dimensions the kinetic motions of 

 the diffusing molecules are restricted by the pore 

 structure; in such membranes the effective pore area 

 per unit path length decreases as a function of mo- 

 lecular size, becoming zero when the test molecules are 

 the same size as the pores. Capillary permeability to 

 lipid-insoluble molecules of graded sizes can be 

 explained, in large part, by restricted diffusion 

 through aqueous channels of molecular dimension. 

 For this reason it will be necessary to discuss physical 

 aspects of restricted diffusion in some detail. 



Figure 7.1 shows apparent pore areas per unit 

 path length for molecules of graded sizes diffusing 

 through a cellulose membrane of the type commonly 

 used for ultrafiltration or dialysis (Visking sausage 

 casing). It is evident that the apparent pore area for 

 free diffusion decreases rapidly as a function of 

 molecular size. The true pore area in the membrane 

 is, of course, constant and it is useful to think of the 

 apparent decrease in terms of a restricted diffusion 

 coefficient, D', such that 



D * D A ,/A. 



(7.7) 



where A s is the apparent pore area for the solute and 

 A v is the true pore area. Substitution of D' for D in 

 equation 7.6 would yield the true membrane pore 

 area per unit path length for all molecular species. 

 The essential theoretical problem is now to relate 

 the observed restriction to diffusion, D' D, to di- 

 mensions of the membrane pores. 



The theory of restricted diffusion proposed by 

 Pappenheimer el al. (281) takes into account two 

 factors impeding the passage of molecules through 

 pores of molecular dimensions. The first factor is 

 concerned with steric hindrance at the entrance of 

 the pore. It is assumed that for entrance into a pore 

 a molecule must pass through the opening without 

 striking the edge as originally suggested by Ferry 

 (95). For the case of cylindrical pores the effective 

 target area, A s , for the solute is then 



A p (l-a/r)* 



A s dn/dt 

 Ax"~ DAC 



VISKING CELLULOSE 



Mean pore radius 



r = 16 S 



-V—if 



MOLECULAR RADIUS, A 

 * 5 6 



H3HO 



—If-, 



GLUCOSE 



SUCROSE 

 ANTIPYRINE 



RAFFINOSE 



fig. 7.1. Apparent pore areas per unit path length as a 

 function of molecular size. The smooth curve is constructed 

 from the theory of restricted diffusion, equation 7.9, assuming a 

 mean pore radius of 16 A. Mean pore radius determined on 

 the same membrane from combination of diffusion and filtra- 

 tion was 19 A (equation 7.13). Similar data for diffusion of 

 lipid insoluble molecules through the walls of muscle capillaries 

 are shown in figure 9.2. [Adapted from Renkin (298).] 



where A v is the true geometrical area of the opening 

 and a r is the ratio of molecular radius to pore 

 radius. 



The second factor takes account of friction between 

 a molecule moving within a pore and the stationary 

 walls of the pore. This factor, first studied by Laden- 

 burg (193), was employed by Friedman & Kraemer 

 (112) to describe the diffusion of sugars through 

 gelatin gels. The Ladenburg treatment of the problem 

 is strictly applicable only to cases where a r < .1 

 and it is preferable to use the more general formula- 

 tion of Faxen (94) 



1-2.10 (£) * 2 09(f) -0.95(f) 



(78) 



where / ,'/o is the frictional resistance to diffusion in 

 the pore relative to that in free solution. Taking into 

 account both steric hindrance (equation 7.7a) and 

 wall effects (equation 7.8), the theoretical restriction 

 to diffusion through cylindrical pores becomes 



£ - |- (I- ° 7 f[-z J0 (f). 2 .09(?) 



4fl 



-0.95 



(7 9) 



(7.7a) 



The last term of the series is negligible when a/r < 

 0.5. During net flow through the membrane the 



