Polymer confined between two surfaces

Transcription

1 Appendix 4.A 15 Polymer confined between two srfaces In this appendix we present in detail the calclations of the partition fnction of a polymer confined between srfaces with hard wall bondary conditions. We shall discss two examples: Gassian chains with infinite extensions and rigid rods with finite extensions. 4.A.1 Gassian chain First we consider Gassian chains. The Green s fnction of a free Gassian chain is governed by the partial differential eqation Doi and Edwards, 1986) ) N b 6 Gr, r ; N) = δ 3 r r )δn). 4.41) The non-adsorbing hard wall) bondary conditions are Gr, r ; N) = if r or r is at the bondary. 4.4) The general reslt is Gr, r ; N) = gx, x ; N)gy, y ; N)gz, z ; N); 4.43) gx, x ; N) = ) ) pπx pπx sin sin exp p π Nb ), L x L x L x 4.44) 1 p and similar reslts for gy, y ; N) and gz, z ; N). Here L x, L y, L z ) is the size of the box containing this polymer. In or system the x and y directions are infinite, hence gx, x ; N) and gy, y ; N) are Gassian = x, y)]: 6L x 3 3 g; N) = πnb e Nb. 4.45) In the z direction gz, z ; N) is confined between and L z with hard-wall bondary conditions. The general expansion for gz, z ; N) is gz, z ; N) = k z = k z ak z )e ikzz z) e k z Nb /6 ak z ) cosk z z ) cosk z z) + sink z z ) sink z z)] e k z Nb /6. To satisfy the Dirichlet bondary condition, we shold choose sink z z ) sink z z); if we want a reflective bondary condition we shold se cosk z z) cosk z z ).

2 16 For the non-adsorbing bondary condition we have gz, z ; N) = L z p sin pπz sin pπz exp L z L z Nb p π 6L z ). 4.46) In or problem, one end of the polymer is anchored at z very close to the srface, therefore the Green s fnction is given by to first order in z /L z. h z; N) = z L z The partition fnction is given by q z = Lz p pπ sin pπz L z dzh z; N) = 4z L z exp Nb p π ) p=1,3,5,... 6L z 4.46 ) e p π /6l, 4.47) and is approximated by q z = 4z e π /6l l 1, L z 6z l 1. πnb 4.48) Similarly for h L z ), the partition fnction of a ligand-receptor bridge we have 14 h L z ; N L + N R ) := h L z z ; N L + N R ) = 1) p+1 sin pπz sin pπz L z L z h L z ; N) can be approximated by z L 3 z p L z 1) p+1 pπ) e pπ) /6l p = z L 3 z e pπ) /6l cos pπpπ) e pπ) /6l. 4.49) p zπ L 3 e π /6l l 1, z 6 z 18 π Nb ) 3/ l e 3l / l ) Assembling the terms together we have e ɛl) ɛ = h L z ; N L + N R ) Nb Lz dzh z; N L ) L z dzh z; N R ) 14 Since h L z) vanishes, we define it to be h L z z ), as is shown later, for or interest this will not case ambigity.

3 1 8l p=k 1 exp and the asymptotic limits are 17 p 1)p+1 pπ) e pπ) /6l )] NL p π N L+N R 6l p=k 1 exp NR p π N L+N R 6l )], 4.51) π l 1, 8l 3 6πl e 3l / NL N R l 1. N L + N R 4.5) 4.A. Rigid rod and variants Here we stdy models with finite extensibility. First we consider a spherical chain model, in which the distribtion of the free end is niform within the hemisphere of radis R and zero otside. R can be identified as the contor length of the polymer, or as an approximation to the Gassian chain model, identified with the mean sqare end-to-end distance of the Gassian chain. For this model the Green fnction of the polymer with one end fixed at the origin is given by and the partition fnction is Gr, θ, φ; N) = 3r sin θ πr 3, 4.53) 1 L z R, q = 1 3L z 3 R ) L z 3 ] R L z < R. 4.54) Slightly different is the model of a freely rotating rod, corresponding to a short polymer whose contor length is smaller than the persistence length. The Green fnction is Gr, r ; R) = 1 ) r πr δ r ) R R is the rod length, which is eqal to the contor length of the polymer. For this model, the partition fnction is L z R L z < R, q = 1 L z R. 4.56) The Green s fnction for the tether chain with two connected rods is conveniently represented by the length of the arc from the intersection circle of the two hemispheres spanned by the rod ends that is confined between the srfaces. The expression can be worked ot, bt is qite lengthy. Two examples are shown in Figre 4.14 on page 18. For ligand and receptor tethers we have R L,R = N L,R b, and the combined tether length is

4 18 Gr, L z ; N L = N R ) r L z Gr, L z ; N L =1.5N R ) r L z Figre 4.14: Green s fnction of joined rods N L + N R )b. Let s define the scaled densities φ L,R = ρ L,R N b.

5 19 From Eqs. 4.4), the binding constant is given by K = ρ K LR A r = L,r R G LR r L, r R ) ρ L ρ R r r L Gr, r L ; N L ) r r R Gr, r R ; N R ) = K G LR r; N L, N R )d r. 4.57) q L q R G LR is the partition fnction of a ligand-receptor bridge. In the qenched case we have cf. Appendix 4.B for the definition of wr)) wr) = K G LR r; N L, N R ) q L q R. 4.58) We immediately recognize from the finite extensibility that in these models binding is present only if L z N L + N R )b. In addition, if molecles are immobile, binding is less probable compared with Gassian chains. For the rigid-rod model, consider a ligand and a receptor with lateral separation r, a necessary bt not sfficient condition for binding to be possible is N L N R b r + L z N L + N R )b. When srfaces come too close, binding becomes less probable. Appendix 4.B Low-density expansion for the qenched problem For an immobile ligand anchored at r L and a receptor at r R, the ratio of the Boltzmann factor of the bond state to that of the nbond state is given from Eqs. 4.4) and 4.5) to be wr L, r R ) = q LR q L q R = K q t LR q t L qt R = K hl z )gr L r R ; N L + N R ). 4.59) q L L z )q R L z ) Note that since molecles are immobile, the integration over r L or r R is removed; bt the translational invariance implies that wr 1, r ) = wr 1 r ). Using w) we can easily write down the first few terms of F m L, m R ) cf. Eq. 4.35)): βf 1, 1) = ln1 + wx 1 y )] + ln q L + ln q R, βf 1, ) = ln q L + ln q R + ln1 + wx 1 y 1 ) + wx 1 y )], βf, ) = ln q L + ln q R + ln 1 + wx 1 y 1 ) + wx 1 y ) + wx y 1 ) + wx y ) + wx 1 y 1 )wx y ) + wx 1 y ) + wx y 1 )]. Here x i and y j are positions of ligands and receptors, respectively.

6 11 Assming that receptors and ligands are randomly distribted on the srfaces and for any receptor or ligand, its position distribtion is independent of the others, we have the qenched average F x1, x,, x m, y 1, y,, y n ) = 1 A i+j F {x i }, {y j }), {x i},{y j} Evalating these averages is straighforward, which gives β F 1, 1) = ln q L + ln q R + 1 A ln1 + w)], 4.6a) β F 1, ) = ln q L + ln q R + 1 A 1, ln1 + w 1 ) + w )], 4.6b) β F, 1) = ln q L + ln q L + 1 A ln1 + w 1 ) + w )], 4.6c) 1, β F 1, m) = ln q L + m ln q R + 1 A m ln1 + w m )], 4.6d) 1, m m β F, ) = ln q L + ln q R + 1 A 3 1,,v 1 ln1 + w 1 ) + w ) + w 1 + v) + w + v) + w )w 1 + v) + w 1 )w + v)]. 4.6e) Sbstitting these back into Eq. 4.35) we have β F 1,1) = β F 1,) = 1 e AρL+AρR m L 1,m R 1 = Aρ L ρ R = Aρ L ρ R F 1,1) ; 1 ln1 + w)] e AρL+ρR) m L 1,m R { 1 A = Aρ Lρ R = Aρ Lρ R β F 1,m) = Aρ Lρ m R m! β F,) = Aρ L ρ R 4 Aρ L ) ml Aρ R ) mr m L!m R! m L m R A Aρ L ) ml Aρ R ) mr m 1 m L!m R! Lm R ln1 + w 1 ) + w )] 1, A } ln1 + w)] ln1 + w)] 1, {ln1 + w 1 ) + w )] ln1 + w 1 )] ln1 + w )]} F 1,) ; 1,, m ln ln 1,,v ] + w 1 )w + v) + w )w 1 + v) 4A L, R 1 + ] w i ) A m k C k mf 1,k) ; i 1 k<m 1 + w 1 ) + w ) + w 1 + v) + w + v) { ln 1 + w 1 ) + w ) ] ln1 + w 1 )] ln1 + w )] } 4.61a) 4.61b) 4.61c)

7 4A = Aρ L ρ R F,). 4 ) ln1 + w)] d) For Gassian chains, the qantity wr) can be rewritten as w) = K h l) q L l)q R l) g; N L + N R ) = 3 ] β ɛl) π exp 3 Nb, 4.6) where the effective binding energy ɛ is defined above as in Eq. 4.9) and the second term acconts for lateral stretching. From Eq. 4.6) we see that a) the effective binding energy has a similar dependence on the srface separation as in the annealed case as reflected in ɛ; b) each integral over gives a factor of Nb, hence F n,m) Nb β) n+m 1, and we see that in Eq. 4.61) the real expansion parameter is φ = ρnb. Similarly one can verify that in the case of rigid rods, the expansion is in terms of φ = ρn b.) For large binding energy β ɛ, each integral over the scaled also contribtes a factor of β ɛ, therefore the asymptotic density expansion is valid only if β ɛφ 1. The density of bond pairs is obtained by taking the derivative of F against ln w. At leading order the binding fraction can be expressed in a close form: f 1,1) = d w) ln1 + w) = d ln w r 1 + w), 4.63) which for Gassian chain with w) e 3, becomes f 1,1) = π w)e w)e 3 d = π 3 ln1 + w)) = π ln1 + w)). 4.64) 3

8 Appendix 4.C 11 Exact reslts for the single-chain qenched problem Consider the problem of one ligand with randomly anchored receptors, the qenched average qantities inclde the free energy F kt = ln 1 + r σr)wr) ] {σ}, and the binding fraction f = r σr)wr) 1 + r σr)wr), {σ} both of which involve Σ = r σr)wr). σr) labels the occpation of each lattice site, namely σr) = 1 if the lattice site is occpied and otherwise. Σ is a random variable with mean EΣ] = ρe βɛ gr)d r. Σ. The only problem is to find the distribtion of Σ. Let s calclate the characteristic fnction of 4.C.1 Ideal soltion model Here we assme that each lattice site has a probability φ to be occpied. The partition fnction of non-interacting system is Q = 1 + e µ ) A, with φ = eµ 1 + e µ. Frther we assme that lattice sites are decopled, i.e., they are independent. The characteristic fnction of one site is ϕ σ t) = 1 φ + φe it. 4.65) Then for Σ = r σr)wr),

9 113 we have ϕ Σ t) = r 1 φ + φe itwr)) = exp log 1 φ + φe itwr))]. 4.66) r In the continm model, the smmation can be replaced by an integral, 1 ϕ Σ t) = exp a log 1 φ + φe itwx,y)) ] dxdy. Assming that φ 1, we can approximate the exponent by This is in fact the φ ) a dx dy e itgx,y) 1. 4.C. Ideal lattice gas model In the ideal lattice gas model, the lattice distribtion variable satisfies the Poisson distribtion P σ = n) = e φ φ n, n! ϕ σ t) = exp e it φ φ ). Then for Σ = r wr)σr), we have ϕ Σ t) = exp φ r e itwr) 1) ]. Alternatively one can define Σ = r wr) i δr r i ), where r i are the positions of the receptors. Ignoring the maximm occpancy constraint, as the receptors position distribtions are independent, we have ϕ Σ = r r 1 a A + a na exp A A eitgr) ) n = exp e itwr) 1) ] exp ) ln 1 ] a A + a A eitwr) 4.67) ) ] n r ρ e itwr) 1 dr. 4.68) The becomes = in the thermodynamic limit A and in the continm limit a.

10 114 We know that wr) is Gassian, and can be written as 15 wx, y) = exp βɛ x + y ) Nb = gx, y)e βɛ. We can drop the expβɛ) term as it is a constant, and ϕ Σ t) = exp ρ dx = exp {ρnb = exp ρnb m 1 = exp πρnb m 1 d 1 m! ) ] dy e itwx,y) 1 it) m e mβɛ m m! ) ] } dv exp ite βɛ +v ) 1 d dv = exp ie ] βɛ te +v m ) { πρnb e βɛ t } e ix 1 dx x. 4.69) The probability distribtion of e βɛ Σ is given by f e Σx) = 1 βɛ π We note that t = 1 π e itx ϕ Σ t)dt = 1 π t Φ exp 1 cos d ] t exp itx + Φ cos tx + Φ t e i ] 1 d dt ) 4.7) dt. 4.71) sin d sin x x dx is an odd fnction of t while t cos 1 d is an even fnction of t. If Φ 1, then the integral has most contribtion from t 1 1 cos = 4 4! + 6 6! + t t 1 cos d t 4 t sin t3 d t In general we have 3 ϕ A t) = exp 4φ X i n t n W n 5 n! n where W n = X r wr) n and the probability distribtion of A is 3 f A x) = 1 Z e itx ϕ A t)dt = 1 Z dt exp 4 itx + φ X i n t n W n 5 π π n! n 3 1 = 1 Z k 1 X ) exp 4φ k t k k 1 X W n 5 ) tx k t k+1 + φ W n A. π n! n! n=k n=k+1

11 f e βɛ Σx) 1 π exp 115 Φt 4 ] cos )] tx + Φ t t3 dt. 4.7) 18 The extra t 3 term is kept as the leading-order correction to Gassian distribtion of Λ. If Φ 1, then the exponent can be expanded as a power series we have { t ϕ e Σt) = exp Φ βɛ 1 + Φ t e i } 1 d Let s choose δ = te R /Nb. Then this eqation becomes t ϕ e Σt) 1 ΦR βɛ Nb + Φ e i 1 d 1 Φ ln t t δ + Φ δ =1 ρπr + ρ e R /Nb t R exp e i d e i d. 4.73) ite r) rdr. 4.74) This corresponds to a niform distribtion of receptors within a circle with radis R. We see that if Φ 1 then each ligand essentially sees only one receptor and the pertrbative expansion in Appendix 4.B is accrate in this regime. Appendix 4.D Mlti-chain qenched problem in the highdensity limit As seen in Appendix 4.C, in the high-density limit, the single-chain qenched problem approaches the annealed case. Will the same conclsion hold for the mlti-chain problem? Assme the area densities of receptors and ligands to be ρ R and ρ L. And we simplify w) to be a step fnction, i.e., e βɛ <, w) =. 4.75) Now within a area S = π ) all ligands and receptors can bind with each other. Assme that ρ i ) 1, the average nmber of molecles within S is roghly Gassian and peaked at ρ i ). Also assme that ɛ 1, therefore as many molecles are bond as possible. The only difference between the qenched case and the annealed case is in their entropy. The partition fnction for each qenched sample satisfies Q{r L }, {r R }) > q ) A/ ), 4.76)

12 116 q is the partition sm within S, whose entropic term is approximately q ρ L ) ]!ρ R ) ]! ρl ) ]! ρ R ) ]! ρ LR ) ]! = φ L!φ R! φ L!φ R!φ LR!. 4.77) In the annealed case the change in the entropic part of the free energy is f s kt =φ L ln φ L + φ R ln φ R + φ LR ln φ LR + φ LR φ L ln φ L φ R ln φ R. 4.78) Therefore one concldes that the qenched free energy within S < f > f. On the other hand, we know that < f > f by definition, therefore this sggests that < f >= f in the thermodynamic limit. This reslt follows from the fact that within S the flctation in the qenched distribtion is negligible, hence the qenched system is essentially annealed within S; therefore their free energies are eqal, as the free energy < f > is self-averaging.