Eur. Phys. J. E (2011) 34: 46
DOI 10.1140/epje/i2011-11046-3

THE EUROPEAN
PHYSICAL JOURNAL E

Regular Article

Dynamics of a polymer chain conﬁned in a membrane
S. Ramachandran1 , S. Komura1,a , K. Seki2 , and G. Gompper3
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397,
Japan
2
National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8565, Japan
3
Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
Received 28 June 2010 and Received in ﬁnal form 18 December 2010c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2011
Published online: 11 May 2011 – 
Abstract. We present a Brownian dynamics theory with full hydrodynamics (Stokesian dynamics) for a
Gaussian polymer chain embedded in a liquid membrane which is surrounded by bulk solvent and walls. The
mobility tensors are derived in Fourier space for the two geometries, namely, a free membrane embedded ina bulk ﬂuid, and a membrane sandwiched by the two walls. Within the preaveraging approximation, a new
expression for the diﬀusion coeﬃcient of the polymer is obtained for the free-membrane geometry. We also
carry out a Rouse normal mode analysis to obtain the relaxation time and the dynamical structure factor.
For large polymer size, both quantities show Zimm-like behavior in the free-membrane case, whereas they
are Rouse-like for the sandwiched membrane geometry. We use the scaling argument to discuss the eﬀectof excluded-volume interactions on the polymer relaxation time.

1 Introduction
Integral membrane proteins play a vital role in a variety ofcell functions such as solute transport, signal transduction
and regulation of membrane composition [1]. Owing to ﬁnite temperatures, such proteins along with other membrane components are constantly undergoing Brownian
motions. The resulting diﬀusive motion plays an important role in determining their transport properties. Hencethe studies of diﬀusion constitute an important basis forunderstanding the physical properties of membrane proteins, and have been an active area of focus on model
systems [2–6] as well as on living cells [7–10].
Although proteins consist of polymeric units of aminoacids, the standard approach is to consider them as rigid
disks moving in a two-dimensional (2D) liquid membrane
under low-Reynolds-number conditions. The diﬀusion coeﬃcient of a rigid disk translating in a membrane whichis embedded in a three-dimensional (3D) bulk ﬂuid was
calculated by Saﬀman and Delbrück (SD) [11,12]. The
obtained logarithmic size dependence is valid in the limitof small disk sizes. The SD theory was formally extended
by Hughes et al. to all disk size ranges [13]. In the case oflarge disk sizes, they showed that the diﬀusion coeﬃcientis inversely proportional to its size, which is analogous
to the Stokes-Einstein relation in 3D. However, it should
be noted that there is no single expression of the diﬀusion coeﬃcient which covers the whole disk size ranges. In
a separate theoretical study, Evans and Sackmann (ES)ae-mail: komura@tmu.ac.jp

employed a phenomenological approach to calculate thediﬀusion coeﬃcient of a rigid disk moving in a membrane
attached to a substrate [14]. The presence of a substrate inclose proximity to the membrane was taken into accountthrough a momentum decay term in the hydrodynamic
equations. An extension of this work taking into accountthe eﬀect of the advective terms has also been done [15].
In addition to these, diﬀusion of rod shaped objects onmembranes [16,17] or on Langmuir monolayers [18] havealso been theoretically analyzed.
Alternatively, the membrane protein can be regarded
as a polymer chain rather than a rigid disk. In this case,the internal degrees of freedom of the polymer should betaken into account, which is the main subject of this paper.
Apart from the protein analogy, hydrophobically modiﬁed
polymers which adhere to the membrane could also be described using our description [19]. There are two theoretical works preceding the current work. One of them is a
study by Muthukumar on the dynamics of a hydrophobic polymer conﬁned in a 2D liquid membrane [20]. In his
theory, the membrane itself was treated as an isolated 2D
system having an anisotropic viscosity. It was shown thatthe mean squared displacement of a monomer obeys a diffusive law. He also pointed out that the mode dependenceof the relaxation time arises from the excluded-volume effect. The second and more direct precursor to the present
work is the analytical calculation of the diﬀusion coeﬃcient of a polymer chain [21] using a 2D hydrodynamic
model with momentum decay [22–25]. Since their hydrodynamic model is essentially equivalent to that used by

Page 2 of 13

The European Physical Journal E

z

wall
η+
s

Fig. 1. (Colour on-line) A transmembrane protein approximated as a polymer chain (red chain) embedded in a liquid membrane (yellow particles). The membrane itself is surrounded by solvent (blue particles). Only a few representativesolvent particles are shown.

ES, the asymptotic size dependencies of the diﬀusion coefﬁcient are the same between disks and polymers; namely,
logarithmic in the small-size limit, and algebraic in thelarge-size limit.
Further motivation is provided by the experiments on
DNA molecules embedded on a cationic supported membrane [26–28]. The negative charge of the DNA molecules
leads to strong adhesion with the membrane so that only
the lateral motions are allowed. The measured diﬀusion
coeﬃcient showed a Rouse-like behavior. More recently,
a similar experiment with DNA on a free-standing membrane has been conducted [29]. Another related situation
can be found in a dilute polymer solution conﬁned between
narrow slits. Based on the scaling argument, such a polymer was predicted to show a Rouse-like behavior [30–33].
In an attempt to verify these predictions, experimentson dilute solutions of DNA conﬁned in narrow slits haveshown that the exponents for conformation and chain relaxation are 0.68 and 2.2, respectively, which lie between
2D and 3D behaviors [34].
In this paper, we discuss a Brownian dynamics theory
for a polymer chain conﬁned in a membrane. As schematically presented in ﬁg. 1, we consider a polymer chain(connected red particles) embedded in a liquid membrane(yellow particles) surrounded by a 3D bulk ﬂuid (blue particles) and walls (not shown). We ﬁrst derive the mobility
tensors for the two geometries, i.e., a free membrane embedded in a bulk ﬂuid, and a membrane sandwiched by
two walls. For these two cases, we shall obtain the corresponding analytical expressions for the polymer diﬀusion coeﬃcient which are valid for all sizes. We further
perform a Rouse normal mode analysis to calculate therelaxation time and dynamical structure factor. For large
polymer sizes, these quantities show Zimm-like behavior
in the free-membrane case, whereas they are Rouse-likein the presence of walls. We also use a scaling theory to
discuss the eﬀect of excluded-volume interactions on therelaxation times for the two geometries. The present work
demonstrates the importance of the outer environment of
the membrane in determining the dynamics of a 2D polymer chain.

ηs

h+
0
-h-

η

wall
Fig. 2. Schematic picture showing a planar liquid membrane
having 2D viscosity η located at z = 0. It is sandwiched by asolvent of 3D viscosity ηs± . Two impenetrable walls are locatedat z = ±h± bounding the solvent.

In the next section, we start to set up the governing
equations for the membrane and the derivation of mobility tensors. With the introduction of the polymer, thegeneral formalism of the problem is constructed in sect. 3.
Sections 4 and 5 describe the results for the polymer dynamics for the free- and conﬁned-membrane limiting cases,
respectively. Excluded-volume eﬀects are discussed using
scaling arguments in sect. 6. We ﬁnally close with several
discussions in sect. 7.

2 Membrane hydrodynamics
Before introducing the polymer, we ﬁrst establish the governing equations for the membrane and its surrounding
environment. The aim of this section is to derive the membrane mobility tensors which will be used in the later sections for the polymer equations of motion. The present
calculation closely follows the formulation by Inaura andFujitani [35]. However we consider a more general situation which will be described below. The details of thecalculation are relegated to appendix A.
As shown in ﬁg. 2, we assume that the membrane is
an inﬁnite planar sheet of liquid, and its out-of-plane ﬂuctuations are totally neglected, which is justiﬁed for typical bending rigidities of bilayers. Relaxation dynamics ofmembrane ﬂuctuations near walls have been previously
considered [36–38]. The liquid membrane is embedded ina bulk ﬂuid such as water or solvent which is bounded byhard walls. Let v(r) be the 2D velocity of the membrane
ﬂuid and the 2D vector r = (x, y) represents a point in theplane of the membrane. We ﬁrst assume the membrane to
be incompressible∇ · v = 0,(1)
where ∇ is a 2D diﬀerential operator. We work in the lowReynolds-number regime of the membrane hydrodynamics

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membraneso that the inertial eﬀects can be neglected. This allows
us to use the 2D Stokes equation given byη∇2 v − ∇p + fs + F = 0,(2)where η is the 2D membrane viscosity, p(r) the 2D inplane pressure, fs (r) the force exerted on the membraneby the surrounding ﬂuid (“s” stands for the solvent), andF(r) is any other force acting on the membrane such asthat due to a polymer chain introduced in sect. 3.
As presented in ﬁg. 2, the membrane is ﬁxed in the xyplane at z = 0. The upper (z > 0) and the lower (z < 0)
ﬂuid regions are denoted by “+” and “−”, respectively.
The velocities and pressures in these regions are written asv± (r, z) and p± (r, z), respectively. Since the 3D viscosityof the upper and the lower solvent can be diﬀerent, wedenote them as ηs± , respectively. Consider the situation inwhich impenetrable walls are located at z = ±h± , whereh+ and h− can be diﬀerent in general. We note that thispoint diﬀers from ref. [35]. Similar to the liquid membrane,
the solvent in both regions are taken to be incompressible
˜ · v± = 0,∇(3)

˜ represents a 3D diﬀerential operator. We also
where ∇
neglect the solvent inertia and hence the solvent obeys
the 3D Stokes equations
˜ ± = 0.
˜ 2 v± − ∇p
ηs± ∇

(4)

The presence of the surrounding solvent is important because it exerts force on the liquid membrane. This force,indicated as fs in eq. (2), is given by the projection of(σ + − σ − )z=0 · êz on the xy-plane. Here êz is the unit
vector along the z-axis, and σ ± are the stress tensors dueto the solvent
˜ ± + (∇v
˜ ± )T ].
σ ± = −p± I + ηs± [∇v(5)

In the above, I is the identity tensor and the superscript“T” indicates the transpose.
Using the stick boundary conditions at z = 0 and z =±h± , we solve the hydrodynamic equations (3) and (4) toobtain fs . Then we calculate the membrane velocity fromeq. (2) asv[k] = G[k] · F[k],(6)where v[k] and F[k] are the Fourier components of v(r)and F(r) deﬁned byd2 kv(r) =
v[k] exp(ik · r),
(7)
(2π)2


and

d2 k
F[k] exp(ik · r),
(8)
(2π)2
respectively, and k = (kx , ky ). After some calculations (seeappendix A for the details), one can show that the mobility tensor G[k] in Fourier space is given by
F(r) =Gαβ [k] =1
ηk 2 + k[ηs+ coth(kh+ ) + ηs− coth(kh− )]


kα kβ
× δαβ − 2,k

(9)Page 3 of 13

with α, β = x, y, and k = |k|. As in ref. [35], we mainly
consider (except in sect. 7) the case when the two walls
are located at equal distances from the membrane, i.e.,h+ = h− = h. Then the above mobility tensor becomes


1
kα kβ
δ
,
(10)
−
Gαβ [k] =αβ
η[k 2 + νk coth(kh)]k2where ν ≡ 2ηs /η, with ηs = (ηs+ + ηs− )/2. An almost
equivalent expression to eq. (10) has also been derived
for Langmuir monolayers in which there is only one wallor a substrate [18,39]. In the following, we discuss the twolimiting situations of eq. (10).
Saﬀman and Delbrück (SD) investigated the case when
the two walls are located inﬁnitely away from the membrane [11,12]. This is called the free-membrane case andall the related physical quantities are denoted by the superscript “SD”. Taking the limit of kh  1 in eq. (10),the mobility tensor becomes [35,40,41]


1
kα kβ
SD
δαβ − 2.
(11)Gαβ [k] =η(k 2 + νk)k
Notice that the length ν −1 is called the SD hydrodynamic
screening length. The real space expression of this mobility
tensor is obtained by the Fourier transform of eq. (11)G(r) =

d2 k
G[k] exp(ik · r).
(2π)2

(12)

Performing the calculations presented in appendix B, we
obtain [18,40,41]1
2
H0 (νr) − Y0 (νr) +
(r)
=
GSD
αβ
4η
πν 2 r2
H1 (νr) Y1 (νr)
+
δαβ
−
νr
νr

4
1
2H1 (νr)− 2 2+
+
4η
πν r
νr

rα rβ
2Y1 (νr)− H0 (νr) + Y0 (νr)
, (13)−
νrr2
where r = |r|. In the above, Hn (z) are the Struve functions and Yn (z) are the Neumann functions or the Bessel
functions of the second kind.
In the opposite kh  1 limit, the membrane is conﬁned between the two walls. Since such a limiting casewas considered by Evans and Sackmann (ES) [14], we denote all the physical quantities for this situation with thesuperscript “ES”. In this case, eq. (10) takes the following
form:


1
kα kβ
ES
δαβ − 2,(14)Gαβ [k] =η(k 2 + κ2 )kwhere κ ≡ (ν/h)1/2 . This new length scale κ−1 is the ES
hydrodynamic screening length. We note that κ−1 is thegeometric mean of ν −1 and h [42]. The above ES mobility tensor was previously used in a phenomenologicalPage 4 of 13

The European Physical Journal E

membrane hydrodynamic model [21,24,25,41,43]. Following the calculations in appendix B, the real space representation of the ES mobility tensor becomes11
K1 (κr)ES
K0 (κr) +− 2 2 δαβGαβ (r) =
2πη
κr
κ r


2
rα rβ
1
2K1 (κr)−K0 (κr)−+ 2 2+, (15)
2πη
κr
κ r
r2
where Kn (z) are the modiﬁed Bessel functions of the second kind. We use either eq. (11) or eq. (14) in our subsequent calculations with a polymer.
Strictly speaking, eqs. (13) and (15) should have beenobtained by taking the limits of νh  1 and νh  1,
respectively, after the integration of eq. (10) over k. However the inverse Fourier transform of eq. (10) is nontrivial.
The present approach serves our purpose and the rigorous
derivation will be given in a separate publication.

3 Dynamics of a 2D Gaussian polymer chain
embedded in a membrane
We are now ready to introduce a polymer into the membrane. For simplicity, we ﬁrst work with a Gaussian polymer chain whose conformation is given by a set of N position vectors denoted as {Rn } = (R1 , . . . , RN ) embeddedin the 2D membrane. The excluded-volume eﬀects will be
discussed later in sect. 6. It is implicitly assumed that thepolymer relaxation time is much longer than that for the
typical hydrodynamic disturbances so that the membrane
can be eﬀectively considered as a 2D liquid. If the polymer
consists of monomers which exert a set of point forces fn
acting at Rn , the external force due to the polymer F(r)in eq. (2) can be written asF(r) =fn δ(r − Rn ).
(16)n

In writing this expression, we have assumed that the superposition principle holds. With the use of the mobility
tensor obtained in the previous section, eq. (2) can beformally solved asG(r − Rn ) · fn .
(17)v(r) =

where ζm (t) is the Gaussian random force acting at Rm ,kB the Boltzmann constant, T the temperature. The potential energy of the 2D polymer has the formU=

where we have used the notation Gnm ≡ G(Rn − Rm ).
The Langevin equation for a polymer chain embedded
in a membrane is written as [44,45]∂U∂Rn (t) =
Gnm · −+ ζm (t)∂t∂RmmkB T  ∂· Gnm ,
(19)+
2 m ∂Rm(20)

where b is the Kuhn length. It can be shown that themobility tensors eqs. (13) and (15) satisfy∂· Gnm = 0.
∂Rm

(21)

Substituting eq. (20) into eq. (19) and using eq. (21), wehave
2kB T ∂ 2 Rm (t)∂Rn (t) 
=
Gnm ·
+
ζ
(t)
. (22)
m
∂t
b2
∂m2
m
Due to the hydrodynamic coupling between diﬀerent
parts of the polymer, the above equation is nonlinear anddiﬃcult to solve analytically. In order to overcome this difﬁculty, we employ the preaveraging approximation which
has been successfully used for a polymer in 3D solvent [45,46]. Assuming that the polymer is close to its equilibrium,
we replace Gnm by its equilibrium value Gnm  such thatGnm  = d{Rn } Gnm Ψ({Rn })


 ∞
1
r2
Gnm (r)dr 2πrexp−=
π|n−m|b2|n−m|b20= g(n − m)I,
(23)
where Ψ({Rn }) is the 2D Gaussian distribution function.
Within this approximation, eq. (22) can be simpliﬁed as
2kB T ∂ 2 Rm (t)∂Rn (t) =g(n−m)+ ζm (t) . (24)
∂t
b2
∂m2
m
The above equation can be rewritten in terms of the
Rouse normal coordinates deﬁned by [45]

as

1
N

 N

 pπn

Rn (t),(25)∂Xp (t) =
gpq [−kq Xq (t) + ζq (t)],
∂t
q

(26)

Xp (t) =n

Since monomers move with the same velocity as the membrane, their velocities are given by∂Rn (t)= v(Rn ) =Gnm · fm ,
(18)
∂t
m

N
kB T 
(Rn − Rn−1 )2 ,b2 n=2dn cos0

withN

4π 2 kB T 2
p .
(27)
N b2
Here p, q = 0, 1, 2, . . . and gpq is the mobility tensor interms of the normal coordinates:

 N
 pπn
 qπm
dn N dmcoscosg(n − m).
gpq =
N 0 N
N
N
0
(28)
kp =

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membrane

If one neglects the contribution from the oﬀ-diagonal components of gpq , we ﬁnally obtain∂Xp (t)= gp [−kp Xp (t) + ζp (t)] ,
∂t

(29)

with the deﬁnition gp ≡ gpp . The Gaussian random forcesζp (t) satisfy the following conditions:ζpα (t) = 0,−1ζpα (t)ζqβ (t ) = 2δpq δαβ (gp )kB T δ(t − t ).

(31)

Therefore the relaxation time of a polymer in terms of the
Rouse modes is given by
τp =

1
.
gp kp

(32)

Furthermore, the polymer diﬀusion coeﬃcient can be calculated according to the following equation:
 N
D = kB T g0 = kB T
0

dn
N

 N
0

dm
g(n − m).
N

(33)

Another useful quantity that we calculate is the dynamic structure factor deﬁned by
1 
exp[ik · (Rn (t) − Rm (0))].
N n,m

(34)

Since Rn (t) − Rm (0) is a linear function of ζn (t) obeying the Gaussian distribution, the distribution of Rn (t) −Rm (0) is also Gaussian [47]. Hence we have(35)

in 2D. Denoting the center of mass by X0 and using theinverse relation of eq. (25)
R n = X0 + 2

∞

p=1

Xp cos

 pπn
N

,

(36)

4.1 Mobility tensor
In order to perform the preaveraging of the mobility tensor, we take the conﬁgurational average of eq. (11) byusing the equilibrium probability distribution function ofa Gaussian polymer in 2D:
I − k̂k̂
d2 k
SDexp[ik · (Rn − Rm )] ,Gnm  =(2π)2 η(k 2 + νk)(38)
where k̂ denotes a unit vector along k. This leads toSDGSD(n − m)I, withnm  = g
g SD (n − m) =


 ∞
1
k
1
exp − b2 k 2 |n − m| =
dk 2
4πη 0
k + νk
4


1 2 2
1
exp − b ν |n − m|
8πη
4





1 2 2
1 
bν |n − m| − Eib ν |n − m| , (39)× π erﬁ
2
4
where erﬁ(z) is the imaginary error function

with erf(z) being the error function
 z
22
erf(z) = √du e−u ,
π 0

(40)

(41)

whereas Ei(−z) is the exponential integral given by [48]
 ∞
e−u
.
(42)
Ei(−z) = −duuz
It should be noted that the obtained g SD (n − m) is real
despite the presence of complex functions.

we can calculate the dynamical structure factor as
1 
1
exp − k 2 Dt − |n − m|b2 k 2
N n,m4

4.2 Relaxation time

 pπm
 pπn
N b2 k 2  1cos[1−exp(−t/τp )] .
cos
π 2 p=1 p2
N
N
∞

−

When the two walls in ﬁg. 2 are located at inﬁnite distancefrom the membrane, one can use the SD mobility tensor
given by eq. (11). We calculate the preaveraged mobility
tensor, relaxation time, diﬀusion coeﬃcient and structure
factor following the recipe described in the previous section.

erﬁ(z) = −i erf(iz),exp(ik · [Rn (t) − Rm (0)]) = 2
kexp − (Rn (t) − Rm (0))2  ,4S(k, t) =4 Polymer dynamics: free-membrane case

(30)



S(k, t) =Page 5 of 13(37)
This completes the general formalism of the 2D polymer
dynamics conﬁned in a liquid membrane. In the next section, we consider the free- and conﬁned-membrane cases
separately by using eqs. (11) or (14) for the mobility tensor.

In order to obtain the polymer relaxation time, we ﬁrst
substitute eq. (39) into eq. (28) to express the mobility
tensor in terms of the Rouse normal coordinates
 ∞
1
k3
dk
gpSD =2 =πηN b2 0(k 2 + νk)[k 4 + (4πp/N b2 ) ]√
1 π 2 p − 2pπ 3/2 δ + 2 ln(πp/δ 2 )δ 2 + ( 2π/p)δ 3,
16πη
π 2 p2 + δ 4(43)Page 6 of 13

The European Physical Journal E
3

10

3

10

2

2

10

1

10

0

10

p=1

-1

10

-2

10

p=10
-2

-1

10

10

0

10

δ, ε

1

10

2

10

SD
(note that gpSD = gpp). In the above, we have deﬁned√
the dimensionless polymer size δ ≡ N bν/2. Since theradius of gyration for the 2D Gaussian polymer is Rg =√N b/2 [45], δ can be also written as δ = Rg ν. Usingeq. (32), the relaxation time becomes

4N b2 ηπkB T

p2 [π 2 p −

√

1

10
10

ε=0.01-1

δ=0.01

0

10

Fig. 3. Scaled relaxation time πkB T τp /4N b2 η as a functionof δ = Rg ν for free membranes (solid lines) or ε = Rg κ for
conﬁned membranes (dashed lines) for p = 1 and 10.

×

δ=100

2

πkBTτp/4Nb η

2πkBTτp/4Nb η

10

τpSD =

ε=100

-2

10

0

10

1

10

p

2

10

Fig. 4. Scaled relaxation time πkB T τp /4N b2 η as a functionof p for diﬀerent values of δ = 0.01, 100 (solid lines) andε = 0.01, 100 (dashed lines). The two curves for δ = ε = 0.01
overlap each other and cannot be distinguished.

SD hydrodynamic screening length ν −1 , i.e., δ ∼ 1. In thelimit of large δ, the p-dependence of the relaxation timeis analogous to that obtained from the Zimm model [45].
The solid lines in ﬁg. 4 show the relaxation time as afunction of the Rouse normal mode p for δ = 0.01 and100. These lines have slopes −1 and −3/2 for δ = 0.01
and 100, respectively. These results indicate the diﬀerentmode dependencies in the two limiting polymer sizes.

π 2 p2 + δ 4

.
2pπ 3/2 δ + 2 ln(πp/δ 2 )δ 2 + ( 2π/p)δ 3 ](44)

This expression shows how the presence of the bulk solventaﬀects the relaxation time.
We consider two asymptotic limits of eq. (44). For
small polymer sizes or δ  1, we have
τpSD ≈

4N b2 η 1.
πkB T p(45)

For large sizes, the condition δ  1 yields
τpSD ≈

4N b2 ηδ√.
πkB T 2πp3/2(46)

It should be noticed that eq. (46) depends only on thesolvent viscosity ηs but not on the membrane viscosity η.
In ﬁg. 3, the scaled relaxation time eq. (44) is plotted
as a function of δ for p = 1 and 10 as solid lines. For
small δ, the relaxation time is independent of the polymer
size, which is consistent with eq. (45). The relaxation timeincreases through a crossover regime towards a linear behavior as given by eq. (46). Such a crossover occurs aroundthe region where the polymer size Rg is comparable to the

4.3 Diﬀusion coeﬃcient

By substituting eq. (39) into eq. (33), the diﬀusion coeﬃcient of the polymer can be obtained as√kB T 1
4 π 3
2
2
δ
(π
erﬁ(δ)−Ei(δ))
exp(−δ)+DSD =4πη δ 43
√+δ 2 − (ln δ 2 + γ)(δ 2 − 1) − 2 πδ ,(47)
where γ = 0.5772 · · · is Euler’s constant. This expression
for the polymer diﬀusion coeﬃcient is valid for all theranges of δ. Equation (47) is one of the main results ofthis paper.
We now discuss the asymptotic limits of eq. (47) forsmall and large polymer sizes. When δ  1, it reduces to


γ
3
kB T
− ln δ − +
.
(48)
DSD ≈
4πη
2 4
Such a logarithmic behavior is consistent with that of an
object in a pure 2D system [11,12]. In the opposite limit

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membrane2

for t  τpSD . In this case, we have

10

1 
1
N b2 k 2
exp − |n − m|b2 k 2 −
N n,m
4
π2

S SD (k, t) ≈0

4πηD / kBTPage 7 of 13

10

∞

1

 pπn

 pπm

[1 − exp(−t/τpSD )] =
p
N
N
p=1
 ∞
8
du exp(−u − k 2 I1 (u)),(51)
b2 k 2 0

×
-2

10

cos
2

cos

with
-4

10

10

I1 (u) =
-3

-1

10

1

δ, ε

10

3

10

Fig. 5. Dimensionless diﬀusion coeﬃcient 4πηD/kB T as afunction of δ = Rg ν (solid line) or ε = Rg κ (dashed line).

of δ  1, we have
D

SD

√
kB T
kB T 4 π= √≈.
4πη 3δ
6 πηs Rg

 ∞
dp
0

1
cos
p2



4πpu
N b2 k 2[1 − exp(−t/τpSD )].

(52)
When δ  1, we can use the limiting expression for τpSD
as obtained in eq. (46). In this case, the above expression
becomes
S SD (k, t) ≈
 ∞


8
SD 2/3SD −2/3
du
exp
−u
−
(Γ
t)
w
((Γ
t)
u)
, (53)
1b2 k 2 0with the decay rate

(49)
Γ

Similar to eq. (46), this expression depends only on ηs . The
obtained 1/ηs Rg -dependence is analogous to that of an
object moving in 3D ﬂuid as well as the result by Hughes
et al. [13].
In ﬁg. 5, we plot the diﬀusion coeﬃcient DSD as afunction of δ (solid curve). With the increase in the polymer size, there is a crossover from logarithmic to algebraic
decay indicated by eqs. (48) and (49), respectively. Thechange in the behavior of a domain of the diﬀusion coefﬁcient with the addition of solvent has also been shown
through recent dissipative particle dynamics simulations
on pure 2D and quasi-2D systems [49].

4.4 Dynamic structure factor
The last quantity calculated for the free-membrane geometry is the dynamical structure factor S SD (k, t) deﬁned ineq. (34). Since the full expression is rather complicated,
we derive several analytical expressions for the limitingcases. For Rg k  1, we have
S SD (k, t) ≈ N exp(−k2 DSD t),

N b2
2π 2

(50)

where DSD is given by eq. (47). This is reasonable because
only the center-of-mass motion of the polymer is captured
in the small-angle regime.
For Rg k  1, on the other hand, we only consider thetime region t  τpSD because S SD (k, t) becomes very small

SD√
kB T k 3
kB T N bk 3
=
,
=
16ηδ
8ην

(54)

and
2
w1 (u) =
π

 ∞
dx
0

√
cos(xu)[1 − exp(−x3/2 / 2)].
2
x

(55)

Notice the k 3 -dependence of the decay rate. For Γ SD t  1,
the above expression is further simpliﬁed to
S SD (k, t) ≈ S SD (k, 0) exp(−1.35(Γ SD t)2/3 ),(56)

since w1 (0) = Γ (1/3) ≈ 1.35. This expression is valid in
the limit of large polymer sizes such that Rg k  1 andδ  1. Large wave vectors (probing the internal motion of
the polymer) and hydrodynamic screening lengths (highsolvent viscosity) will also lead to the same expression.
The applicable time window for eq. (56) is 1/Γ SD  t 
τpSD . These expressions for large δ are analogous to thatobtained from the Zimm model [45] in the Rg k  1 regime
for polymer in 3D.

5 Polymer dynamics: conﬁned-membrane
case
When the thickness of the solvent layers is very small, themembrane is now almost conﬁned by the two walls. However, there is a thin lubricating layer between the membrane and the walls so that h = 0. In this case, we use theES mobility tensor given by eq. (14). Several quantities
for this case are obtained below.

Page 8 of 13

The European Physical Journal E

5.1 Mobility tensor

5.3 Diﬀusion coeﬃcient

By using eq. (14), the preaveraged mobility tensor is calculated from

With the use of eq. (58), the diﬀusion coeﬃcient for the
conﬁned-membrane geometry is written asI − k̂k̂
d2 k
exp[ik · (Rn − Rm )] .
(2π)2 η(k 2 + κ2 )(57)
ESThis results in GES(n − m)I, withnm  = g

DES =

GES
nm  =(n − m) =


 ∞
k
1 2 2
1
dk 2
exp − b k |n − m| =4πη 0
k + κ2
4

 

1 2 2
1 2 2
1
exp
b κ |n − m| Ei − b κ |n − m| , (58)
−
8πη
4
4
g

ES

kB T 1[(1 + ε2 )(2 ln ε + γ) − ε24πη ε4− exp(ε2 ) Ei(−ε2 )].

This equation was also obtained before [21]. The limiting
expression for ε  1 is


γ
3
kB T
ES
− ln ε − +
,
(64)
D ≈
4πη
2 4
which coincides with eq. (48) as long as ε is replaced by
δ. When ε  1, eq. (63) reduces towhich was previously derived in ref. [21].

DES ≈5.2 Relaxation time
The polymer relaxation time can be obtained by substituting eq. (58) into eq. (28). Then we have
gpES =

1πηN b2

 ∞
dk

k3
2

(k 2 + κ2 )[k 4 + (4πp/N b2 ) ]1 π 2 p + 2ε2 ln(ε2 /(πp)).
16πηπ 2 p2 + ε4
0

=
(59)

In the above,√ we have deﬁned the dimensionless polymer
size as ε ≡ N bκ/2 = Rg κ which should be distinguishedfrom δ in the previous section. Then the relaxation time
can be written as
τpES =

4N b2 ηπ 2 p2 + ε4.
22πkB T p [π p + 2ε2 ln(ε2 /(πp))]4N b2 ηε2.
πkB T 2p2 ln(ε2 /(πp))(65)

This 1/Rg2 -dependence is a characteristic of a system inwhich there is momentum loss from the membrane to thesurrounding environment [50]. An intuitive understandingof the large-size behaviors of the diﬀusion coeﬃcient interms of the conservation principles will be described insect. 7. The dashed line in ﬁg. 5 shows the plot of DES
as a function of ε, showing the logarithmic and algebraic
behaviors as derived above.
Accordingto the deﬁnition of δ and ε, one obtains√
ε = δ/ νh. This correspondence leads to a √rescaling ofthe ε-axis in ﬁgs. 3 and 5. Here the value of νh cannot
be taken arbitrarily since the condition for the conﬁned
membrane is given by νh  1 as explained in sect. 2.
5.4 Dynamic structure factor
The dynamic structure factor can be calculated in thesame manner as before. For Rg k  1, we have
S ES (k, t) ≈ N exp(−k 2 DES t).

(61)

(62)

In ﬁg. 3, we plot τpES as a function of ε for p = 1 and 10
in dashed lines. The algebraic ε2 -dependence in eq. (62) isseen for large ε. The dashed lines in ﬁg. 4 are the plots of
τpES as a function of p for ε = 0.01 and 100. For ε = 100,the slope −2 is consistent with eq. (62) neglecting thelogarithmic correction. Notice that this p-dependence isin contrast to that for the free-membrane case given ineq. (46).

(66)

For Rg k  1 and t  τpES , we getwhich coincides with eq. (45). In the opposite limit of ε 
1, one gets
τpES ≈

kB T 1
kB T h=.
4πη ε28πηs Rg2(60)

In the limit of ε  1, it reduces to
4N b2 η 1,
τpES ≈
πkB T p

(63)

S

ES

8
(k, t) ≈ 2 2
b k

 ∞

du exp(−u − k 2 I2 (u)),

(67)

0

with
N b2
I2 (u) =
2π 2

 ∞
0

1
dp 2 cosp

4πpu
N b2 k 2[1 − exp(−t/τpES )].

(68)
Considering ε  1 and neglecting the logarithmic dependence in eq. (62), the above expression becomes
S ES (k, t) ≈
 ∞


8
ES 1/2ES −1/2
du
exp
−u
−
(Γ
t)
w
((Γ
t)
u)
, (69)
2b2 k 2 0

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membranewith the decay ratekB T N b2 k 4
kB T k 4=,32πηε28πηκ2Γ ES =and2w2 (u) =
π

 ∞
0

(70)

Table 1. Comparison of the relaxation times between a Gaussian polymer chain and a polymer with excluded-volume interactions.
Limits

Ideal chain (νF = 1/2)δ1

cos(xu)dx[1 − exp(−x2 )].
x2

ES

(71)

Real chain (νF = 3/4)

−1

τ SD ∼ p−3/2

−1

∼p

τ

SD

ε1

τ

ES

∼p

τ ES ∼ p−3/2

δ1

τ SD ∼ p−3/2

τ SD ∼ p−9/4−2

τ ES ∼ p−3

ε14Page 9 of 13

τ

ES

∼p

Note that Γ
is proportional to k as for the Rousemodel. For Γ ES t  1, it can be further simpliﬁed to(72)
S ES (k, t) ≈ S ES (k, 0) exp(−1.13(Γ ES t)1/2 ),√since w2 (0) = 2/ π ≈ 1.13. This expression is valid whenRg k  1 and ε  1. The relevant time interval for thisexpression is 1/Γ ES  t  τpES .

other words, we replace N with N 2νF in these equations.
On the other hand, the relaxation time in the presence ofthe excluded-volume eﬀects can be obtained by replacing
N/p with (N/p)2νF in the Gaussian chain cases. In thesmall size limits, i.e., δ, ε  1, we haveτSD

6 Excluded-volume eﬀects
So far, we have treated only a 2D Gaussian polymer chain.
In this section, we brieﬂy discuss the eﬀects of excluded
volume on the dynamical quantities. Even if the polymer
is not a Gaussian chain, one can still use the preaveraging approximation for the mobility tensor. For the freemembrane case, one can generally show thatg SD (n − m) =1
H0 (νrnm ) − Y0 (νrnm ),
8η

(73)

where rnm = |Rn − Rm | (see appendix B). Then the diffusion coeﬃcient can be expressed as
kB T N dn N dm
H0 (νrnm ) − Y0 (νrnm ).
DSD =8η 0 N 0 N(74)
Similarly for the conﬁned-membrane case, we ﬁndkB T N dn N dm
ES
K0 (κrnm ),(75)
D =4πη 0 N 0 N(see eq. (B.14)). These expressions are the 2D analog of
the Kirkwood formula [45]. It should be emphasized thatthey are rigorous even in the presence of excluded-volume
eﬀect.
For excluded-volume chains, however, the appropriate
equilibrium distribution function Ψ({Rn }) needed to calculate averages is not known [45]. Therefore, we cannot
obtain rigorous forms of diﬀusion coeﬃcient. Instead, weshall make use of scaling arguments to infer the eﬀects of
excluded-volume interactions. Here, we limit our discussion to small and large polymer sizes. A simple argument
is that the excluded-volume eﬀects lead to a rescaled polymer radius of gyration Rg . Within the Flory theory, theradius of gyration scales as Rg ∼ bN νF using the Flory
exponent νF . Up to a numerical factor, the diﬀusion coeﬃcient in the limiting cases will still show the same size
dependence as in eqs. (48), (49), (64) and (65) in which Rg
is now replaced with that of excluded-volume chains. In

ηb2
∼
kB T

and
τ ES ∼

ηb2
kB T





N
p
N
p

2νF
,

(76)

.

(77)

2νF

In the opposite of large polymer size limits, i.e., δ, ε  1,
we get 3νFηb3 ν N,(78)
τ SD ∼
kB T
p
and
τ

ES

ηb4 κ2
∼
kB T



N
p

4νF
.

(79)

These are the scaling predictions. Notice that the above
results can also be obtained by using the relation τ ∼Rg2 /D, where Rg includes the excluded-volume eﬀect [51].
Table 1 shows the comparison between a Gaussian
chain polymer (νF = 1/2) and a chain with excludedvolume interactions (νF = 3/4) in 2D. The former Gaussian case recovers all the relations in the previous sections (see eqs. (45), (61), (46), (62)). For the chain with
νF = 3/4, the relaxation time shows a p−3/2 -dependence
for both the free- and conﬁned-membrane cases whenδ, ε  1. This is consistent with the result in ref. [20]. Forδ  1 and ε  1, we have τ SD ∼ p−9/4 and τ ES ∼ p−3 , respectively. These exponents are unique for polymers conﬁned in a membrane.

7 Discussion
In this paper, we have investigated the dynamics of a
Gaussian polymer chain conﬁned in a liquid membrane
taking into account the surrounding environment. For themost general geometry with the membrane, solvent andwalls (see ﬁg. 2), we have derived the mobility tensor ineq. (9). We obtained the analytical expressions for the relaxation time, diﬀusion coeﬃcient and dynamic structure
factor of a Gaussian chain for the two limiting cases ofPage 10 of 13

The European Physical Journal E

free and conﬁned membranes. These quantities were calculated within the preaveraging approximation of the mobility tensor in order to avoid the nonlinearity introducedby the hydrodynamic coupling. We shall summarize anddiscuss the results obtained in this paper.
Our theoretical analysis relies on the preaveraging approximation, which is used to decouple the polymer ﬂuctuations from hydrodynamics. The validity of this approximation has been studied in some detail for polymers in
a 3D bulk ﬂuid, where the preaveraging approximation(Zimm model) yields results which are not very diﬀerent
from more sophisticated calculations [45]. The responseof a dilute polymer solution to an external ﬁeld has also
been experimentally veriﬁed to follow the predictions of
the Zimm model [52]. Deviations from the Zimm model
have been attributed to non-Gaussian distributions, theslowness to reach asymptotic behaviors or the eﬀect of hydrodynamic ﬂuctuations, although a clear conclusion has
not been reached [53]. Up to now, there have not been any
experimental studies of the dynamics of polymers conﬁned
to membranes, to the best of our knowledge. Experiments
using a combination of Langmuir trough as well as lightscattering techniques should be able to test our predictions. The success of preaveraging in three-dimensional
polymer solutions encourages us to expect the same tohold for quasi-2D systems.
We ﬁrst compare the relaxation times between the twocases. As seen from eqs. (45) and (61) for small polymer
sizes, i.e., δ, ε  1, both τpSD and τpES show exactly thesame mode dependence. This is due to the fact that forpolymer sizes smaller than the hydrodynamic screening
lengths (ν −1 or κ−1 ), the outer environment does not affect the polymer dynamics. The behavior τpSD ∼ p−3/2 forδ  1 (see eq. (46)) is analogous to the Zimm relaxation
time of a Gaussian polymer in a 3D solvent. On the other
hand, the dependence τpES ∼ p−2 for ε  1 (see eq. (62))
is similar to the Rouse relaxation time in 3D.
One of the important results of this paper is the derivation of eq. (47) which provides the diﬀusion coeﬃcientvalid for all size ranges in the free-membrane case. According to eqs. (48) and (64), the diﬀusion coeﬃcients
DSD and DES show the same logarithmic dependence
when δ, ε  1. Again this occurs when the polymer size
is smaller than ν −1 or κ−1 . A crossover from logarithmic to algebraic behavior takes place when the polymer
size Rg is comparable to these hydrodynamic screening
lengths. Thus ν −1 or κ−1 determines the length scale atwhich the outer environment surrounding the membrane
becomes important. For a pure 2D system, the screening
length is inﬁnite, which causes the Stokes paradox.
When the polymer size becomes much larger than thehydrodynamic screening lengths, the interactions are nolonger only through the membrane. In the free-membranecase, the outer ﬂuid plays the role of the mediator of
the hydrodynamic interactions. This attributes a 3D nature to the polymer dynamics and hence the scaling
DSD ∼ 1/ηs Rg is recovered (see eq. (49)). In the conﬁnedmembrane case, the presence of the walls takes away
the momentum from the membrane. Owing to the stickboundary condition, a linear shear velocity gradient is set
up in the intervening ﬂuid layer between the membrane
and the walls leading to a momentum leakage from themembrane. As given in eq. (65), the resultant diﬀusion
coeﬃcient behaves as DES ∼ 1/Rg2 . These asymptotic behaviors are consistent with the results of correlated diﬀusion obtained in ref. [41].
The diﬀerent large-size behaviors of the diﬀusion coefﬁcient can be better understood by focusing on the mobility tensors. It is essentially described as a consequence ofthe conservation of mass and momentum principles [50].
The mobility tensor G(r) acts as a Green’s function relating a source of disturbance at the origin to the velocity
at a point r. At suﬃciently large distances, the sourceof disturbance in the ﬂuid can be thought of as a force
monopole which introduces a source for momentum in the
ﬂuid. A moving object also causes a perturbation in themass density and this can be regarded as a mass dipole(a source and a sink of mass density). We now apply theconservations of momentum and mass to the 3D and 2D
cases separately.
For the 3D case, the presence of a force monopole atthe origin should cause the momentum ﬂux or stress σ decay as 1/r2 in order to conserve the total momentum (r2being proportional to the area of a sphere surrounding
the force monopole). Since the shear stress is related to
the ﬂuid velocity through σ ∼ ηs v/r, we have v ∼ 1/ηs r.
This implies that the mobility tensor should also scale as
1/ηs r. Concerning the mass conservation principle, a massmonopole would create a ﬂow velocity that decays as 1/r2 .
Since we have a mass dipole, the resulting velocity now decays as 1/r3 . Comparing these two eﬀects, the contribution to the velocity due to momentum conservation (which
varies as 1/r) always dominates at large distances in 3D.
This essentially explains the behavior DSD ∼ 1/ηs Rg .
Let us now consider the 2D case in which the membrane is in contact with the walls leading to a loss of
momentum from the membrane. This implies that the
momentum is not conserved, and the only contribution
to the velocity is from the mass conservation. In 2D, amass monopole will create a velocity which decays as 1/r(r being the perimeter of a circle surrounding the massmonopole). Hence the velocity and the mobility tensor due
to a mass dipole decays as 1/r2 . This explains the scaling
DES ∼ 1/Rg2 . These two diﬀerent behaviors of the mobility tensors are reﬂected in the diﬀusion coeﬃcients forlarge-size polymers. Incidentally, for the pure 2D case, thestress decays as 1/r due to the momentum conservation.
Since the stress scales as σ ∼ ηv/r, we have v ∼ 1/η. This
explains the logarithmic size dependence of the diﬀusion
coeﬃcient.
The dynamic structure factor is a quantity readily accessible through scattering experiments. For small wave
numbers Rg k  1, only the center-of-mass motion of thepolymer can be captured. When Rg k  1 and t is muchless than the relaxation times, the screening lengths become important. In the limit of δ  1, S SD (k, t) shows astretched exponential decay with an exponent 2/3, and sodoes S ES (k, t) for ε  1 with an exponent 1/2. Again the

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membranerole of the outer environment is reﬂected in the decay rates
Γ SD and Γ ES given by eqs. (54) and (70), respectively.
In the free-membrane case, the dependence Γ SD ∼ k 3
resembles that of the Zimm model in 3D. For conﬁned
membranes on the other hand the behavior Γ ES ∼ k 4 isanalogous to that obtained from the Rouse model.
The dynamics of a hydrophobic polymer embedded in a 2D membrane was previously investigated by
Muthukumar both for a Gaussian polymer and a polymer with excluded-volume eﬀects [20]. In his treatment,
the membrane was regarded as an isolated entity without
any couplings to the outer environment. The membrane
2D nature was taken into account through an anisotropic
viscosity. It was shown that the longest relaxation timeis proportional to p−1 for a Gaussian chain. This agrees
with our limiting expressions of the relaxation times in
eqs. (45) and (61) obtained for small δ and ε, respectively.
In ref. [20], the excluded-volume eﬀects were taken into
account by a mode-dependent polymer blob size. In thiscase, he showed that the relaxation time scales as p−3/2 .
As discussed in sect. 6, we ﬁnd that the mode dependencies of the relaxation times are altered in the presence of
excluded-volume interactions. In the small-size limit, we
indeed recover the scaling p−3/2 for the free- and conﬁnedmembrane cases (see table 1). Notice again that the polymer dynamics in this limit remains unaﬀected by the outerenvironment. For large polymer sizes, we obtain eitherτ SD ∼ p−9/4 or τ ES ∼ p−3 for real polymer chains.
A related situation to a polymer in a conﬁned membrane is a dilute polymer solution trapped in a slit-likegeometry whose width is much smaller than the polymer
blob size. Using scaling arguments, Brochard calculated
the polymer relaxation time scales as p−5/2 [31]. An experimental realization of such a geometry was done by
Lin et al. who conﬁned dilute DNA solution in quasi-2D
110 nm wide slits [34]. The relaxation times measured inthis case was found to scale as p−2.2 . However, it should beemphasized that this scenario is diﬀerent from the model
discussed in this article. In our model, the polymer chain
is strictly conﬁned to the 2D plane of the membrane whichitself is embedded in a 3D bulk ﬂuid.
At this stage, a rough estimate of the screening
lengths would be useful. As reported in ref. [2], the membrane viscosity of dimyristoylphosphatidylcholine bilayers at 32 ◦ C (rounded to the nearest order) is approximately 0.1 N s/m2 and the viscosity of water is ηs ≈10−3 N s/m2 . For supported membranes we can approximate the height of the intervening solvent region to beh ≈ 10−8 m [54]. Hence we obtain ν −1 ≈ 2.5 × 10−7 m
and κ−1 ≈ 0.5 × 10−7 m. As described in the Introduction, the experiments with DNA adsorbed on supported
membranes point towards a Rouse-like behavior [26,27].
The strong electrostatic attraction between the negatively
charged DNA and the positively charged membrane prevents any out-of-plane motions of the polymer. Because
the typical length scales of the DNA molecules used in theexperiments were of several microns, the scenario should
be close to the case of ε  1. Hence our result is consistent
with the experimental observations showing the Rouse-likePage 11 of 13

behavior. A more recent experiment on DNA adsorption
on free-standing cationic giant unilamellar vesicles showedthat the diﬀusion coeﬃcient of DNA molecules lies in the
crossover region between the logarithmic to the algebraic
regimes [29].
We ﬁnally discuss the mobility tensor for a supported
membrane. In this case, a membrane sits at the bottomof a trough ﬁlled with a bulk solution having an inﬁnite depth when compared to the membrane thickness. In
ﬁg. 2, this corresponds to the case where h+ is inﬁnitely
large while h− is ﬁnite. Using the general expression of
the mobility tensor in eq. (9), we obtain


1
kα kβ
δαβ − 2.
Gαβ [k] = 2kηk + k[ηs+ + ηs− coth(kh− )](80)
If we further assume that h− is very small, the above
equation reduces to


1
kα kβ
δ
. (81)−Gαβ [k] = 2
αβk2
ηk + ηs+ k + (ηs− /h− )This mobility tensor for a supported membrane contains
two screening length scales, i.e., η/ηs+ and (ηh− /ηs− )1/2 .
The investigation using the above mobility tensor is left
as our future work.
We thank H. Diamant, Y. Fujitani, M. Imai, T. Kato and N.
Oppenheimer for useful discussions. This work was supported
by KAKENHI (Grant-in-Aid for Scientiﬁc Research) on Priority Area “Soft Matter Physics” and Grant No. 21540420 from
the Ministry of Education, Culture, Sports, Science and Technology of Japan.

Appendix A. Derivation of the generalmobility tensor
Our derivation of the mobility tensor for the membraneclosely follows that by Inaura and Fujitani [35]. As presented in ﬁg. 2, we consider a more general case where
the two walls are located at diﬀerent distances from themembrane, i.e., h+ = h− . The membrane is assumed to
be impermeable.
Our purpose is to derive the in-plane force fs on themembrane due to the bulk solvent and walls (see eq. (2)).
We ﬁrst take the Fourier transform of v± (r, z) by
v± [k, z] = d2 r v± (r, z) exp(−ik · r),(A.1)
where r = (x, y) and k = (kx , ky ). As shown in ﬁg. 6, theprojection of the vector v± [k, z] on the xy-plane can be±
expressed as v± [k, z]k̂ + v⊥
[k, z]k̄ where k̂ = (kx /k, ky /k)and k̄ = (−ky /k, kx /k) with k = |k|. The subscriptsand ⊥ indicate the components parallel and perpendicularto k̂, respectively. From eq. (4), the vertical component±[k, z] obeys the equation
v⊥


∂2
±
2
[k, z] = 0.
(A.2)
−k + 2 v⊥∂z

Page 12 of 13

The European Physical Journal EzAppendix B. Mobility tensors in real space
Free-membrane case

v
y
k
x

vk
vk

Fig. 6. Projection of v± [k, z] onto the xy-plane which is de±k̄.
composed as v± k̂ + v⊥

The solution to the above equation can be written as±v⊥[k, z] = A1 e−kz + A2 ekz ,where v⊥ [k] = v[k] · k̄ and see eq. (7) for the deﬁnition ofv[k]. Thus we have

(B.1)

with two coeﬃcients B1 and B2 . By considering the diagonal and oﬀ-diagonal parts of eq. (B.1) separately, we
have
 ∞
1
J0 (kr)(B.2)
dk
2B1 + B2 =
2πη 0
k+ν
1
=
[H0 (νr) − Y0 (νr)] ,
(B.3)
4η
and
 ∞
1
J1 (kr)dk
2πη 0kr(k + ν) ∞J0 (kr) + J2 (kr)
1
dk
=
4πη 0
k+ν


2
1
H1 (νr) Y1 (νr)− 2 2+
−
.
=
4η
πν r
νr
νr

B1 + B2 =

(B.4)
(B.5)
(B.6)

See ref. [55] for the integral of eq. (B.3). In evaluating the
integral of eq. (B.6), we have made use of the following
relations:d2
J1 (z)= J0 (z) + 2 J0 (z),
z
dz
 ∞
 ∞
d2
d2 1
J0 (z)=.
dz J0 (z) 2dz
2dz z + a
da 0
z+a
0

(B.7)
(B.8)

Solving eqs. (B.3) and (B.6) for B1 and B2 , we get(A.6)

Since the incompressibility condition of the membrane
ﬂuid implies v [k] = v[k] · k̂ = 0, the perpendicular component of the Fourier transform of eq. (2) gives−ηk2 v⊥ [k] + fs⊥ [k] + F⊥ [k] = 0,

rα rβ
,
r2

(A.5)

From eq. (A.5), the in-plane force on the membrane
due to the outer solvent and walls becomesfs⊥ [k] = fs [k] · k̄ =∂  + +−
ηs v⊥ [k, z] − ηs− v⊥[k, z] z=0 =∂z−ηs+ v⊥ k coth(kh+ ) − ηs− v⊥ k coth(kh− ).

GSD
αβ (r) = B1 δαβ + B2

(A.3)

with unknown coeﬃcients A1 and A2 . These coeﬃcientsare determined by the stick boundary conditions imposedat the membrane-solvent and solvent-wall boundaries:v⊥ [k], at z = 0,±
(A.4)
v⊥ [k, z] =0,
at z = ±h± ,sinh(k(z ∓ h± ))±.
v⊥[k, z] = ∓v⊥ [k]sinh(kh± )The mobility tensor in Fourier space is given by eq. (11).
The expression for GSD
αβ (r) can be found by assuming(A.7)

with F⊥ [k] = F[k] · k̄. Hence the mobility tensor deﬁned
by v[k] = G[k] · F[k] is given by eq. (9).


1
2
H0 (νr) − Y0 (νr) +4η
πν 2 r2
H1 (νr) Y1 (νr)
+
,
−
νr
νr

4
1
2H1 (νr)− 2 2+
B2 =
4η
πν r
νr

2Y1 (νr)− H0 (νr) + Y0 (νr) .
−
νr

B1 =

(B.9)

(B.10)The preaveraging of eq. (B.1) yields
g SD (n − m) =1
2B1 + B2 .
2Hence we obtain eq. (73) using eq. (B.3).

(B.11)

S. Ramachandran et al.: Dynamics of a polymer chain conﬁned in a membrane

Conﬁned-membrane case
Now the mobility tensor in the Fourier space is eq. (14).
Similar to the free-membrane case, its real space expression can be written asrα rβ,(B.12)GES
αβ (r) = C1 δαβ + C2
r2
with two coeﬃcients C1 and C2 . Then it follows that ∞1
kJ0 (kr)dk 2(B.13)
2C1 + C2 =2πη 0
k + κ2
1=
K0 (κr),(B.14)
2πη
and

 ∞
1
J1 (kr)C1 + C2 =dk
2πη 0r(k 2 + κ2 )
1
1
K1 (κr)=.
−2πη κ2 r2
κr

Solving these equations, we obtain1
1
K1 (κr)C1 =K0 (κr) +− 2 2 ,
2πη
κr
κ r


2
1
2K1 (κr)−K0 (κr) −+ 2 2 .
C2 =
2πη
κr
κ r

(B.15)(B.16)(B.17)(B.18)

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