Transmission Line Matrix (TLM network analogues of reversible trapping processes Part B: scaling an consistency Donar e Cogan * ANC Eucation, 308-310.A. De Mel Mawatha, Colombo 3, Sri Lanka * onarecogan@gmail.com
Abstract In this paper the influence of asorption an trapping processes as escribe by TLM are investigate as a function of scaling of the spatial iscretisation. It is observe that there is a continuum between the macro-scale an the micro-scale that can be moelle by TLM. The resolution of some initial ifficulties in interpreting the effect of scaling of the 'capacity ratio' has provie useful insights into the nature of that parameter.
1. Introuction The first paper in this series [1] introuce the process of constructing TLM analogues for iffusion with asorption/trapping. The electromagnetic network can be solve exactly, so the only approximation in this approach is how well the construct escribes the physical process being moelle. In general we get reasonable agreement between theory an experiment at the macro-level. This paper is concerne with consistency; whether the unerlying PDE which is escribe by a particular scheme remains unchange as we scale own a moel from macro level to micro level. In many moelling scenarios these two levels of abstraction are consiere as istinct an the intermeiate cases ten to be avoie, which is unfortunate. It has been shown that simple iffusion processes which are escribe by Fick's secon law reuce to a Telegraphers' equation as Δx an Δt ten towar zero when the ratio Δx/Δt is hel constant [,3]. So long as the correct space/time scaling is use it appears that there is inee a continuum between the macro- an micro-levels of moelling. Here we will exten this work an emonstrate that the network moels for iffusion with asorption an trapping are inee consistent with the physical processes as space an time iscretisations are scale with constant velocity. It will be seen that although the parameters which escribe scattering in the presence of asorption/trapping appear complicate, they possess an unerlying simplicity which can be relate to more funamental moels. The paper starts by consiering the relationship between the effective iffusion constant an the TLM scattering parameters. This is followe by an analysis of the scattering processes as moelle by TLM. We are then in a position to consier what happens when such network moels are subjecte to space/time scaling. It will be shown that although the starting point may appear more complicate the results are not out of line with what has been previously observe [1,,3,].. The effective iffusion constant an the TLM scattering parameters In part A of this paper we ha an effective iffusion constant, which for the Langmuir isotherm is D eff D / (1 + K, where K was the constant or proportionality relating the concentration of immobilise iffusant to the concentration of free iffusant. This can be relate to the TLM moel for an iniviual pulse arriving at a noe which we can now consier as a scattering/asorbing site. We use the same efinitions as before [1] for the reflection, transmission an retention coefficients, which for ease of reference are reprouce in Appenix I. We can express the conservation of mass flux in a lossy network which inclues stubs as where ρ ( ρ LL I τ ( τ LL I an σ ( τ LS I ρ + τ + σ 1 (1 Thus σ is the fraction that is absorbe an ρ + τ represents fraction of free iffusant (1 σ For the Langmuir isotherm the factor 1 the fraction of free-iffusant (1 σ (1+ K
Thus the effective iffusion constant which has been erive strictly from a TLM formulation D eff D(1 σ is consistent with Eqn ( in [1], D eff D / (1+ K, which was erive from conventional calculus consierations. 3. An insight into the physical processes as moelle by TLM We can rearrange the expression for the one-imensional line-line reflection coefficient in an informative way Since ρ LL ( S + Z S + Z ( + Z( + Z + S We rewrite this as ρ LL ( S + Z S ( + Z( + Z + S + ( + Z( Z ( + Z( + Z + S or ρ LL ( S + Z S ( + Z( + Z + S + ( + Z ( + Z( + Z + S Z ( + Z + S The first two terms can be amalgamate an reference to the efinition of τ LS gives ρ LL ( S + ( + Z ( + Z( + Z + S τ LS ( which reuces to the one-imensional reflection coefficient in the absence of iffusion with an aitional term ρ LL ( + Z τ LS ( + Z σ (3a Using an ientical line of argument we can show that one imensional transmission coefficient can be represente as τ LL Z ( + Z τ LS Z ( + Z σ (3b These results o not require that ρ SS 0 an hence are applicable to both asorption an trapping processes. We have effectively prove that the one-imensional scattering coefficients in the presence of asorption/trapping is exactly what it woul be in the absence of asorption/trapping, but with half of the absorption component apportione to reflection an half to transmission, inepenent of the relative magnitues of these two coefficients.
. The scaling of the lossy TLM scattering parameters in the presence of asorption e Cogan [] has shown how the reflection coefficient (in the absence of asorption can be scale an how this provies a continuum of results between macro an micro-scale moels. This means that if we start with the reflection coefficient in the absence of absorption ρ LL ( + Z 1 1+ D Δt (a Δx an if we choose a unit propagation velocity which we assume remains constant uner scaling. Then, as Δx Δx/f an Δt Δt/f, Eqn (a becomes ρ LL 1 1+ Df (b The reflection coefficient scales as shown, an as f it tens towars zero:- the problem moves from a iffusion which can be expresse by a parabolic equation, through the telegraphers' equation an ultimately to a purely wave equation. The succeeing sections of this paper investigate how the various scattering terms which are involve in the process of iffusion plus asorption/trapping behave as we move from the macro- to the micro-scale. The basic noe of the transmission line network analogue is shown in figure 1 where an Z represent the components which contribute to iffusion while s an Z s both contribute to trapping an storage. In our treatment we will move from these values to istribute values (base on resistance or capacitance per unit length. These are summarise below Figure 1 A one-imensional lossy noe (length Δx with an open-circuit stub of length Δx/ an impeance Z S, which is in series with a resistor, S. Z Δt C Δx Z s Δt Δx s Δx Δx (because there are two resistors within a noe (because the stub-length is Δx/ (because the stub-length is Δx/
.1 The transmission line to transmission line reflection coefficient ρ LL ( S + Z S + Z ( + Z( + Z + S (5a When the above substitutions are mae in the expression for the line-to-line reflection coefficient an when the scalings Δx Δx/f an Δt Δt/f are inclue, then the ivien (top part of eqn (5a becomes while the ivisor becomes 1+ + 8 f D f (5b 1 + + D + + D f + D D + C f (5c ( ρ LL I 1+ + 1 + + D + 8 f D f + D f + D D + C f (5 This is consistent in that if either is infinite or if is zero (Z s is infinite then eqn (5 reverts to eqn (b. However, because of the nature of the ivien the reflection coefficient can assume negative values an we nee to investigate this an what it means. As a first step we can consier the case of infinite scaling when Δx an Δt ten towars zero (f. In the extreme ( ρ LL I D D + Z Z + Z s (5e The relationship on the right above is obtaine by multiplying the mile expression above an below by Δx/Δt an using D 1/( C, Z Δt/C an Z s Δt/( Δx/. If stub capacitance is zero (stub impeance is infinite then the reflection coefficient is zero, which is consistent with the case in eqn (b when f. Using D 1/( C, Z Δt/C an Z s Δt/( Δx/ simple rearrangement an multiplication above an below by Δx/Δt yiels the result on the right of (5e, namely that in the extreme as f, an s play no part. The negative sign in (5e is then the result of a signal travelling on a lossless line Z encountering a pair of lossless lines, one of which has impeance Z an the other Z s.
. The scaling of the line-to-line transmission coefficient ( τ LL I Z( S + Z S ( + Z( + Z + S (6a The ivien is D f +16D 1 f (6b while the ivisor is ientical with eqn (5c Thus ( τ LL I 1+ + D + D f + 16D 1 f + D f + D D + C f (6c This is consistent in that if either is infinite or if is zero (Z s is infinite then the line-to-line transmission coefficient becomes unity. When the scaling becomes very large most terms in (6c isappear an we are left with a situation where s plays no part. ( τ LL I (6 D + If stub capacitance is zero (stub impeance is infinite then the coefficient is unity. Simple rearrangement an multiplication above an below by Δx/Δt yiels (6e, which shows that also plays no part. We simply have lossless line-to-line transmission at the iscontinuity where the stub is joine to the main network. ( τ LL I Z s Z + Z s (6e.3 The scaling of the line-to-stub transmission coefficient ( τ LS I Z ( + Z + S The ivisor is ientical to (5c, while the ivien is Df+8D f (7a (7b so ( τ LS I 1 + + D + Df + 8D f + D f + D D + C f (7c It is clear that if either is infinite or if is zero (Z s is infinite then the line-to-stub transmission coefficient becomes zero. However if is infinite an/or if is non-zero then as
f we have a situation which is inepenent of the stub resistance, but epens only on the ratio of the line an stub istribute capacitances. D ( τ LS I D + This can be translate into network impeances as above an we have Z ( τ LS I Z + Z z (7 (7e. The scaling of the stub-to-stub reflection coefficient ( ρ SS I + Z + Z S S + Z + S (8a The ivien is 1+ + D - + D f + D D - C f (8b so ( ρ SS I 1 + + D - 1+ + D + + D + D f + D D - C f f + D D + C f (8c This is perhaps the most variable of the scattering coefficients in terms of the way in which the various parameters can influence its polarity. However, as we scale to large values of f then we once again get a situation where s plays no part D - ( ρ SS I (8 D + In fact, plays no part at the extreme where f an this can be seen by transforming (8 back into impeances Z ( ρ SS I - Z s (8e Z + Z s A signal travelling along the stub an approaching the iscontinuity sees two impeances (each of magnitue, Z in parallel.
.5 The scaling of the stub-to-line transmission coefficient ( τ SL I Z S + Z + S (9a 1 The ivien is 8 f + 16D 1 f (9b 1 8 f + 16D 1 f so ( τ SL I 1+ + D + + D f + D D + C f Which is positive sign for all network parameters. As f the above reuces to (9c 1 ( τ SL I (9 D + which is ientical to (6, the expression for the line-to-line transmission coefficient. Similarly, the transformation to impeances is ientical to (6e. Z ( τ SL I s (9e Z + Z s.6 The scaling of eqn (1 Having evelope scaleable expressions for sum is always unity, inepenent of scaling. ( ρ LL I,( τ LL I an ( τ LS I it is easy to show that their.7 Scaling of the capacity ratio In orer to scale this quantity correctly we must recognise that is represents the ratio between the concentration of trappe species to the concentration of free species. Sections.1 -.6 have been concerne with the scaling of flux-relate parameters which are not applicable here. Instea we must consier the voltage scattering parameters which are reprouce in Appenix I. We consier the single event where species enters the noe. Some is transmitte, some reflecte an some trappe. The ratio of trappe to free is then K ( τ LS V ( ρ LL V + τ LL ( V (10a
If we use the same approach to scaling as use in the previous sections we obtain for this ratio K( f 8 f + + 8 f Df (10b In the extreme, as f this reuces to K Z s Z s Z (10c Which is exactly what we woul get if we were to take the expressions for the parameters in (10a an set both an s 0. As elsewhere, the lossy components are reuce as the problems is scale own an tens towars zero as f.8 Discussion on scaling It is known that in the extreme of infinite scaling even a simple iffusion formulation reuces to a lossless wave-propagation treatment. Of course we are never at quite this extreme, but rather at the level where the physical processes can be moelle by the telegraph equation. It has been emonstrate here that all of the scattering parameters which are involve in iffusion with asorption/trapping/recombination o inee scale an in the extreme, they too behave like propagation on a lossless network. We have emonstrate that matter is conserve at all levels of scaling an although it has come as a surprise that the capacity ratio oes not remain constant with scaling, we have been able to make sense of what has been obtaine. In spite of what has been sai above we live in a finite worl. We o not expect real physical processes to operate at the level where Δx an Δt are actually zero. However we woul like to be able to treat them where the iscretisations are at an atomic level an the question of an appropriate moification to the telegraph equation must now be consiere. 5. TLM moels for iffusion with asorption an the telegraphers' equation It is easiest to evelop an appropriate set of equation by using a parallel arrangement of two stubs of length Δx, instea of the one of length Δx/ which we have been using up to now. This equivalence allows us to use a metho similar to that given by Christopoulos [5]. A pseuo twoimensional representation is shown in figure where the arms in the y-irection represent the one-imensional line an the arms in the x-irection represent the lossy stub-line which is terminate at both ens by infinite (open-circuit impeances. We will assume that all transmission lines have the same istribute inuctance, L an that ifferences in impeance are expresse in the ifferent capacitances, C for Z an C S for Z S. In all treatments up to now we have use s which is given by s Δx However, since the stub length here is Δx we use s ' Δx
Figure : A simplifie network for one-imensional iffusion with asorption, where all lengths are reuce to the same iscretisations, Δx. In the x-irection we have V x L i x t i x an i x x C ' i x t (11a where C ' C + C S Δx In the y-irection we have V y L i y t i y an i y y C ' i y t (11b These can be manipulate so that the currents (fluxes are eliminate to give V x L ' C V t + ' C V t an V y L ' C V t + ' C V t (1 The sum is V x + V y L ( C + V t + ( + ( C + V t (13
Now since we are measuring the voltage (concentration at the centre of the noe in figure an since Δx Δy we have the telegraph equation for one-imensional iffusion with asorption/trapping V x L ( C + V t + ( + ( C + V t (1 This equation is entirely consistent for the stanar lossy telegraph equation if s 0 an C s 0. V x L C V t + C V t (15 We see that the inclusion of a provision for asorption has two effects. In the first case the iffusion constant is reuce, which is generally accepte to be the case, but at the microscopic level we also see that the wave-like character is slowe own ue to loaing by the capacitance in the stub. 6. Conclusions It has been emonstrate here that all of the TLM scattering parameters which are involve in iffusion with asorption/trapping/recombination o inee scale in a way which is at all times consistent with the unerlying physics. It has been particularly interesting to see how the associate TLM networks can provie fresh insights into the physical processes which they are being use to moel. This inclues the 'effective iffusion constant' which is applicable in chromatography/electrophoresis. This in turn utilises a parameter calle the capacity ratio an the way in which this scales was not obvious at first sight. This eficiency probably has more to o with the inconsistencies of efinition of the parameter itself, which effectively fuses ieas of surface concentration (trappe species an volume concentration (free species, which only become resolve as scaling tens towars infinity. While this paper has commente on the fact that much research elsewhere iscusse either macro scale or micro scale processes while ignoring the intermeiate cases, in the en it has been guilty of the same fault. However, expressions for the scattering parameters an unerlying equations as a function of scaling have been evelope, but their investigation must be part of another paper.
eferences 1. D. e Cogan Transmission Line Matrix (TLM network analogues of reversible trapping processes (Part A: introuction an methoology D. e Cogan, The relationship between parabolic an hyperbolic transmission line matrix moels for heat-flow Microelectronics Journal 30 (1999 1093-1097 3. D. e Cogan, Some observations on spurious effects in Numerical Moels International Journal of Mathematical Algorithms 1( (1999 153-163. D. e Cogan, X. Gui an M. ak, Some Observations on the TLM Numerical Solution of the Laplace Equation, Journal of Mathematical Moelling an Algorithms, vol. 8, no., pp. 363-385, Dec. 009 5. C. Christopoulos, The Transmission Line Moelling Metho IEEE/OUP 1995, p. 171
Appenix I The various one-imensional TLM current/flux scattering coefficients which were erive as Eqn (5 in [1] are reprouce here for ease of reference ( ρ LL I ( τ LL I ( τ LS I ( S + Z S + Z ( + Z( + Z + S Z( S + Z S ( + Z( + Z + S Z ( + Z + S line-to-line reflection line-to-line transmission line-to-stub transmission (asorption ( ρ SS I + Z + S Z S + Z + S ( τ SL I Z S + Z + S stub-to-stub reflection (retention stub-to-line transmission (release The voltage/concentration scattering parameters are ( ρ LL V ( ρ LL I line-to-line reflection ( τ LL V ( τ LL I line-to-line transmission ( τ LS V Z S ( + Z ( + Z( + Z + S line-to-stub transmission (asorption ( ρ SS V ( ρ SS I stub-to-stub reflection (retention ( τ SL V Z( + Z ( ( + Z + Z + S stub-to-line transmission (release note that some of these parameters have been ivie above an below by ( + Z in orer to clarify some of the erivations in the text.