Structure of Artificial Cytoskeleton Containing ... - ACS Publications

Biomacromolecules 2002, 3, 565-578

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Structure of Artificial Cytoskeleton Containing Liposomes in Aqueous Solution Studied by Static and Dynamic Light Scattering Oliver Stauch and Rolf Schubert Department of Pharmaceutical Technology, Albert Ludwigs University of Freiburg, 79104 Freiburg, Germany

Gabriela Savin†,‡ and Walther Burchard*,† Institute of Macromolecular Chemistry, Albert Ludwigs University of Freiburg, 79104 Freiburg, Germany, and Transilvania University, 2200 Brasov, Romania Received January 10, 2002; Revised Manuscript Received March 4, 2002

The structure of three types of liposomes (egg yolk phosphatidylcholine (EPC) without modification and EPC vesicles containing cross-linked N-isopropylacrylamide (NIPAM) networks of low and a high concentration inside the vesicles) were analyzed by static and dynamic light scattering. Upon polymerization the network was assumed to become attached to the membrane by reactive anchoring monomers. For the sample of high poly(NIPAM) content the polymer network was assumed to fill the whole space in the vesicles. The issue of the present study was to examine hard and hollow sphere behavior of the liposomes with networks of high and low poly(NIPAM) content. The theoretical scattering curves differ markedly for uniform hard and uniform hollow spheres by the presence of specific peaks. However, polydispersity washed out the peaks and led to smoothed asymptotes with fractal dimensions of df ) 2 for hollow and df ) 4 for hard spheres. The experimental data could efficiently be fitted with weakly polydisperse hollow spheres. No clear conclusion could be drawn from the angular dependence alone for the liposome of high poly(NIPAM) content. The two wavelengths from the HeNe and Ar lasers proved to be too long for the studied liposomes of about 100 nm in radius. However, evidence for hollow sphere behavior was found for fractionated liposomes from the ratio F ) Rg/Rh ) 1.04 ( 0.02 (theory F ) 1.00 for hollow spheres). Finally, from the molar mass and the sphere radius, an apparent density was determined. The analysis gave the expected density for the pure EPC lecithin vesicles and a poly(NIPAM) network density of 0.244 g/mL. For the liposome of low poly(NIPAM) content the network appeared to be attached to the inner surface of the lecithin shell to form a layer of about 18 nm thickness. Introduction In a previous paper1 we reported some results of attempts to stabilize the bilayer of phospholipid vesicles via network polymerization around the inner surface of the phospholipid shell. On the preparation of such vesicles, three components were added which were (i) an anchoring monomer 1,2distearyl-3-octaethyleneglycol ether methacrylate (DOGM) that was integrated in the double layer of egg yolk phosphatidylcholine (EPC), (ii) N-isopropylacrylamide (NIPAM), and (iii) tetraethyleneglycol dimethacrylate (TEGDM). The two encapsulated monomers were copolymerized by UV-C irradiation, where TEGDM acted as cross-linker, using 2,2diethoxyacetophenone (DEAP) as photoactive initiator. During this reaction the pendant double bonds of the DOGM component acted as anchoring couplers for the poly(NIPAM) network onto the inner surface of the vesicle. Depending on the molar ratio of the components, three main types of vesicle structures should be realized. These were (i) a threedimensional network that filled the whole space of the † ‡

Albert Ludwigs University of Freiburg. Transilvania University.

vesicle, (ii) small microgels in the interior space which appeared not being connected to the inner vesicle surface, and (iii) a pseudo-two-dimensional network attached to the inner surface of the vesicle. The pseudo-two-dimensional network may be considered as a mimic for the spectrin network that stabilizes the membrane of red blood cells, and the network, which fully spans the inner space of the vesicle, can be taken as a model for cytoskeletons which keep a cell in its swollen state.2-5 The biological aspect and possible application of liposomes in pharmacy and medicine stand in the foreground.6-9 Interest was focused in particular on thermal- and pH-dependent systems which would allow triggering the shape and structure of the liposomes for release of encapsulated drugs. Such thermal sensitivity was obtained with water-soluble polymers, notably poly(NIPAM), which exhibits phase separation on heating. Effects on the liposomes were achieved with such polymers, attached or nonattached onto the surface of the vesicles.6-9 The present paper focused on structure and examined procedures for characterizing the chemically modified lip-

10.1021/bm0200074 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/18/2002

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Biomacromolecules, Vol. 3, No. 3, 2002

osomes. In the previous publication the characterization was mainly made by transmission and atomic force microscopy. In these cases the liposomes had to be made water-free, and the samples had to be fixed onto a supporting grid. Liposomes are fairly stable and do not break into pieces during the preparation and the ensuing determination of structure in the electron microscope. The highly advanced cryo-transmission electron microscopy (cryo-TEM) certainly helped to avoid artifacts, but it gives only information on the outer shape. The question remained as to whether the same structure is observed in the aqueous medium and how detailed information can be gained on the internal structure mainly of the poly(NIPAM) network. Structures in solution can be determined by ultracentrifugation procedures, by combination of static and dynamic light scattering, and by small-angle neutron (SANS) or small-angle X-ray scattering (SAXS). In this paper we applied light scattering techniques. From these data we tried to extract information on the shell thickness in order to find out whether the liposomes consist only of a single bilayer or of several ones. After polymerization inside the liposome it was of interest to find out whether this network was attached to the bilayer as a thin shell or whether it fully extends through the available space in the vesicle. Sometimes electron microscopic photographs unconsciously give the impression that all particles may have the same principle structure. This, however, is only rarely the case. Mostly, selected examples are chosen to demonstrate structures of interest. The possibility of choosing special examples is an advantage of electron microscopy. In solution, however, ensembles of particles are probed. Superimposed to the Brownian motion, internal motions with respect to the center of mass can be present.10-12 For the liposomes these could be fluctuations in the outer shape. These will be observable in particular with giant unilamellar vesicles (GUV),13 1-10 µm in radius, but will not be significant with large unilamellar vesicles (LUV),14,15 50-100 nm in radius, which in this paper were the subjects of research. Still, deformation of the spherical shape was predicted under the action of osmotic pressure inside and outside the vesicles16,17 and was confirmed experimentally by various techniques,18-20 recently, in a careful light scattering study by the group of Hallett.19,21 The ensemble behavior brought up problems in the present study which were not expected to occur with these liposomes. Hollow spheres have much lower mass than the corresponding compact spheres of the same sphere radius. Therefore, the difference between these two structures should easily be distinguishable by static light scattering, but only if the absolute scattering intensity is measured. For a quantitative analysis the knowledge of the concentration and the refractive index increment is required, which remained a serious problem in the present case (see below). Furthermore, both architectures display a markedly different angular dependence of the scattered light that is particularly distinct for uniform particles. The present liposome system possessed a significant polydispersity and demanded much additional work for the analysis. Hallett et al.22 were faced with this problem already 12 years ago when they realized that at least two types of

Stauch et al.

heterogeneity have to be taken into account. (i) Clustering of several liposomes can occur keeping intact the spherical shape of the individual liposomes. (ii) In addition, the actual variation in radius of the unilamellar vesicles will be present. This observation was confirmed in our measurements. To our regret, until recently, we had no knowledge of Hallett’s fundamental work,19,21,22 which would have been helpful in planning our experiments. Our interest was not focused on the action of osmotic forces onto the shape of the vesicles but rather on the internal structure of the liposomes. To receive the desired information, the angular dependence of the particle scattering factor was more thoroughly studied than was possible with the fairly small vesicles used by Pencer et al.21 The paper is organized as follows. The preparation of the samples and estimations of their composition are given in the experimental part. This part also contains details on the used equipment. After an outline of the theoretical basis for the present type of light scattering, the results are presented together with some preliminary comments. The complexity of the obtained structures is discussed then in detail in a separate section. Substantial mathematical work had to be invested to attain a consistent interpretation. Details are given in an Appendix. Experimental Section The experimental procedures of vesicle preparation and polymerization of encapsulated monomers were already given in paper 1.1 Vesicle size and lamellarity thickness and the size distribution of the liposomes sensitively depend on the preparation technique.21,22 Therefore, some essential details are given once again. Preparation of Monomer Entrapped Liposomes. The preparation of the vesicles was achieved by the detergent removal technique using N-octyl-β-glucopyranoside (OG) from Fluka, Switzerland. The technique has been known for about 20 years, is well documented, and was critically tested by several groups.23-25 In brief, the procedure was as follows: A thin film of a mixture of EPC, 2 mol % DOGM, and the surfactant was prepared. The lipid/detergent ratio was 0.2 mol of lipid/1 mol of surfactant. The film was suspended in 1 mL of an aqueous solution of 1 M NIPAM and 0.025 M TEGDM. The final lipid concentration was 20 mM. The mixed micelle solution was dialyzed at room temperature for 12 h against deionized water (100 mL of water per 1 mL of mixed micelles) using a dialysis chamber of MiniLipoprep (Diachema AG, Switzerland). A cellulose membrane from Diachema was used with a cutoff of Mw ) 10 000 g/mol. Water was replaced after 1, 3, and 5 h. Upon dialysis the surfactant became removed and vesicles were formed from the mixed micelles after about 90 min as monitored by light scattering. The loss of NIPAM and TEGDM for the vesicles during the vesicle formation was estimated by UV spectrometry being less than 50%. After dialysis the residual amount of surfactant and of nonencapsulated monomer proved to be negligibly small. This detergent removal technique gave more reproducible vesicle sizes than the more often applied extrusion method.23-25

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Structures of Chemically Modified Liposomes

Cross-Linking Polymerization of Encapsulated Monomers. The monomer entrapped vesicles were transferred into a 5 mL round-bottom quartz flask of 2.3 mm wall thickness. 2,2-Diethoxyacetophenon (DEAP) was added as photoactive initiator to a final concentration of 10-5 M. The initiator had to penetrate the shell of the vesicle which required a certain amount of time. Optimum conditions for the preparation of polmer networks which filled the whole space of the vesicle were obtained with incubation times of about 12 h at 4 °C in the dark. The amount of penetrated initiator was estimated by the amount of network formed by the action of the initiator. To this end thin slices from the lipomes were made after polymerization and atomic force microscopy was performed. (For details, see paper 1.) For the vesicle with a network to be formed underneath the inner shell of the vesicle, the polymerization was started immediately after adding the photoinitiator. Finally the aqueous system was degassed five times and flushed with argon. The polymerization was performed by UV irradiation at a distance of 5 cm for 3 h with a wavelength of λ ) 254 nm from a 30 W low pressure mercury lamp of Bresemann+Schorpp, Heiligenstadt, Germany. During the irradation the suspension was magnetically stirred and cooled by flushing the reaction vessel with a constant flow of cold air to avoid phase separation by keeping the system below 30 °C. The final phospholipid concentration ranged between 12 and 20 mM and was determined by their phosphorus content.26 Fractionation of Vesicles by Gel Permeation Chromatography (GPC). Nonencapsulated monomers were removed by preparative gel permeation using SEPHAROSE 4B-CL of Pharmacia Fine Chemicals, Sweden, and deionized water (Ultrapure Water System, Millipore) as eluent. Static Light Scattering. Static light scattering measurements were carried out with two modified SOFICA photogoniometers. The one was equipped with a He-Ne laser (Uniphase) which gave red light of λ0 ) 632.8 nm and the other was an Ar laser (Uniphase) which gave blue light of λ0 ) 488 nm. Both instruments were computer driven (G. Baur, Instrumentenbau, Hausen, Germany). Measurements were made in the angular range from 30 to 145° in steps of 5°. The temperature was 20 °C. The refractiVe index increments were determined with a Brice-Phoenix differential refractometer. Freshly isolated egg phosphatidylcholine (EPC-lecithin) and a radically polymerized N-isopropylamide (NIPAM) were used. The lecithin is soluble in ethanol and in tetrahydrofuran (THF) and was measured in these two solvents. The dn/dc data were then plotted against the refractive index of the solvents, and the curve was linearly extrapolated to the refractive index of water from where the dn/dc of the lecithin in water was obtained. The corresponding refractive index increments of poly(NIPAM) could directly be measured in water. The results are listed in Table 1. The extrapolation of the lecithin data against dn/dc ) 0 also gave a rough estimation of the lecithin refractive index whose value is nlecithin ) 1.482 ( 0.008. Dynamic Light Scattering. Dynamic light scattering was performed with an ALV goniometer that was equipped with an ALV-5000 correlator (ALV-Laservertrieb, Langen, Ger-

Table 1. Refractive Index Increment dn/dc of Lecithin and NIPAM in Various Solvents and the Refractive Indices of These Solventsa

a

solvent

dn/dc (mL/g)

n0

ethanol THF water

Lecithin (EPC) 0.1335 ( 0.0005 0.0844 ( 0.0044 0.1611 ( 0.0019

1.359 1.404 1.333

water

NIPAM 0.1427

1.333

The errors are estimations from the results from two different people.

many). The red light of a krypton ion laser with λ0 ) 647.1 nm was used. Measurements were made in the angular range of 30-145° in steps of 15°. The “Static and Dynamic” light scattering mode was chosen which enabled recording the static light scattering intensity in terms of Rayleigh ratio and simultaneously the scattering intensity time correlation function (TCF). We then evaluated the time correlation function with two techniques, i.e., (i) the KohlrauschWilliams-Watts27,28 (KWW) procedure of stretched exponential and (ii) the CONTIN Laplace inversion method,29 where the latter gave the frequency of relaxation times as a function of the reciprocal relaxation time in a logarithmic scale. The reciprocal relaxation time is related to the translational diffusion coefficient as (1/τ) ) Dq2, where q ) (4πn0/λo) sin(θ/2) is the magnitude of the scattering vector. The viscosity of water was taken as 1.0063 cP ) 10.063 mPa‚s. The water was taken from the Ultrapure Water System of Millipore. The solutions were considered as being dust-free and were not filtered. Theoretical Section There are two methodologies to decide which of the two structures, hard or hollow spheres, is present, when scattering techniques are applied. In the first procedure the angular dependence of the particle scattering factors is considered; in the second, the actual density of the dissolved material is determined, which requires the knowledge of the molar mass of the particle, Mw, and the radius, R, of the sphere. The conclusion can be further confirmed by comparing the radii Rg and Rh as determined by static and dynamic light scattering, respectively. No absolute intensity measurements are required in the first method. The scattering intensity is measured in arbitrary units at a number of selected scattering angles θ and is normalized by the scattering intensity at the scattering angle θ ) 0. This normalized angular dependence P(θ) ≡

i(θ) i(θ)0)

(1)

is the particle structure or particle scattering factor, sometimes also called form factor.30 P(θ) gives information on the shape and internal structure of the particles in solution. Specific differences between various shapes become apparent in the large qRg regime. Careful absolute scattering intensity measurements are needed in the second procedure. A quantitative measure for this scattering intensity is the Rayleigh ratio31,32

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Rθ ≡ (i(θ)/I0)r2

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(2)

where I0 is the primary beam intensity. The ratio i(θ)/I0 is multiplied by the square of the distance r of the detector from the scattering volume, because the scattering intensity decreases with the square of the distance, which is compensated in the Rayleigh ratio by this r2 product. At very dilute solutions the Rayleigh ratio at zero scattering angle is proportional to concentration c times molar mass Mw of the particles31,33 Rθ)0 ) KcMw for cf0

(3)

The proportionality constant K is a contrast factor which significantly depends on the square of the refractive index increment dn/dc. The latter has to be measured separately with an appropriate differential refractometer. The two possibilities of how to distinguish hollow and hard spheres are now outlined in detail. Particle Scattering Factors. The particle scattering factors for uniform hollow and hard spheres are long known and are given by eqs 4 and 5. Hollow sphere (infinitely thin shell)34 P(θ) )

(sinX X)

2

Figure 1. Dependence of the particle scattering factors P(u) ≡ i(u)/ i(u)0) on the parameter u ) qRg of uniform hollow34 and hard spheres,35 of uniform random coils40 and according to the DebyeBueche41 space correlation function, where q ) (4πn/λ0) sin(θ/2) is the value of the scattering vector, n the refractive index of the medium, λ0 the wavelength used, and θ the scattering angle. Both, P(u) and u are dimensionless parameters and allow the presentation of light and SANS experiments in one common diagram.

(4)

X ) qR ) qRg Hard sphere (homogeneous density)35 P(θ) )

[

3 (sin X - X cos X) X3

]

2

(5)

X ) qR ) (5/3)1/2qRg where R is the sphere radius and Rg is the radius of gyration. The value of the scattering vector q is given by q)

(4πλ) sin θ2

(6)

Here, λ ) λ0/n is the wavelength in the solution, where λ0 is that in the vacuum and n is the refractive index of the solution. Note, the parameter X ) qR is a number and has no dimension. Figure 1 shows these two functions in dependence on qRg ) u. Both functions rapidly decay to very small values when u is increased. The difference between the two particle scattering factors would scarcely be detectable in such a plot. Figure 1 shows that the initial part at small X values looks very similar for both structures. In fact, as was first shown by Guinier36 and later extensively used by Zimm and Stockmayer37 and Debye38 the initial part has the same angular dependence for all structures and is given by 1 P(q) ) 1 - Rg2q2 + ‚ ‚ ‚ 3

(7)

For the infinitely thin hollow sphere the radius of gyration Rg equals the sphere radius R, but for the homogeneous hard sphere, Rg ) (3/5)1/2R is smaller. It is appropriate to use Rgq ) u as a universal scaling parameter, because then measure-

Figure 2. The same functions as shown in Figure 1 in the normalized Kratky presentation.39 The weak differences between hollow and hard sphere are now clearly distinguishable.

ments at different wavelengths and even results from SANS and SAXS can be presented in the same plot. Figure 2 shows the angular dependencies of the two sphere structures as a function of u, now in the form of a Kratky plot,39 in which u2P(u) is plotted against u. This form of presentation has the advantage that the very low scattering intensities at large scattering angles are amplified by the weight u2. This makes the difference between the two sphere structures more evident. For comparison Figures 1 and 2 also contain the particle scattering factors of uniform random coils40 and structures which obey the Debye-Bueche space correlation function.41 Apparent Particle Density. Generally speaking, the density is a quantity defined as the ratio of mass to volume. In the present case the volume refers to the material of the liposome (shell + polymer network) excluding the solvent inside the vesicle. For hard, homogeneous spheres one finds dhardsph. )

Mw Mw Mw ) ) 3 4π 3 N 1.947 R 4π 3 A g NA NA R R 3 3 h

(8)

The corresponding relationship for the hollow sphere is



somewhat more complex, since the material is assembled only in a thin shell (∆R ) R - Rin). Thus, one has Mw

dhollowsph. ) NA

4π 3 (R - Rin3) 3

(9a)

Let h(Ri) be the probability distribution of finding a certain radius Ri. This distribution is usually derived from a histogram H(Ri) which represents the frequency of finding radii in the range of Ri to Ri+1. Then the distribution h(Ri) is given as h(Ri) )

or, since for thin shells Rin3 ) (R - ∆R)3 = R3 - 3R2∆R dhollowsph. =

Mw 2

NA4πR ∆R

H(Ri) n

(14)

H(Ri) ∑ i)1

(9b) such that

The density remains still an apparent one, notably for liposomes, because of the water-swollen thin bilayer shell of 3.5 nm in thickness,42 which results in a considerably lower value of the apparent density than the actual density of liquid alkanes. F-Parameter (F ) Rg/Rh). When in addition to static light scattering also dynamic light scattering is applied, the results obtained from the particle scattering factor and from the density calculation can be checked. In dynamic light scattering the translational diffusion coefficient D is measured. According to the Stokes-Einstein relationship43

n

h(Ri) ) 1 ∑ i)1

(14′)

The upper limit, n, in the sums is the number of the different classes in the histogram. With this distribution the following averages are obtained n

h(Rgi)Rgi2)(1/2) ∑ i)1

Rgz ) (

(15)

n1

kT at c f 0 D) 6πη0Rh

(10)

the diffusion coefficient is related to a hydrodynamically effective radius Rh. The hydrodynamic particle radius differs characteristically from the geometrically defined radius of gyration Rg. It is useful to consider the ratio44-46 F ≡ Rg/Rh

(11)

because the molar mass dependencies cancel largely. The F-parameter depends on the average segment density in the particle and thus gives valuable information on the internal structure. For the hollow sphere one has Rh ) R ) Rg, but for the homogeneous sphere, Rg is smaller and one has Fhollow-sph. ) 1.0

(12)

Fhard-sph. ) 0.775

(13)

The difference appears small. However, the hydrodynamic radius can be measured with accuracy of (3% and the radius of gyration with an average error of (6% resulting in an error for the F-parameter of (7%. Therefore, a cross check of the results from the particle scattering factor P(q) and the apparent segment density dapp should be possible. Note, ellipsoids and other deformed spherical structures will give F-parameters slightly different from those in eqs 12 and 13, smaller as well as larger ones.46 Therefore, the F-parameter permits only an approximate estimation of shapes. Polydispersity. Most colloidal and macromolecular objects are not uniform in size. The dimensions and the molar masses in the system vary somewhat in value. Such polydispersity has an effect on measurable quantities, notably on the radius of gyration and the particle scattering factor. Often the effect is of minor relevance, but in a quantitative analysis the influence of polydispersity has to be taken into account.21

Rhz ) [

h(Rhi)(1/Rhi)]-1 ∑ i)1

(16)

n

Pz(uz) )

h(Rgi)Pi(ui) ∑ i)1

(17)

in which subscript z indicates the z-averages. Note, the scaled scattering vector ui ) qRgi for individual fractions now differs from uz ) qRgz of the polydisperse system. In the present study the distribution of the hydrodynamic radii is determined by Laplace inversion of the measured time correlation function in dynamic light scattering using the CONTIN program by Provencher.29 This distribution can be considered as being identical in shape with that for the radii of gyration, but it differs from the size distribution as determined from electron microscopic photographs which represents a number distribution. Once the size distribution is known, the influence of polydispersity on the particle scattering factor can be calculated from the functions of the uniform structures. This will be compared with the experimental findings. Results Three series of measurements have been performed. In the first one the liposomes were not fractionated but only freed from the polymers and monomers outside the vesicles. We compared vesicles containing no, a small, and a large amount of poly(NIPAM), respectively. For the vesicles of low poly(NIPAM) content an attached pseudo-two-dimensional gel was expected. The vesicles should show behavior of hollow spheres, similar to that of the natural EPC vesicles. On the contrary, for high poly(NIPAM) content the crosslinked gel will be extended all over the inner compartment of the vesicle and could resemble hard spheres. Because of the much lower density of the poly(NIPAM) network than

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Table 2. Molecular Parameters of Different Liposomesa

Mw × 10-6 (g/mol)

composition EPC/NIPAM

mEPC

dn/dc (mL/g)

EPC EPC/NIPAMlow EPC/NIPAMhigh

25 mM 25 mM/30 mM 12 mM/300 mM

1.0000 0.8118 0.2159

0.161 0.158 0.147

EPC FR 7 EPC FR 10

25 mM 25 mM

1.0000 1.0000

Fractionated Vesicles without NIPAM 0.161 92.4 0.161 96.6

EPC/NIPAMhigh FR 4 EPC/NIPAMhigh FR 5

12 mM/300 mM 12 mM/300 mM

0.2159 0.2159

Fractionated Vesicles with NIPAMhigh 0.147 86.2 ( 1.8 0.147 98.2 ( 3.0

sample

Rg (nm)

Nonfractionated Vesicles 444 131 792 ( 197 175 ( 25 616.5 ( 69.3 140.3 ( 18.2

s ) σ2/Rav2 (a) µ2 ) (〈R2〉 - 〈R〉2)/〈R〉2 (b)

Rh (nm)

F ) Rg/Rh

112.3 111.5 86.5

1.167 1.574 ( 0.229 1.622 ( 0.211

0.1510 (b) 0.2678 (b)

92.0 91.1

1.004 1.060

0.0904 (a) 0.1128 (a)

82.6 85.9

1.044 ( 0.021 1.141 ( 0.038

0.1111 (b) 0.1023 (b)

a EPC ) egg yolk phosphatylcholine, m EPC ) massEPC/(massEPC + massNIPAM) denotes the mass fraction of EPC, dn/dc ) mEPC(dn/dc)EPC + (1 mEPC)(dn/dc)NIPAM is the refractive index increment of the liposome (see Table 1). Mw is the weight average of the vesicle, Rg and Rh are the corresponding radius of gyration and hydrodynamic radius, and F ) Rg/Rh is a structure-sensitive parameter. The parameters (a) s ) σ2/Rav2 and (b) σ2 ) (〈R2〉 〈R〉2)/〈R〉2 represent the normalized standard deviation of the Gauss distribution of the Appendix and of the actual distribution.

Figure 3. Angular dependence of scattered light Rθ from an egg yolk phosphatodylcholine (EPC) vesicle and a similar liposome (EPC/ NIPAMlow) that contained a NIPAM network of low NIPAM concentration (NIPAM ) N-isopropylacrylamide). Because of the large size and the expected shape the Berry representation (Kc/Rθ)1/2 instead of the more common Zimm presentation59 was chosen. The two curves correspond to the data after extrapolation to zero concentration and were fitted by a special analytic function.60 The two liposome systems were not fractionated.

that of the lipid shell, the behavior is not precisely predictable. In all these cases the anchor DOGM was present. If NIPAM was encapsulated, the cross-linker TEGDM was always added in a ratio of TEGDM/NIPAM ) 1/40. In the second series two fractions of pure lecithin vesicles, containing no DOGM anchoring compound, were prepared, fractionated by GPC and then measured by LS. These data were compared in the third series with two other GPC fractions of liposomes with a high poly(NIPAM) content. The purpose of the two latter series was to get an impression on the effect of polydispersity on the angular distribution of the scattering intensity. Angular Dependence. Measurements were made with the two wavelengths of a HeNe laser (λ0 ) 632.8 nm) and an Ar laser (λ0 ) 488.2 nm). In the first series of experiments, only the red light was used. Measurements were made at five concentrations, and the data were then extrapolated toward c ) 0. The result of the extrapolated curves at c ) 0 is shown in Figure 3 as Berry plots, i.e., (Kc/Rθ)1/2 against q2, for two of the nonfractionated vesicles. The seemingly rather small variation in the angular dependence became

Figure 4. Kratky plot of the data in Figure 3 compared with the scattering curve of hollow spheres. The dash-dotted lines were obtained from the particle scattering factor of uniform hollow spheres after calculating the average with a size distribution that was measured by dynamic light scattering. The solid line was derived in the same manner but with particle scattering factor of uniform hard spheres. The red light of a HeNe laser was used (λ0 ) 632.8 nm).

more exaggerated when the particle scattering factors P(q) were calculated and plotted in the Kratky representation, i.e., u2P(u) against u ) qRg (Figure 4). The curves exhibit no oscillation as was to be expected for uniform hollow spheres. They also seem to show no similarity to uniform hard sphere behavior. The full and dash-dotted lines in this figure correspond to theoretical curves and will be discussed below. The data for Mw, Rg, and Rh and of the F-parameter, F ) Rg/Rh, are collected in Table 2 where Rh was obtained from dynamic light scattering. The corresponding results from the fractionated samples FR7 and FR10 without poly(NIPAM) content are shown in Figure 6 as Berry plots and in Figure 7 as Kratky plots. The corresponding curves for fractions FR4 and FR5 from liposomes with a large NIPAM content are given in Figure 8. The various lines correspond to fit curves and will be discussed later. The molecular data are given in Table 2. Radii Distributions. The observed strong deviations from the expected behavior of uniform hollow and hard spheres made it necessary to study the influence of a radii distribution on the angular dependence of the static light scattering intensity. For the first two examples, EPC and


Figure 5. The size distributions of the two liposome systems (EPC) and EPC/NIPAMlow. The distributions were obtained by inverse Laplace transformation of the field time correlation functions measured by dynamic light scattering. The distributions can roughly be described by the mean and the standard deviation which are as follows: EPC, R ) 112.3 ( 43.6 nm; EPC/NIPAMlow, R ) 111.5 ( 57.7 nm.

EPC/NIPAMlow (Figure 5) and for the FR4 and FR5 (Figure 9) of the liposome fractions, containing a large amount of poly(NIPAM), the distributions were determined by Laplace inversion of the field time correlation functions measured in dynamic light scattering. The CONTIN program of Provencher29 was used. The diffusion coefficients were converted into hydrodynamic radii via the Stokes-Einstein relationship (eq 10). For the two fractionated vesicles without poly(NIPAM), FR7 and FR10, a fit was made with a special type of distribution which permitted an analytic calculation for the averaged particle scattering factor. The fit parameters indicate the width of the distribution. For details see Appendix. More familiar is a rough characterization of a distribution by giving the mean value of radius ( the standard deviation. This notation is made in the captions to Figures 5 and 9. The radii distributions were determined at a few selected angles and mostly displayed only little and


nonsystematic variation. This was not the case for EPC/ NIPAMhigh. Up to three different peaks were found with this sample which became especially pronounced at small scattering angles and indicated agglomerated vesicles to clusters of different size (compare findings in ref 22). This distribution was not applied to further analysis. The distributions for the nonfractionated EPC and EPC/NIPAMlow are shown in Figure 5, and those for the fraction FR4 and FR5 from EPC/NIPAMhigh are shown in Figure 9. The latter figure contains two further curves which will be commented on in the forthcoming discussion. In all cases the Laplace inversion with the CONTIN program led to a symmetric distribution in the logarithmic scale for the radii, indicating a log-norm type distribution. With these distributions for the radii the averaged scattering curves were calculated for the hollow and hard spheres as described in the theoretical section. The calculated particle scattering factors are shown in Figure 4 as dash-dotted and solid lines, respectively. Within the range that could be covered by measurement with the red light, the obtained results agreed well with the hollow but also with the hard sphere models. In fact, differences between these two types of sphere structures will become apparent only at larger u ) qRg > 4. This requires the use of a shorter wavelength or the preparation of considerably larger vesicles. For this reason all further measurements were made with both, the red and blue monochromatic light of the HeNe and Ar lasers by which the u-range could be expanded by a factor of 1.295. Apparent density. The determination of the density is connected with some uncertainties; because not the whole amount of monomers could be expected to become incorporated in a network, a certain amount of undercritically cross-linked polymers may be encapsulated in the vesicle. These polymers are highly branched, but the clusters remain too small as to span the whole internal space of the vesicle. Unfortunately, the actual concentration of the GPC fractions

Figure 6. Berry plots from two GPC fractions of EYC measured with the red light of a HeNe laser (λ0 ) 632.8 nm) and the blue light of an Ar laser (λ0 ) 488 nm). The liposome concentration could not be determined. Thus the data of the ordinate represent only relative values made by an unreliable estimation of the concentration. The full lines are fit curves with a model function.60

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of the molar mass the scattering theory of copolymers has to be applied,47,48 which would require measurements in solvents of different refractive indices. However, as was shown by Benoit et al.,47-49 a good approximation for the molar mass is obtained if the refractive index increments of both components are fairly large and an average dn/dc is used that is derived from the two components according to the equation dn/dc ) mLipo(dn/dc)Lipo + (1 - mLipo)(dn/dc)NIPAM

Figure 7. The same data as in Figure 6 in the Kratky representation. The solid and broken lines were made with functions which were derived for hollow spheres with a Gaussian size distribution (see Appendix). The parameter s ) σ2/R02 represents the normalized variance of the Gaussian distribution.

FR4, FR5, FR7, and FR10 could not be determined. The estimated lipid concentration was 4 nM. The whole fractions consisted of only a few drops, and this was not sufficient for carrying out both light scattering and concentration determination. For a first estimation of the apparent shell density in EPC and EPC/NIPAMlow liposomes, a homogeneous density was assumed. In this estimation we also included the EPC/ NIPAMhigh (nonfractionated) liposome but now made the density estimation for a homogeneously filled sphere. The estimated data are given in Table 3. Even for this rough estimation the copolymer character of the EPC/NIPAM liposomes had to be taken into account. The lipid and NIPAM polymers possess somewhat different refractive index increments. For an accurate determination

(20)

The corresponding data of the components are given in Table 1, and the calculated average dn/dc is given in Table 2. The data in columns 2 and 3 of Table 3 are, of course, only numbers but gave valuable indications to the trend of the poly(NIPAM) concentration. As will be shown in the discussion, these numbers allowed us to draw more detailed conclusions on the density profile. Discussion Angular Dependence The first experiments with the nonfractionated liposomes were rather disappointing. The scattering curves showed no similarity to hollow or hard spheres. This unexpected behavior turned out to be an effect of polydispersity by which even strong differences become largely washed out. A similar effect is known for randomly branched samples with their huge polydispersities which made it very difficult to distinguish polydisperse linear chains from such branched materials.50 Two types of heterogeneity had to be considered. The system can consist of individual single-bilayered vesicles which may display a weak variation in the radius, but the vesicles also can be clustered together.22 Furthermore, there may be strong fluctuations in the com-

Figure 8. Kratky plots of the scattering curves from two GPC fractions of EYC/NIPAMhigh. The large symbols correspond to measurements with red light (filled) and with blue light (open). The solid lines with small filled and open symbols were derived from the distributions, shown in Figure 9, for hollow spheres. The broken line with small open symbols is a fit curve for hollow spheres with the Gaussian distribution of the Appendix. The solid and broken lines without symbols correspond to the scattering curves of hard spheres derived with the same distributions as given in Figure 9.


Structures of Chemically Modified Liposomes Table 3. Overall Apparent Density of the Liposome 〈dapp〉, Lecithin Apparent Density NIPAM a dlecithin app , and Overall Apparent Density of the NIPAM Network in the Liposome 〈dapp 〉

a

liposome

〈dapp〉 (g/mL)

dlecithin app (g/mL)

〈dNIPAM app 〉 (g/mL)

R ) Rout (nm)

Rin (nm)a

EPC EPC/NIPAMlow EPC/NIPAMhigh

0.477 0.226 0.377

0.477 0.477 0.477

0.125 0.244

112.3 111.5 86.5

102.55 101.25 76.75

shell thickness (nm) 3.5 17.8

Rin is the inner shell radius of the lecithin.

Figure 9. Radii distribution obtained by inverse Laplace transformation of the time correlation functions of dynamic light scattering on the basis of the CONTIN program. The symbols represent experimental data. The dash-dotted line corresponds to the Gaussian distribution of the Appendix for hollow spheres, which gave the best fit of the scattering curves. The dotted line corresponds to the same type of Gaussian distribution, however now with the same µ2 ) (〈R2〉 - 〈R〉2)/〈R〉2 variance (FR 4, R ) 82.6 ( 27.5 nm; FR 5, R ) 85.6 ( 27.5 nm). The µ2 variance is always larger than s ) σ2/R02 of the model distribution in the Appendix.

position, for instance by variation of the encapsulated poly(NIPAM) but also by vesicles which consist of more than one single bilayer. Most of the oligolamellar bilayered vesicles were avoided by the preparation method used.1,23-25 The good agreement of the calculated Kratky plots obtained

with the aid of the size distribution, which was determined by dynamic light scattering, gives some evidence that no significant clustering of vesicles occurred. However, in the third sample EPC/NIPAMhigh (not fractionated) the effect of clustering was noticeable in the time correlation function of dynamic light scattering, which after Laplace inversion showed three different modes (curves not shown). In this case the calculation of the angular dependence led to a strong discrepancy between the measured and calculated Kratky plots. More definite conclusions could be drawn from the angular scattering envelope obtained with the fractions. Evidently clustering of vesicles now could be excluded, and only the size variation of individual vesicles contributed to the size distribution. For the fraction with a high poly(NIPAM) content this size distribution was determined from dynamic light scattering. The Laplace inversion always led to a symmetric Log-Norm distribution. This effect may be an effect of the CONTIN inversion program. Ambiguity became apparent when the effect of polydispersity on the scattering envelope was examined with analytic size distributions which permitted exact averaging of uniform hollow or uniform filled spheres. The plots in Figure 10 from the Appendix for hollow spheres and Figures 11 and 12 for hard spheres impressively demonstrate that even a very weak polydispersity gradually levels out the pronounced peaks. Eventually, for large u ) qRg, smooth curves are obtained which indicate

Figure 10. Kratky plots of the particle scattering factors of hollow spheres with increasing width of the size distribution. The figure demonstrates how quickly the pronounced peaks become smoothed out as the polydispersity increases. The q-independent plateau corresponds to a fractal dimension of df ) 2 and indicates a two-dimensional object.51

574


Stauch et al.

Figure 11. Kratky plots of hard sphere of increasing polydispersity. For clarity the ordinate is presented in a logarithmic scale. As in Figure 10, the peaks become quickly washed out for increasing polydispersity.

Figure 12. The same data as shown in Figure 11 but now in a double logarithmic plot. The smoothed curves approach asymptotically a power law that corresponds to a fractal dimension51 of df ) 4 as was predicted by Porod.52

objects of fractal dimension51 df ) 2 for hollow spheres but df ) 4 for the filled hard spheres. In fact, at large u values only the short distances of the bilayer are probed, and because of the small curvature of the vesicles, the membrane can be considered as a two-dimensional object. The fractal dimension of df ) 4 was already predicted by Porod52 for polydisperse hard spheres in the asymptotic region of large u values. The scattering curves for the fraction (FR7, FR10) of EPC liposomes and EPC/NIPAMhigh liposomes (FR4, FR5) could very efficiently be fitted by the analytic function of eq A5 from the Appendix. Reproducible results were obtained for the absolute values of the radii and for the angular dependencies of the scattered light. Even for the width of the radii distributions, a reasonable agreement was observed. Despite

GPC fractionation the width of these distributions remained fairly large (R = 100 ( 30 nm). Two further comments have to be added here. The size distribution from dynamic light scattering for the fractions FR4 and FR5 gave a fairly satisfying agreement with the measured Kratky plot. However, a much better fit was possible with the Gaussian distribution of the Appendix, but now with half the width of the Log-Norm distribution from the dynamic light scattering. The distributions obtained by inverse Laplace transformation with the CONTIN program were always found to be symmetric in the logarithmic scale. This symmetry in the logarithmic scale possibly may be an effect of the Laplace inversion procedure, which requires some smoothing of the experimental data, and this again may cause a broader distribution than actually is present. Second,


it appears likely that the shape of the radii distribution will have a non-negligible influence on the averaging procedure. The Gaussian distribution was more efficient in leveling out the peaks in the uniform particle scattering factors than the Log-Norm distribution (compare the two theoretical curves in Figure 8). The experimental curves for the fractions FR4 and FR5 (Figure 8) of the liposomes with a high NIPAM content give some indication for filled spheres. However, the indication is too weak for a safe conclusion. Evidently, measurements have to be made with shorter wavelength to reach the expanded u range that may allow a better distinction between structures. For a filled sphere, the asymptotic scattering curve will sharply decrease in the Kratky plot whereas for the hollow sphere the Kratky plot will reach a plateau, whose value, Hshell plateau, will increase with polydispersity but not beyond Hshell plateau > 1.0 (Figure 10). This plateau height differs significantly from Hmonodsp.coil ) 2.0 of monodiperse plateau flexible linear chains and Hpolydisp.coil ) 3 for polydisperse plateau flexible linear chains and randomly branched or cross-linked flexible chains.53 G-Parameters. Despite the attained high accuracy in the angular dependencies, no clear distinction between hollow and filled spheres could be made. As was outlined in the theoretical section the ratio of Rg/Rh ) F should permit a further check. With the exception of the nonfractionated liposomes the fractions FR7, FR10, and FR4 gave values close to F ) 1.04 ( 0.02, which are indicative for hollow spheres. Somewhat unexpected is the higher value of F ) 1.14 for FR5. For the expected filled sphere, it should give a value around F ) 0.775. It appears, that the poly(NIPAM) gel, even for the liposomes with a high poly(NIPAM) content, may be attached to the inner surface of the vesicle and are condensed in a fairly thin layer. However, see further discussion below. Density Profile A last check for hollow or filled spheres can be made on the basis of the apparent densities, which was possible only for the nonfractionated samples. For the EPC liposome without poly(NIPAM), an apparent density of 0.477 g/mL was found for the lipid shell using eq 9b. This value can be checked when using the information on the shell thickness and the structure of the bilayer which was determined by Lis et al.42 According to these authors the bilayer has a thickness of d ) 3.5 nm and an headgroup area of A ) 0.756 nm2, which gives a volume of V ) 2.646 nm3, and finally with the molar mass of 780 g/mol for the egg yolk phosphatidyl choline, an effective density of the bilayer of calculated: dapp, shell ) 0.489 g/mL measured: dapp,shell ) 0.477 g/mL This value agrees well with the apparent density determined with the present light scattering data when Rh ≈ R was assumed. Knowing this density, one can calculate for other sizes of liposomes (i) the molar mass of the lipid shell in the


liposome, (ii) the molar mass of the encapsulated poly(NIPAM) network, and (iii) the apparent density of the encapsulated poly(NIPAM) network. For the pure EPC a molar mass of Mshell ) 444 × 106 g/mol was found. This gives for the encapsulated poly(NIPAM) network in the EPC/ NIPAMlow liposome a molar mass of Mpoly(NIPAM) ) (792 408) × 106 g/mol ) 384 × 106 g/mol. Finally, with the inner sphere radius of Rin ) 108.8 nm, an average density of dNIPAM(low) ) 0.125 g/mL is obtained, where the different size of the liposomes with and without poly(NIPAM) was taken into consideration. Similar for the EPC/NIPMhigh liposomes and the data of Table 2, one finds for the shell Mshell ) 263.4 × 106 g/mol, for the poly(NIPAM)high network Mpoly(NIPAM) ) 352.6 g/mol and an average density of dNIPAM(high) ) 0.244 g/mL. These density data correspond to networks, which in both cases the networks span the whole available space of the vesicle. The latter density is fairly high compared to common networks. If in the EPC/NIPAMlow liposome this high segment density is assumed being concentrated in a network shell attached to the inner surface of the lipid, a shell thickness of 17.8 nm for the pseudo-two-dimensional NIPAM network is found. The total shell thickness would add up to about 21.3 nm. NIPAM has a smaller refractive index increment and a 27% lower contrast. In addition the network density is about 2 times smaller than that of the lipid, which together gives by factor 2.5 a weaker light scattering signal than the lipid layer. Such a vesicle with an inner radius of 108 nm will very much show behavior of a hollow sphere in good agreement with the observed angular dependence of the scattered light. The data of the mentioned densities are collected in Table 3. Conclusions In the present light scattering study the possibility of distinguishing liposome structures without and with an enclosed polymer network was examined. The discrimination of hollow spheres from filled or hard spheres was strongly inhibited by the polydispersity of the liposomes in the ensemble. The characteristic peaks became almost completely smoothed out by the existent polydispersity. Significant differences in the asymptotic decay still remained. Unfortunately, the wavelengths of the used red and blue laser light proved to be too large for reaching this asymptote. Despite a good fit for hollow spheres, a definite discrimination between filled and hollow spheres could not be made. The value of the structure sensitive parameter F ) Rg/Rh ) 1.04 ( 0.02 gave preference to hollow spheres. Further support to hollow spheres was obtained from estimations of the apparent density. Within experimental error the same density was measured for the lipid shell (without poly(NIPAM) network) as calculated from the shell thickness and the headgroup area of the phosphatidylcholine.42 A layer thickness of about 18 nm was found for the liposomes with low poly(NIPAM) content if the high segment density of the poly(NIPAM) gel is used. The EPC/NIPAMlow liposome appears to have a pseudo-two-dimensional network anchored to the inner shell surface. On the other hand, also a network of lower segment density that spans the whole available space would be compatible with the measured light scattering data.

576


For a more definite conclusion the measurements have to be expanded to a larger u range. This requires either the preparation of well-fractionated large liposomes with radii of about 200 nm or more, or the choice of shorter wavelengths, i.e., application of small angle neutron scattering (SANS). A combination with light scattering is essential. For a conclusive interpretation the radius of gyration has to be determined accurately, and this can be done by light scattering only. SANS measurements have the advantage of probing short distances. This may permit characterization of the poly(NIPAM) gel structure in the interior of the liposome. The fractal dimension of well-swollen networks is close to df,gel ) 2.0,51 which is the same as for a thin shell structure. However, the plateau of the asymptote in the normalized ) 3,53 which is Kratky plot will be around Hrandom-network plateau considerably higher than that for the shell with Hshell plateau e 1. Such measurements will require a high accuracy in determining the absolute intensity of the scattered neutrons. Neutron scattering is less sensitive to polydispersity, but a conclusive interpretation of the results will remain difficult. So far known to us, there exist no cross-linking theory of monomers within the strictly confined space of the vesicle interior. Some attempts were made several years ago by Gordon et al.,54 who predicted how the sol fraction, i.e., the undercritically cross-linked macromolecules, disappear when the gel point is exceeded. An experimental work by Nerger and Burchard55 with network formation in a latex particle revealed a slight contraction of the network when the gel point was exceeded. Similar effects should be operative with the network in a vesicle, in particular when it is anchored to the inner surface of the vesicle. The present study gives some indications for this effect, the sphere radii of the EPC/NIPAM vesicles were found consistently to decrease when the poly(NIPAM) content was increased. All conclusions were based on two essential approximations. The first is the validity of the Rayleigh-Gans-Debye approach, and the other is the assumption of a negligible shell thickness compared to the hollow sphere radius. For large objects of high density and large differences in the refractive indices between the dissolved material and the solvent, the more exact Mie scattering theory has to be applied. According to our experience with latex particles the effect of the Mie correction should be small for spheres with R < 100 nm. A check with our data could not be made since a suitable program was not in our hands. We also hesitated to apply this complex theory because the unexpected strong influence of polydispersity seems to be far more dominant than the corrections by the Mie theory. This assumption may be in question and certainly requires a thorough check. The approach of Wyatt61 is of particular interest. He extended the Mie theory to inhomogeneous systems and calculated as a special example the scattering behavior of a sphere with a density that decayed exponentially toward zero at larger radii. This theory will have to be modified to the hollow sphere with a similar density profile, but now extending from the inner surface toward the center. Polydispersity has to be taken into account. The mathematical effort will be large and requires a separate treatment. The assumption of a thin shell thickness, on the other hand, appears with d/R ≈ 0.035 to

Stauch et al.

be justified. The effect of polydispersity will be much stronger than the small correction by the use of the more correct equation by Kerker et al.56 for hollow spheres. SANS measurements are presently in progress, and the results will help to obtain a clearer picture of this rather complicated structure. Addendum. After this paper was already written, we came across two recent papers by McKelvey et al.57 who studied synthetic bilayered vesicles by SANS. They concentrated their attention on the structure of the bilayer and chose the range of large q values. In both papers the linear asymptote of power 2 proved the existence of the fractal dimenison df ) 2 that was predicted as asymptotes in the above conclusions. To obtain an overlap with the light scattering data, the SANS measurement have to be extended to very small q values. Determination of behaVior will be not sufficient, but absolute intensity measurements are required, which give the correct value of the apparent molar mass and the radius of gyration. Only then can the correct height of the asymptote be recorded, which is decisive for a discrimination between the fractal behavior of the lipid shell and that of the polymer NIPAM network. Acknowledgment. The work was gratefully supported by the Deutsche Forschungsgemeinschaft within the scheme SFB 428 “Strukturierte makromolekulare Netzwerksysteme”. Appendix: Influence of Polydispersity on the Angular Distribution of Scattered Light from Hollow and Hard Spheres The theoretical relationships for the particle scattering factors for uniform hard and hollow spheres are well-known. They show a strikingly characteristic angular dependence that becomes especially well noticeable when the Kratky plot is applied. A significant periodic behavior is observed which is very regular for the hollow sphere but is strongly damped for the hard sphere. Figure 10 shows this feature. In the Kratky plot the angular dependence of the particle scattering factor P(u) is multiplied by (qRg)2 ≡ u2. This causes a magnification of the function in the large u range and makes the structure dependence asymptotic range particularly evident. Rg is the radius of gyration and q ) (4πn0/λ) sin(θ/2), the magnitude of the scattering vector; n0 is the refractive index of the solvent, λ0 the wavelength of the light in a vacuum, and θ the scattering angle. Note: the particle scattering factor P(u) ≡ I(θ)/I(θ)0) is a dimensionless quantity and the same holds for parameter u. Therefore, all plots in terms of the parameter u are universal, and the results become applicable to visible light and to small-angle neutron or X-ray scattering. The mentioned particle scattering factors for hollow and hard spheres can be written in two alternative, but mathematically equivalent, ways as follows P(u)hollow-sphere ) (sin(u)/u)2 P(u)hollow-sphere ) where

1 [1 - cos(2u)] 2u2

(A1a) (A1b)



X ) qRhollow-sphere ) qRg,hollow-sphere P(u)hard-sphere ) P(u)hard-sphere )

[

(A1c)

]

3 (sin(X) - X cos(X)) X3

2

(A2a)

9 [1 - cos(2X) - 2X sin(2X) + X6 X2(1 + cos(2X)] (A2b)

where X ) qRhard-sphere ) 1.291qRg,hard-sphere

(A2c)

In favorite cases, when dynamic light scattering measurements have been performed with the same sample, the radii distribution can directly be obtained from the Laplace inversion of the corresponding time correlation function. Since in the ALV 5000 correlator the relaxation times are recorded in a logarithmic scale, one obtains after application of the Stokes-Einstein relationship a logarithmic radii distribution h(Rh) d log(Rh). Actually, however, the distribution is presented as a histogram, because the various frequencies are presented pointwise equally spaced in the logarithmic space, and this histogram can be used in the same manner for calculating the influence of polydispersity on the particle scattering factor. In practice this means that the exact averaging procedure by integration is replaced by a sum over the histogram as follows

∫

∞

〈P(u)〉 f Aq-2

hj(R)Pj(uR) ∑ j)1

w(R) dR )

[

]

(R - R0) R2 1 exp w(R) dR ) dR (A4) (2πσ2)1/2 R02(1 + s) 2σ2 2

which after integration of P(u) gave the average 〈P(u)〉 as 〈P(u)〉hollow-sphere ) 1 [1 - cos(2X0) exp(- 2X02s)] (A5) 2X02(1 + s) with X0 ) qR0 1 + 6s + 3s2 ) (qRg)2 1+s s ) σ2/X02 The Rg is the radius of gyration of the polydisperse hollow sphere and parameter s describes the width of the distribution. Figure 10 shows the effect of the polydispersity on the shape

1 R6 1 × (2πσ2)1/2 (1 + 15s + 45s2 + 15s3) R06

[

exp -

]

(R - R0)2 2σ2

dR (A7)

and with this distribution the particle scattering factor of the polydisperse hard spheres is found to be58 〈P(u)〉hard-sphere )

9 (F1 - F2 + F3) 2NX06

(A8)

F1 ) 1 - [cos(2X0) - 2X0 sin(2X0)] exp(-2X02s) (A9) F2 ) [2X0 sin(2X0) + 4sX02] exp(-2X02s)

(A3)

This technique proved to be useful and efficient for fairly broad distribution, but it became inaccurate for rather narrow fractions. A description of the time correlation function by a stretched exponential with subsequent Laplace transformation gave no real insight on the polydispersity influence. For special size distributions the required integration according to eq A3 can be carried out analytically.58 For the hollow spheres the corresponding distribution is

(A6)

and indicates a fractal dimension of df ) 2. This value had to be anticipated, because in the large q-region the thin shell of a hollow sphere corresponds to a two-dimensional sheet. For hard spheres the corresponding radii distribution that can be used and is given by

n

w(R)P(uR) dR = -∞

u2 ) X02

of the angular dependence in the Kratky representation for s-values from 0.02 to 1.0. Already at very low polydispersity the undamped periodic appearance is strongly effected, and eventually at large polydispersities a constant asymptotic behavior is obtained which for the particle scattering factor corresponds to the power law

(A9b)

F3 ) X02(1 + s) + X02{cos(2X0)(1 + 0.5s - 4s2X02) 2sX0 sin(2X02)} exp(-2X02s)

(A9c)

N ) 1 + 15s + 45s2 + 15s3

(A9c)

with

u2 ) X02

1 + 28s + 210s2 + 420s3 + 105s4 (A10) N

References and Notes (1) Stauch, O.; Uhlmann, T.; Fro¨hlich, M.; Thomann, R.; El-Badry, M.; Kim, Y.-K.; Schubert, R. Biomacromomolecules, in press. (2) Shen, B. W. Ultrastructure and Function of Membrane Skeleton. In Red Blood Cell Membrane; Agre, P., Ed.; Marcel Dekker: New York, 1989; Chapter 10. (3) Janmey, P. Cell Membranes and the Cytoskeleton. In Handbook of Biological Physics; Lipowsky, R., Sackmann, E., Eds:; Elsevier Science: New York, 1995; Vol. 1. (4) Sackmann, E. (a) Physical Basis of Self-Organization and Function of Vesicles. (b) Biological Membranes, Architecture and Function. In Handbook of Biological Physics; Lipowsky, R., Sackmann, E., Eds.; Elsevier Science: New York 1995; Vol. 1. (5) See also: Alberts, B.; Bray, D.; Lewis, M.; Roberts, K.; Watson, J. D. Molecular Biology of the Cell; Garland Publishing: New York, 1989. (6) Ringsdorf, H.; Sackmann, E.; Simon J.; Winnik, F. M. Biochim. Biophys. Acta 1993, 1153, 335. (7) Kim, J. C.; Bae, S. K.; Kim, J. D. J. Biochem. 1997, 121, 15. (8) Hayashi, H.; Kono, K.; Takagishi, T. Bioconjugate Chem. 1998, 9, 382. (9) Kono, K.; Nakai, R.; Morimoto, K.; Tagagishi, T. Biochim. Biophys. Acta 1999, 1416, 239. (10) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley & Son: New York, p 1076.

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