Rayleigh and Rayleigh-Debye-Gans Light Scattering

Intensities and Spetroturbidimetry of Dispersions of

Unilamellar Vesicles and Multilamellar Liposomes

An-Hsuan Hsieha, David S. Cortib, Elias I. Fransesc

May 20, 2020

Supporting Material (SM)

a: hsieh25@purdue.edu

b: dscorti@purdue.edu

c: franses@purdue.edu

Davidson School of Chemical Engineering

Purdue University

West Lafayette, IN, 47907-2100

A. Refractive Index of an Inhomogeneous Spherical Particle

The derivation of Eq. (13) in the main text is given as follows. From the theory of classical

electrostatics, when a uniform electric field E is applied to a homogeneous dielectric sphere of

radius a with a relative refractive index ms it induces an electric dipole moment p given by [1, p.

139]

p = 4πϵm

(ms

2 − 1

ms2 + 2

)a3E = 4πϵmMsa

3E (SM.1)

where ϵm is the permittivity of the medium, in SI units of C/Vm. The dipole moment is related

to the excess polarizability αs (in units of m3) of the homogeneous sphere relative to that of the

medium as follows

p = ϵmαsE (SM.2)

Thus,

αs = 4πMsa3 (SM.3)

The specific excess polarizability αs of a homogeneous sphere of volume Vs =4π3a3 is therefore

equal to

αs ≡αs

Vs= 3Ms (SM.4)

The expression in Eq. (SM.4) resembles the Clausius-Mossotti equation, which relates the specific

polarizability to the permittivity and the refractive index of the material [2, p. 162]. The specific

excess polarizability applies to any particle shape, and hence to inhomogeneous particles, such as

vesicles or liposomes.

Because the Rayleigh (R) scattering amplitudes from each individual volume element of a parti-

cle are in phase, and hence additive, the spatial distribution of the scattering elements in the particle

has no effect on the resulting scattered intensities in the R regime. Thus, for a vesicle, the excess

polarizability is

αv = αsVv,b (SM.5)

1

where Vv,b is the volume of the bilayer in a vesicle. Equivalently, an “effective” polarizability based

on a homogeneous spherical shape with its mass uniformly distributed throughout the total volume

Vv can be defined, with the same αv as in Eq. (SM.5). In other words, an “effective” specific

polarizability αv of the whole vesicle is given as

αv = αsVv,b = αvVv (SM.6)

where Vv is the volume of the entire vesicle. Thus,

αv =Vv,b

Vvαs = 3ϕv,bms (SM.7)

where ϕv,b = Vv,b/Vv is the volume fraction of the bilayer in a vesicle. To obtain the effective

relative refractive indexmv of a vesicle, one can introduce the parameterMv, as in Eq. (SM.4),

Mv = ϕv,bMs ≡mv

2 − 1

mv2 + 2

(SM.8)

For a liposome comprised of K bilayers with alternating water layers (see Fig. SM1), and an

outer radius al, the volume fraction of the bilayers in the particle depends on ac, db, dw, and K as

follows

ϕl,b =

K∑j=1

[aj3 − (aj − db)

3]

al3(SM.9)

The term aj = ac + (j − 1)dw + jdb is the outer radius of the jth bilayer (j = 1, 2, 3, ...), ac is

the radius of the central water core, and dw is the thickness of each water layer. If we assume, for

convenience, that ac = dw, then aj = j(dw + db) and al = K(dw + db), and Eq. (SM.9) becomes

ϕl,b =

K∑j=1

(3aj2 − 3ajdb + db

2)

K3(dw + db)3

=(K + 1)(2K + 1)

2K2

dbdw + db

+3(K + 1)

2K2

(db

dw + db

)2

+1

K2

(db

dw + db

)3

(SM.10)

2

For a large number of bilayers,K ≫ 1, ϕl,b approaches the limiting value of

ϕl,b ≈db

dw + db(SM.11)

which is independent of the outer radius al.

Similarly, for a liposome (see Fig. SM1),

αl = αsVl,b (SM.12)

where Vl,b is the volume of the bilayers in a liposome. Equivalently, the effective polarizability αl

of the whole liposome is given as

αl = αsVl,b = αlVl (SM.13)

where Vl is the volume of the entire liposome. Then,

αl =Vl,b

Vlαs = 3ϕl,bMs (SM.14)

To obtain the effective refractive indexml of a liposome, one can also introduce the parameterMl,

as in Eq. (SM.4),

Ml = ϕl,bMs ≡ml

2 − 1

ml2 + 2

(SM.15)

B. Rayleigh Scattered Intensities for Dispersions of Particles

From Eq. (12) in the main text, the volume fraction of the vesicles ϕv in the dispersion is, when

av ≫ db and ρdisp ≈ ρsurf,

ϕv =ϕsurf

ϕv,b=

wsurf

ϕv,b

(ρdispρsurf

)≈ av

3dbwsurf (SM.16)

3

Fig. SM1. (a) A liposome of outer radius al, total volume Vl, bilayer thickness db, specific bilayerpolarizability αs, and relative refractive index ms; (b) a homogeneous sphere with radius al, totalvolume Vl, and effective relative refractive index ml < ms, and specific polarizability αl < αs.The homogeneous sphere yields the same combined R scattering intensities as the liposome.

The number density of the vesicles is

Nv =ϕsurf

Vv,b=

ϕsurf4π3[av3 − (av − db)3]

(SM.17a)

where Vv,b is the volume of the bilayer in a vesicle; approximately, for av ≫ db,

Nv ≈ϕsurf

4πav2db(SM.17b)

When ρdisp ≈ ρsurf, and hence ϕsurf ≈ wsurf,

Nv ≈wsurf

4πav2db(SM.17c)

Similarly, the volume fraction of the liposomesϕl in the dispersion is, whenK ≫ 1 and ρdisp ≈ ρsurf,

ϕl =ϕsurf

ϕl,b=

wsurf

ϕl,b

(ρdispρsurf

)≈

(dw + db

db

)wsurf (SM.18)

4

The number density of the liposomes is

Nl =ϕsurf

Vl,b=

ϕsurf

4π3

K∑j=1

[aj3 − (aj − db)3]

(SM.19a)

where Vl,b is the volume of the bilayers in a liposome; approximately, forK ≫ 1 and ρdisp ≈ ρsurf,

Nl ≈3ϕsurf

4πal3

(dw + db

db

)≈ 3wsurf

4πal3

(dw + db

db

)(SM.19b)

For reference, we also present the number densityNs for a dispersion of homogeneous spheres with

radius a, when ρsurf ≈ ρdisp, and hence ϕsurf ≈ wsurf

Ns =ϕsurf

Vs

=3ϕsurf

4πa3≈ 3wsurf

4πa3(SM.20)

Then, for homogeneous spheres, using Eqs. (5), (11) in the main text and (SM.20), we find

is,N =3

4πϕsurfγa

3 ≈ 3

4πwsurfγa

3 (SM.21)

For vesicles, using Eqs. (9), (11) in the main text and (SM.17), we find

is,N =3

4πϕsurfγav

3ϕv,b ≈3

4πwsurfγav

3ϕv,b (SM.22a)

or, in the limit of large vesicles,

is,N ≈ 9

4πϕsurfγav

2db ≈9

4πwsurfγav

2db (SM.22b)

Finally, for liposomes, using Eqs. (10), (11) in the main text and (SM.19), we find that

is,N =3

4πϕsurfγal

3ϕl,b ≈3

4πwsurfγal

3ϕl,b (SM.23a)

5

or for K ≫ 1,

is,N ≈ 3

4πϕsurfγal

3

(db

dw + db

)≈ 3

4πwsurfγal

3

(db

dw + db

)(SM.23b)

C. Rayleigh Ratio and Specific Rayleigh Ratio of Spherical Particles

As defined in Eq. (14) in the main text, the expression of the Rayleigh ratio, in general, from Eq.

(13) in the main text, we find for R scattering

Rθ = ap3

(nm

λ0

)4

Ms2(1 + cos2 θ)ϕsurfϕp,b (SM.24)

With Eqs. (12) and (14) in the main text, the specific Rayleigh ratio, defined in Eq. (15) in the

main text, is found to be

Rθ∗∗ =

Rθ

ϕsurf

(ρdispρsurf

)(SM.25)

and, from Eq. (SM.24), for R scattering

Rθ∗∗ = ap

3

(nm

λ0

)4

Ms2(1 + cos2 θ)ϕp,b

(ρdispρsurf

)(SM.26)

When ρdisp ≈ ρsurf,

Rθ∗∗ ≈ ap

3

(nm

λ0

)4

Ms2(1 + cos2 θ)ϕp,b (SM.27)

For Rayleigh-Debye-Gans (RDG) scattering, using Eq. (31) in the main text, the specific

Rayleigh ratio for homogeneous spheres is, when ρdisp ≈ ρsurf,

Rθ∗∗ = a3

(nm

λ0

)4

Ms2(1 + cos2 θ)

(ρdispρsurf

)fs

2 ≈ a3(nm

λ0

)4

Ms2(1 + cos2 θ)fs2 (SM.28)

6

For vesicles, when ρdisp ≈ ρsurf, it is,

Rθ∗∗ = av

3

(nm

λ0

)4

Ms2(1 + cos2 θ)ϕv,b

(ρdispρsurf

)fv

2 ≈ 3av2db

(nm

λ0

)4

Ms2(1 + cos2 θ)fv2

(SM.29)

For liposomes, when ρdisp ≈ ρsurf, it is,

Rθ∗∗ = al

3

(nm

λ0

)4

Ms2(1 + cos2 θ)ϕl,b

(ρdispρsurf

)fl

2 ≈ al3

(nm

λ0

)4

Ms2(1 + cos2 θ)

(db

dw + db

)fl

2

(SM.30)

D. Turbidities and Specific Turbidities for Dispersions of Spherical Particles

The turbidity τ is found by integrating Eq. (16) in the main text with Ps,N(z) from Eq. (18) in the

main text, from z = 0 to ℓ

τ ≡ − ln(ItI0

)= ℓβϕsurfap

3ϕp,b (SM.31)

where Ps,N(z) is the total energy per unit time of the intensity I(z) that is scattered, or “lost” per

unit pathlength and ℓ is the pathlength. Then, since the absorbance A is defined as follows

A ≡ − log(ItI0

)(SM.32)

Therefore, with Eqs. (SM.31), (SM.32), and (SM.31), we obtain

τ =A

log(e)= ℓβϕsurfap

3ϕp,b (SM.33)

The turbidity per unit pathlength τ ∗, in units of m-1, is defined as

τ ∗ ≡ τ

ℓ(SM.34)

7

The turbidity per unit pathlength and per unit of surfactant weight fraction, or the specific turbidity

τ ∗∗, is defined as

τ ∗∗ ≡ τ ∗

wsurf(SM.35)

Then, using Eq. (12) in the main text and Eqs. (SM.33) to (SM.35), the specific turbidity, for

homogeneous spheres, in which ϕp,b = 1,

τ ∗∗ = β

(ρdispρsurf

)a3 ≈ βa3 (SM.36)

where the approximation ρdisp ≈ ρsurf was used. For vesicles, or when ρdisp ≈ ρsurf,

τ ∗∗ = β

(ρdispρsurf

)av

3ϕv,b ≈ βav3ϕv,b (SM.37a)

and for large vesicles,

τ ∗∗ ≈ 3βav2db (SM.37b)

Finally, for liposomes, when ρdisp ≈ ρsurf,

τ ∗∗ = β

(ρdispρsurf

)al

3ϕl,b ≈ βal3ϕl,b (SM.38a)

and for large liposomes,

τ ∗∗ ≈ βal3

(db

dw + db

)(SM.38b)

In the RDG regime, the loss of power term Ps,N(z) is found to be equal to the product of Ps,N(z)

for R scattering (Eq. (18) in the main text) and the “dissipation factor” Qp, or

Ps,N(z) = βI(z)ϕsurfap3ϕp,bQp (SM.39)

8

The turbidity in the RDG regime is found to be equal to the R turbidity (Eq. (SM.33)) timesQp

τRDG = τRQp (SM.40a)

Then, from Eqs. (12) in the main text and (SM.33),

τRDG = ℓβϕsurfap3ϕp,bQp = ℓβwsurf

(ρdispρsurf

)ap

3ϕp,bQp (SM.40b)

When ρdisp ≈ ρsurf,

τRDG ≈ ℓβwsurfap3ϕp,bQp (SM.40c)

where Qp is the integral of the square of the form factor, defined as Eq. (33) in the main text.

E. Size Range of Vesicles Applicable to the Rayleigh-Debye-Gans Regime

For vesicles, the second condition for the validity of the equations of the RDG scattering regime

(Eq. (4) in the main text) is given by the value of the term Xv,RDG,

Xv,RDG ≡ 4πnmavλ0

(mv − 1) ≪ 1 (SM.41)

where av is the outer radius of the vesicle and mv is the effective relative refractive index of the

vesicle. Since as av → ∞, ϕv,b → 0, Eq. (SM.8) indicates that mv − 1 → 0. Hence, the term

av(mv − 1) may approach a finite limit as av → ∞. To determine this limit, we consider the

following alternate form of Xv,RDG,

Xv,RDG =4πnm

λ0

(mv − 1)

1/av(SM.42)

9

and then apply l’Hôpital’s rule, in which one needs to evaluate the following limit

4πnm

λ0

limav→∞

(−av2)d(mv − 1)

dav(SM.43)

To determine the derivative in Eq. (SM.43), we rearrange Eq. (SM.8) to yield

mv =

(1 + 2ϕv,bMs

1− ϕv,bMs

)1/2

(SM.44)

Therefore,d(mv − 1)

dav=

−9

2

(1− ϕv,bMs

1 + 2ϕv,bMs

)1/2 Ms(dbav2

− 2db2

av3+ db

3

av4)

(1− ϕv,bMs)2(SM.45)

where db is the bilayer thickness of the vesicle. Using Eqs. (SM.43) and (SM.45), for av → ∞,

shows that Xv,RDG in Eq. (SM.41) approaches the limit of

Xv,RDG → 18πnm

λ0

dbMs (SM.46)

The RDG scattering regime is valid in practice when Xv,RDG = 0.1 or 0.2. As an example, for

λ0 = 350 nm, nm = 1.333, Ms = 0.0654 (ms = 1.10), and db = 2.4 nm, Xv,RDG = 0.0338.

Hence, the equations of the RDG scattering regime are applicable to all finite values of av for

typical parameter values and for λ0 ranging from 350 to 700 nm.

F. Roots of the Form Factor for Homogeneous Spheres

The first nine roots of Eq. (21) in the main text

3

q3a3[sin (qa)− qa cos (qa)] = 0 (SM.47)

are listed in Table SM1. Thus, the scattered intensity is not zero for qa < 4.4934.

10

Table SM1. The values of qa for the first nine roots of Eq. (21) in the main text.

Root no. qa-root

1 4.49342 7.72533 10.9044 14.0665 17.2216 20.3717 23.5208 26.6669 29.812

G. Comparison of the Exact and Approximate Form Factors of Vesicles

As shown in Fig. SM2, the squares of the exact form factors (Eq. (23) in the main text) and the

approximate form factors (Eq. (24) in the main text) of vesicles with av = 25, 100, or 524 nm

are plotted vs. θ from 0 to π for λ0 = 700 nm. For each size, the two curves overlap to better

than 0.1%. This indicates that the approximate equation is adequately accurate, at least for these

examples.

H. Approximate Form Factor of Liposomes

This form factor is determined from Eq. (25) in the main text [3]

fl =

K∑j=1

Vv,jfv,j

K∑j=1

Vv,j

(SM.48)

where Vv,j is the volume of the jth bilayer, fv,j is the form factor of the jth bilayer, and K is the

number of bilayers in the liposome. The volume of the jth bilayer is given by

Vv,j =4π

3(aj

3 − aji3) (SM.49)

11

3(/)0 30 60 90 120 150 180

f v2

0

0.2

0.4

0.6

0.8

1

av = 25 nm

(i) approximate equation(i*) exact equation

av = 100 nm

(ii) approximate equation(ii*) exact equation

av = 524 nm

(iii) approximate equation(iii*) exact equation

Fig. SM2. Squares of the exact and approximate form factors of vesicles, fv2, vs. scattering anglesfrom 0 to π for λ0 = 700 nm at av = 25, 100, or 524 nm.

where aj = ac + (j − 1)dw + jdb is the outer radius of the jth bilayer (j = 1, 2, 3,...), aji = aj − db

is the inner radius of the jth bilayer, ac is the radius of the central water core, dw is the thickness of

each water layer, and db is the thickness of each bilayer. If we assume ac = dw, for convenience,

then aj = j(dw + db).

As with vesicles, the form factor of the jth bilayer is given approximately by Eq. (24) in the

main text

fv,j ≈sin (qaj)

qaj(SM.50)

where aj = aj − db2is the average radius of the jth bilayer. The magnitude of the scattering vector,

q, is

q ≡ 4πnm

λ0

sin(θ

2

)(SM.51)

in which nm is the refractive index of the medium, λ0 is the wavelength of light in vacuum, and θ

is the scattering angle.

After substituting Eqs. (SM.49) and (SM.50) into Eq. (SM.48), the form factor of a liposome

12

is approximately

fl ≈

K∑j=1

(aj3 − aji

3)sin (qaj)

qaj

K∑j=1

(aj3 − aji3)

(SM.52)

For the small bilayer thickness approximation (aj ≫ db), which is used to obtain Eq. (SM.50), the

term (aj3 − aji

3) becomes

aj3 − aji

3 ≈ 3aj2db = 3j2(dw + db)

2db (SM.53)

Since the term 3(dw + db)2db cancels out of the numerator and denominator in Eq. (SM.52), one

obtains

fl ≈

K∑j=1

j2sin (qaj)

qaj

K∑j=1

j2=

6K∑j=1

j2sin (qaj)

qaj

K(K + 1)(2K + 1)(SM.54)

where [4, p. 1]K∑j=1

j2 =K(K + 1)(2K + 1)

6(SM.55)

In the limit of very large liposomes, or forK ≫ 1, Eq. (SM.52) is simplified as follows:

fl ≈3

K3

K∑j=1

j2sin (qaj)

qaj(SM.56)

I. Integrals of the Form Factors of Homogeneous Spheres and of Vesicles

The dissipation factor of a homogeneous sphere Qs is defined as (see Eq. (33) in the main text)

Qs ≡3

8

∫ π

0

fs2(1 + cos2 θ) sin θdθ (SM.57)

13

For a homogeneous sphere, the form factor fs is given as (see Eq. (21) in the main text)

fs =3

q3a3[sin (qa)− qa cos (qa)] =

3j1(qa)

qa(SM.58)

where a is the radius of a homogeneous sphere, q is defined in Eq. (SM.51), and j1 is the first order

spherical Bessel function,

j1(x) =sinxx2

− cosxx

(SM.59)

To evaluate the integral of Qs in Eq. (SM.57) analytically, a change of variable from θ to q is

used. Moreover, for convenience, the term h is defined as

h ≡ 4πnm

λ0

(SM.60)

Then,

q = h sin(θ

2

)(SM.61)

Thus, for θ ranging from 0 to π, q ranges from 0 to h. From the trigonometric formula, sin θ2=

±(1−cos θ

2

)1/2, the terms (1 + cos2 θ) and sin θdθ in Eq. (SM.57) are related to q as follows

1 + cos2 θ = 2− 4q2

h2+

4q4

h4(SM.62)

and

sin θdθ =4q

h2dq (SM.63)

Then, after substituting Eqs. (SM.58), (SM.62), and (SM.63) into Eq. (SM.57), the integral of Qs

in Eq. (SM.57) becomes

Qs =3

8

∫ h

0

9[j1(qa)]2

q2a2

(2− 4q2

h2+

4q4

h4

)4q

h2dq (SM.64)

14

After splitting the above integral into three separate integrals, we get

Qs =27

h2a2

∫ ha

0

[j1(qa)]2

qad(qa)− 54

h4a4

∫ ha

0

qa[j1(qa)]2d(qa) +

54

h6a6

∫ ha

0

q3a3[j1(qa)]2d(qa)

(SM.65)

From tables of integrals for the spherical Bessel functions [5], the first integral in Eq. (SM.65) is

∫ ha

0

[j1(qa)]2

qad(qa) =

−1− 2xs2 + 2xs

4 + cos (2xs) + 2xs sin (2xs)8xs4

(SM.66)

where

xs = ha =4πnma

λ0

(SM.67)

The second integral in Eq. (SM.65) is [5]

∫ ha

0

qa[j1(qa)]2d(qa)

=−1− 2xs

2 + 2xs2[γE + ln(2xs)− CI(2xs)] + cos (2xs) + 2xs sin (2xs)

4xs2(SM.68)

where γE is the Euler gamma constant and CI(x) is the cosine integral function.

CI(x) = −∫ ∞

x

cos tt

dt (SM.69)

And, the third integral in Eq. (SM.65) is [5]

∫ ha

0

q3a3[j1(qa)]2d(qa)

=−5 + 2xs

2 + 4[γE + ln(2xs)− CI(2xs)] + 5 cos (2xs) + 2xs sin (2xs)8

(SM.70)

15

After substituting Eqs. (SM.66), (SM.68), and (SM.70) into Eq. (SM.65) and rearranging the terms,

we obtain the final analytical expression for Qs in Eq. (35) in the main text.

Qs =27

xs6

{xs

4

4+

5xs2

4+ (1− xs

2)[γE − CI(2xs) + ln(2xs)] +7

8cos (2xs)−

xs4sin (2xs)−

7

8

}(SM.71)

For vesicles, the dissipation factor Qv is from Eq. (33) in the main text,

Qv ≡3

8

∫ π

0

fv2(1 + cos2 θ) sin θdθ (SM.72)

From Eq. (24) in the main text, the approximate form factor fv is

fv ≈sin (qav)

qav(SM.73)

in which, av = av+avi2

, is the average radius of the bilayer in a vesicle, and avi, the inner radius of

the vesicle. After substituting Eqs. (SM.73), (SM.62), and (SM.63) into Eq. (SM.72), the integral

in Eq. (SM.72) becomes

Qv ≈3

8

∫ h

0

sin2 (qav)q2a2v

(2− 4q2

h2+

4q4

h4

)4q

h2dq (SM.74)

This integral is separated into three integrals,

Qv ≈3

h2a2v

∫ hav

0

sin2 (qav)qav

d(qav)−6

h4a4v

∫ hav

0

qav sin2 (qav)d(qav)

+6

h6a6v

∫ hav

0

q3a3v sin2 (qav)d(qav) (SM.75)

From tables of integrals [4, p. 220], the first integral is

∫ hav

0

sin2 (qav)qav

d(qav) =1

2[γE + ln(2xv)− CI(2xv)] (SM.76)

16

where

xv = hav =4πnmav

λ0

(SM.77)

The second integral is [4, p. 217]

∫ hav

0

qav sin2 (qav)d(qav) =x2v

4− xv

4sin (2xv)−

1

8cos (2xv) +

1

8(SM.78)

And, the third integral is [4, p. 217]

∫ hav

0

q3a3v sin2 (qav)d(qav)

=x4v

8− x3

v

4sin (2xv) +

3

8xv sin (2xv)−

3

8x2v cos (2xv) +

3

16cos (2xv)−

3

16(SM.79)

After substituting Eqs. (SM.76), (SM.78), and (SM.79) into Eq. (SM.75) and rearranging the terms,

the final analytical expression of Qv in Eq. (81) is obtained

Qv ≈3

x6v

{− x4

v

4− x2

v

4+

x4v

2[γE − CI(2xv) + ln(2xv)] +

3− 4x2v

8cos (2xv) +

3xv4

sin (2xv)−3

8

}(SM.80)

17

J. Wavelength Exponents for Large Vesicles and Liposomes

As shown in Fig. SM3, for vesicles with radii larger than 1000 nm, the wavelength exponent gv for

vesicles oscillates between 2.0 to ca. 2.5 or 2.0 to ca. 2.4 for av = 1048 or 2620 nm, respectively.

The amplitudes gradually decrease with increasing particle size, as explained in the main text. If

we use the RDG equations beyond the range of validity of ca. 800 nm for liposomes, we can

predict gl approximately. The gl values are nearly constant at the chosen sizes, between 2.004 to

2.04 for al = 1048 nm, and between 2.000 to 2.004 for al = 2620 nm. This result is consistent

with the analytical results in Section 3.4 in the main text, for the case in which nm, ns, and ms are

independent of the wavelength. The value of gp ranges between 4 and 2 for any size of vesicles or

liposomes in the RDG regime.

60(nm)400 500 600 700

g p

1.8

2

2.2

2.4

2.6

(i)

(i*)

(i) al = 1048 nm (i*) av = 1048 nm(ii)

(ii*)

(ii) al = 2620 nm (ii*) av = 2620 nm

Fig. SM3. Wavelength exponents vs. wavelengths from 350 to 700 nm for the RDG regime fordispersions of monodisperse liposomes with the number of bilayers K = 20 and 50 and vesiclesof av = 1048 and 2620 nm.

18

K. Specific Rayleigh Ratios at Various Scattering Angles

The results of the specific Rayleigh ratio Rθ∗∗ for dispersions of monodisperse homogeneous

spheres, vesicles, and liposomes at θ = 45◦, 90◦, and 135◦ and λ0 = 700 nm are shown in Figs.

SM4 to SM9. For homogeneous spheres with a ≤ 30 nm, the R and RDG predictions are nearly

the same, as expected. The values ofRθ∗∗(45◦),Rθ

∗∗(90◦), andRθ∗∗(135◦) for vesicles are smaller

than those for homogeneous spheres of the same radius, again as expected. For vesicles, in the RDG

regime, Rθ∗∗(45◦), Rθ

∗∗(90◦), and Rθ∗∗(135◦) oscillate with increasing av, because of substantial

intraparticle interference, which is so strong that fv reaches zero values. The zeros result from the

sin (qav) term. For liposomes, Rθ∗∗(45◦), Rθ

∗∗(90◦), and Rθ∗∗(135◦) are calculated for discrete

values of the number of bilayers K. These specific Rayleigh ratios also oscillate, and their am-

plitudes gradually decrease with increasing al, because of the increasing intraparticle interference

as more bilayers are added to the liposome. Such results may be useful in analyzing vesicle sizes

and choosing the appropriate scattering angle or the wavelength to achieve significant scattered

intensities for DLS experiments.

For liposomes with al = 524 nm, the results of Rθ∗∗ are shown in Fig. SM10. There are three

zero values of Rθ∗∗, at θ ≈ 40◦, 72◦, and 112◦. For comparison, vesicles with the same size have

zero values of Rθ∗∗ at θ ≈ 29◦, 60◦, and 98◦.

19

av or a(nm)0 200 400 600 800 1000

R3$$(4

5/)(

m!

1)

0

2

4

6

8

10(i) vesicles, RDG

(ii) spheres, RDG

(i)

(ii)

(i*) vesicles, R

(ii*) spheres, R

(i*)

(ii*)

Fig. SM4. Specific Rayleigh ratios vs. particle radii at θ = 45◦ and λ0 = 700 nm for the R andRDG regimes for dispersions of monodisperse vesicles and homogeneous spheres. The R modelfor vesicles deviates significantly from the RDG model for av ≥ 80 nm.

av or a(nm)0 200 400 600 800 1000

R3$$(9

0/)(

m!

1)

0

2

4

6

8

10

(i) vesicles, RDG

(ii) spheres, RDG

(i)

(ii)

(i*) vesicles, R

(ii*) spheres, R

(i*)

(ii*)

Fig. SM5. Specific Rayleigh ratios vs. particle radii at θ = 90◦ and for λ0 = 700 nm for the R andRDG regimes for dispersions of monodisperse vesicles andhomogeneous spheres. The R modelfor vesicles deviates significantly from the RDG model for av ≥ 50 nm.

20

av or a(nm)0 200 400 600 800 1000

R3$$(1

35/)(

m!

1)

0

2

4

6

8

10(i) vesicles, RDG

(ii) spheres, RDG

(i)

(ii)

(i*) vesicles, R

(ii*) spheres, R

(i*)

(ii*)

Fig. SM6. Specific Rayleigh ratios vs. particle radii at θ = 135◦ and λ0 = 700 nm for the R andRDG regimes for dispersions of monodisperse vesicles and homogeneous spheres. The R modelfor vesicles deviates significantly from the RDG model for av ≥ 40 nm.

al or av(nm)0 200 400 600 800

R3$$(4

5/)(

m!

1)

0

5

10

15

20

25

30

(i) liposomes, RDG

(ii) vesicles, RDG

Fig. SM7. Specific RDG Rayleigh ratios vs. particle radii at θ = 45◦ and λ0 = 700 nm fordispersions of monodisperse liposomes and vesicles. Predictions are shown for discrete values ofal, corresponding to the number of bilayersK = 1 to 15.

21

al or av(nm)0 200 400 600 800

R3$$(9

0/)(

m!

1)

0

0.5

1

1.5

2

2.5

3

(i) liposomes, RDG

(ii) vesicles, RDG

Fig. SM8. Specific RDG Rayleigh ratios vs. particle radii at θ = 90◦ and for λ0 = 700 nm fordispersions of monodisperse liposomes and vesicles. Predictions are shown for discrete values ofal, corresponding to the number of bilayersK = 1 to 15.

al or av(nm)0 200 400 600 800

R3$$(1

35/)(

m!

1)

0

0.5

1

1.5

2

2.5

3

(i) liposomes, RDG

(ii) vesicles, RDG

Fig. SM9. Specific RDG Rayleigh ratios vs. particle radii at θ = 135◦ and λ0 = 700 nm fordispersions of monodisperse liposomes and vesicles. Predictions are shown for discrete values ofal, corresponding the number of bilayersK = 1 to 15.

22

3(/)0 30 60 90 120 150 180

R3$$(m

!1)

10-3

10-2

10-1

100

101

102

103 (i) liposomes, al = 524 nm

(ii) vesicles, av = 524 nm

(i)

(ii)

Fig. SM10. Specific RDG Rayleigh ratios vs. scattering angles for λ0 = 700 nm for dispersions ofmonodisperse liposomes with al = 524 nm and vesicles with av = 524 nm. For liposomes, thereare three zero values at θ ≈ 40◦, 72◦, and 112◦. For vesicles, there are three zero values at θ ≈ 29◦,60◦, and 98◦.

23

L. Specific Turbidities andWavelength Exponents of Vesicles and Liposomes with Interme-

diate Sizes

For liposomes with al = 262 or 524 nm, τ ∗∗ ranges from 2.3×104 to 1.0×105 m−1 or 5.0×104

to 2.0×105 m−1. For vesicles with av = 262 or 524 nm, τ ∗∗ ranges from 8.5×103 to 4.3×104

m−1 or 1.1×104 to 5.2×104 m−1 with a larger oscillation than liposomes because of the stronger

dependence of Qv on wavelength (see Fig. SM11). For liposomes with al = 262 nm, for λ0 =

350 to 700 nm, τ ∗∗ = 1×105 to 2.3×104 m−1. The ratio of the specific turbidities is 4.3, which

significantly deviates from the R limit of 16.

For liposomes with al = 262 or 524 nm, for λ0 = 350 to 700 nm, gl ranges from 2.1 to 2.3

or from 2.0 to 2.1 (see Fig. SM12). For λ0 = 350 to 700 nm, gl oscillates and increases slightly.

For vesicles with av = 262 or 524 nm, gv oscillates from 2.0 to 2.7 or 2.0 to 2.6. The amplitudes

of the oscillating gl and gv decrease with increasing particle radius (Fig. SM12), as can be inferred

from Eqs. (43) and (44) in the main text. As av increases at a fixed λ0, the ratio of the leading

terms in the numerator and the denominator approaches 4. This results in gv ≈ 2 for very large

vesicles. For liposomes, gl approaches the limit of 2 with increasing al because of a combined

intraparticle interference effect between the bilayers, as discussed in Section 3.4 in the main text.

Some additional sample calculations of gp for very large monodisperse vesicles and liposomes are

shown in SM, J.

24

60(nm)400 500 600 700

=$$(m

!1)

104

105

(i) al = 262 nm

(i*) av = 262 nm

(ii) al = 524 nm

(ii*) av = 524 nm

Fig. SM11. Specific turbidities vs. wavelengths from 350 to 700 nm for the RDG regime fordispersions of monodisperse liposomes with the number of bilayers K = 5 and 10 and vesicleswith av = 262 and 524 nm.

60(nm)400 500 600 700

g p

1.6

1.8

2

2.2

2.4

2.6

2.8

3

(i)

(i*)

(i) al = 262 nm (i*) av = 262 nm

(ii)

(ii*)

(ii) al = 524 nm (ii*) av = 524 nm

Fig. SM12. Wavelength exponents vs. wavelengths from 350 to 700 nm for the RDG regime fordispersions of monodisperse liposomes with the number of bilayers K = 5 and 10 and vesicleswith av = 262 and 524 nm.

25

M. Sensitivity of the Estimation of Vesicle Sizes to the Relative Refractive Index and the

Bilayer Thickness

The average DDAB vesicle sizes were also calculated from slightly different values of the relative

refractive index ms, 1.09 or 1.11, and of the bilayer thickness db, 2.3 or 2.5 nm. For ms = 1.09

at db = 2.4 nm, the values of the average radii av∗ for all wavelengths range from 130 to 167 nm

for wDDAB = 0.0025, and from 129 to 162 nm for wDDAB = 0.0010. For ms = 1.11 at db = 2.4

nm, av∗ for all wavelengths range from 64 to 79 nm for wDDAB = 0.0025, and from 65 to 79 nm

for wDDAB = 0.0010, for all wavelengths. For db = 2.3 nm at ms = 1.10, av∗ ranges from 104 to

113 nm for wDDAB = 0.0025, and from 101 to 116 nm for wDDAB = 0.0010. For db = 2.5 nm at

ms = 1.10, the av∗ values range from 89 to 95 nm for wDDAB = 0.0025, and from 91 to 95 nm for

wDDAB = 0.0010. The values of av∗ for the two surfactant weight fractions are consistent. Hence,

av∗ is quite sensitive to the value ofms, but less sensitive to the value of db. Therefore, the results

indicate that for accurate estimation of vesicle sizes one needs to use more accurately measured

values of ns, nm, and ms for each wavelength. Such issues need to be resolved in future detailed

studies of the inverse scattering problem, which is out of the scope of this article.

26

Table SM2. Average radii of DDAB vesicles estimated from specific turbidities for samples at (iii)wDDAB = 0.0025 and (iv) 0.0010 at several wavelengths for various values of ms (1.09, 1.11) anddb (2.3, 2.5 nm).

λ0 (nm) av∗ (nm) (ms = 1.09, db = 2.4 nm) av

∗ (nm) (ms = 1.11, db = 2.4 nm)

(iii) (iv) (iii) (iv)

700 162±2 162±2 79±1 79±1650 157±1 156±7 76±1 75±3600 151±1 152±5 73±1 73±2550 143±1 144±3 69±1 70±2500 135±1 137±3 66±1 66±2450 130±1 130±2 64±1 65±1400 135±1 129±1 68±1 66±1350 167±1 161±2 79±1 75±1

λ0 (nm) av∗ (nm) (ms = 1.10, db = 2.3 nm) av

∗ (nm) (ms = 1.10, db = 2.5 nm)

(iii) (iv) (iii) (iv)

700 110±2 111±2 94±1 94±2650 111±2 111±10 92±1 91±5600 113±1 116±8 90±1 93±6550 113±1 114±4 89±1 91±4500 109±1 111±4 89±2 91±6450 107±1 107±1 93±1 93±2400 104±1 103±1 94±1 93±1350 105±1 101±1 95±1 92±1

27

References

[1] C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley,

New York, 1983.

[2] J. D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, New York, 1999.

[3] J. S. Pedersen, Analysis of small-angle scattering data from colloids and polymer solutions:

Modeling and least-squares fitting, Advances in Colloid and Interface Science 70 (1-3) (1997)

171–210. doi:10.1016/S0001-8686(97)00312-6.

URL https://www.sciencedirect.com/science/article/pii/S0001868697003126

[4] K. S. Kolbig, I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, I. Scripta Technica, Table of Integrals,

Series, and Products., 7th Edition, Elsevier, Amsterdam ; Boston, 2007. doi:10.2307/2153347.

[5] J. K. Bloomfield, S. H. P. Face, Z. Moss, Indefinite Integrals of Spherical Bessel Functions,

arXiv preprint arXiv:1703.06428 (2017).

28