INTRODUCTION

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A gel can be defined as any material that displays both the elastic properties of a solid and the viscosity of a liquid. In biotechnological applications, high-molecular-weight poly(ethylene oxide) [(PEO, (OCH₂CH₂)_N)], which has a low immunogenicity, is a standard coating for more immunogenic tissues and materials (Lee, 1991; Peppas and Langer, 1994). Recent studies show that attaching low-molecular-weight (N < 150) PEO [poly(ethylene glycol) or PEG] to a biological macromolecule can also substantially increase blood circulation times. In particular, "stealth" liposomes, membrane sacks consisting of closed bilayer shells of phospholipids covered with PEG-lipids hydrophobically anchored to the membrane, show promise as a drug carrier system (Allen and Chonn, 1987; Lasic, 1993; Lasic and Martin, 1995; Lasic and Papahadjopoulos, 1995). These results suggest that PEG-lipids might be useful as a new material in the growing biotechnology industry.

The inhibition of the body's immune response to PEG-coated liposomes has been attributed to a polymer-brush steric repulsion between PEG-coated membranes and the antigenic molecules found in the bloodstream (Alexander, 1977; DeGennes, 1980; Lasic and Martin, 1995). This repulsion has been measured between PEG-coated membranes in the chain-frozen phase (Kuhl et al., 1994) and in the chain-melted fluid phase in multilamellar L systems (Kenworthy et al., 1995; Needham et al., 1992). In these systems, PEG-lipids were incorporated into very stiff membranes (bending rigidity   k_BT) with intermembrane distances d  R_g, where R_g is the PEG radius of gyration.

The present work concentrates on a different regime: extremely flexible membranes in a multilamellar L system for which d  R_g. Separated by water, these membranes contain the zwitterionic lipid DMPC, pentanol, and either 0-45 mol % PEG550-DMPE, 0-25 mol % PEG2000-DMPE, or 0-25 mol % PEG5000-DMPE (polyethylene glycol of MW 576, 2053, or 5181 g/mole covalently attached to the headgroup of the zwitterionic lipid DMPE). The chemical structure and molecular weight information for these molecules is summarized in the table of Fig. 1. These membranes are always in the chain-melted fluid state, permitting the polymer-lipid to diffuse or aggregate within the plane of the membrane (Fig. 1). Previous x-ray diffraction work has shown that membranes composed of DMPC and pentanol alone have a very low bending rigidity ( ~ k_BT), swelling via the Helfrich undulation repulsion to intermembrane separations >200 Å (Safinya et al., 1989). We show that the addition of PEG-DMPE to these undulation-stabilized membranes extends the stability of the lamellar phase to even larger separations. At low water concentrations, mixtures possess the rheological properties of a fluid but at higher water concentrations a lamellar hydrogel phase forms, hereafter called L_,g. The gel appears whether brine or water is used to swell the membranes, indicating that gel formation is not an electrostatic phenomenon. Polarized light and freeze fracture electron microscopy of the L_,g reveals a significant increase in spherulitic and layer dislocation defects that resist both temperature and shear annealing (Keller et al., 1997). X-ray data from powder samples confirm the lamellar structure of the L and L_,g phases, but indicate a significant decrease in the domain size of gel samples compared to fluid samples, consistent with an increased defect density. We infer that PEG-DMPE, added to flexible membranes, significantly increases the membrane spontaneous curvature, leading to the proliferation and stabilization of curved regions (defects).

View larger version (35K): [in this window] [in a new window]   FIGURE 1   Schematic of two undulating membranes composed of DMPC, pentanol, and PEG-DMPE. The membranes are in the fluid state, thus all components are free to diffuse within the plane of the membrane. In the lower right, the non-zero spontaneous curvature of the PEG-DMPE molecule is emphasized in comparison with that of DMPC and pentanol. In the table, the polymerization number N, PEG molecular weight, and total molecular weight are summarized for each of the three PEG-DMPE molecules.

An earlier work (Warriner et al., 1996) presented preliminary results on lamellar hydrogels induced by the addition of PEG2000 and PEG5000 to a highly dilute, flexible, L phase. A model for the formation of the L_,g was proposed, based on the idea that PEG-DMPE stabilizes layer dislocation defects by aggregating to regions of high membrane curvature, creating an effective 3-D structure with gel-like elasticity. In a later study of four other PEG-surfactants, it was shown that the hydrophobic moiety of the PEG-surfactant used is not a key factor in the gelation transition (Warriner, 1997; Warriner et al., 1997). In this paper, we present more complete phase diagrams for PEG2000 and PEG5000, as well as the first data for the smaller PEG550. We compare these experimental phase diagrams to the model cited above in terms of 1) the inverse relationship between the intermembrane separation d and both PEG-DMPE concentration and polymer molecular weight at the transition, and 2) the effect of polymer molecular weight on the L-L_,g transition. Within the framework of this model we estimate , the effective bending rigidity of the polymer-coated bilayers.

In addition, we report the first quantitative fits to small-angle x-ray scattering from polymer-decorated L phases. From these fits, we estimate the value for the single membrane bending rigidity  and the bulk compressional modulus B as a function of both water and PEG-DMPE concentration. Within the limits of error for this technique, we find that membrane bending rigidity is unaffected for polymer-lipid coverages below a monolayer. We also find that the  value obtained through the fits agrees well with that extracted from the gelation model. Lastly, our analysis shows that the addition of PEG-DMPE to flexible lamellae leads to a strongly enhanced intermembrane repulsion which is not described either by the Helfrich undulation theory or a polymer mushroom-brush model. However, we demonstrate that this enhanced repulsion is actually only incidental to the L-L_,g transition.

	MATERIALS AND METHODS

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Materials

PEG-DMPEs consisting of dimyristoylphosphatidylethanolamine (MW 635.86) with PEG covalently attached to the amine group (1,2-dimyristoyl-sn-glycero-3-phosphoethanolamine-N-(poly[ethylene glycol]) were purchased and used without further purification from Avanti Polar Lipids (Alabaster, AL). The three PEG-DMPEs used in our experiments contained PEGs with average molecular weights of 576, 2053, and 5181 g/mole (N = 13, 45, and 113) as summarized in the table of Fig. 1. They are referred to individually as PEG550, PEG2000, and PEG5000, respectively; in discussing them collectively we use the term PEG-DMPE.

Dimyristoylphosphatidylcholine (DMPC) and didodecyldimethyl ammonium bromide (DDAB) were also purchased from Avanti Polar Lipids; pentanol (99% purity) was purchased from Sigma Chemical Corp. (St. Louis, MO). Purified 18 M water was obtained via a Milli-Q Plus unit (Millipore Corp., Bedford, MA). Molecular weights of all membrane components were DMPC, 677.94 g/mole; PEG550, 1212 g/mole; PEG2000, 2689 g/mole; PEG5000, 5817 g/mole; pentanol, 88.15 g/mole. Densities used in all calculations were DMPC and lipid moiety of PEG-DMPE, _lipid = 1.1 g/cc (Small, 1986; Trauble and Haynes, 1971); polymer moiety of PEG-DMPE in solution with water, _H2O+PEG = 1.03 g/cc (Gonzalez-Tello et al., 1994); pentanol, _pentanol = 0.81 g/cc; water in 0 mol % samples, _H2O = 1.0 g/cc. Headgroup areas were estimated in the manner described in the Definitions part of the Materials and Methods section and the Phase diagram part of the Results section: for lipids, the headgroup area A_lipid was found to be 72.8 ± 0.1 Å²; for pentanol, the headgroup area A_pent was 12.7 ± 0.1 Å².

Sample preparations

All samples were prepared in 13-mm diameter glass test tubes. The tubes were first cleaned with a 2:1 vol/vol chloroform/methanol solution, rinsed once with spectroscopic grade ethanol, multiple times with Millipore water, and dried in an oven.

Samples were made with a molar ratio of pentanol to lipid molecules of 4.0 ± 0.5. A high ratio of cosurfactant to surfactant was used to ensure that the lipid chains would always be in the melted state and that the multilayers formed would be sufficiently flexible to display a large undulation repulsion. We also wished to fix the pentanol-to-lipid ratio in order to isolate the effect of PEG-DMPE on membrane fluidity, bending rigidity, and shape. With this ratio fixed, the two remaining compositional degrees of freedom are water content _W (water weight fraction) and the molar ratio of PEG-DMPE to total lipid molecules, c_PEG (expressed in %).

Samples were prepared by weighing in the appropriate amounts of lipid, pentanol, PEG-DMPE, and water. Pentanol was always added last, and the test tube top was wrapped in Teflon tape to prevent evaporation. Samples were centrifuged to collect all material at the bottom of the tube and then subjected to ~1/2 h of sonication to break up any clumping. After mixing with a Vortexer (Fisher Scientific, Tustin, CA) the samples were centrifuged again and left to stand for 1-4 weeks before phase determination. After the initial phase determination, samples were checked for any changes every few months.

For a "line" of increasing water samples we often made one large, low water content "seed" sample. After thorough mixing and at least a week of equilibration time, this sample would be "split" into the desired number of samples and the appropriate amount of water added to each new sample to achieve the needed water concentrations. This procedure was most successfully followed for seed samples with "low" PEG lipid concentrations (i.e., c_PEG < c_gel for the seed water amount) because the long equilibration time for highly viscous samples made it difficult to verify that gel seed samples were at equilibrium. In cases where we used a gel seed sample to create an increasing water line, we spot-checked our results with "singly made" samples along the same line.

Definitions and formulas

Phase diagrams for all PEG-DMPEs were constructed both in terms of weight fraction water versus mol % PEG-DMPE (_W vs. c_PEG), and intermembrane spacing d versus volume fraction of PEG-DMPE in the bilayer membrane, _memPEG. The time and expense required for rheological tests on more than 400 samples were prohibitive, thus to construct phase diagrams we adopted the following "operational" definition of a gel: any sample that does not flow for at least 5 s after inversion of the test tube. In the two series of samples for which we did run quantitative rheological tests (described below) we found that gelation occurred somewhat earlier than the concentrations indicated by the above criterion (e.g., at lower _W or lower c_PEG). However, a consistent application of this simpler, operational definition results in a phase diagram close to that obtained through more quantitative rheological data.

For each of the three phase diagrams, the relevant definitions given below were used. For all equations, g_x is the weight in grams of material x. In Eqs. 5 and 7, the factor of 1  g_H2O(0.026/0.974) multiplying the pentanol volume takes into account the 2.6 w/w % solubility of pentanol in water (Stephen and Stephen, 1963-1979).

(1)

where q₀ is the peak position of the first harmonic in the x-ray diffraction pattern

(2)

(3)

(4)

(5)

We also use the classical relationship between the intermembrane spacing d and the volume fraction of membrane _mem to find , the bilayer thickness:

(6)

where

(7)

In Eq. 5, the volume fraction of PEG-DMPE in the bilayer includes the polyethylene glycol moiety of the polymer-surfactant as part of the bilayer, while in Eq. 7 the polymer part of PEG-DMPE is excluded from the calculation of the volume fraction of the membrane. This paradox results from the dual nature of PEG-DMPE: 1) it acts to swell the intermembrane distance like an equivalent volume of solvent and so must be counted as part of the solvent in Eq. 7, but 2) the effective headgroup-to-chain area ratio of PEG-DMPE surfactant is controlled by the size of the polymer moiety (Fig. 1); hence, this part of the polymer-lipid must be taken into account in calculations relating to average bilayer properties or composition.

We determine the area per headgroup for the molecules in the membrane using standard relationships between area and volume for a predominately flat membrane. Taking the areas of the DMPC and bare DMPE headgroups to be approximately equal, we calculate the average headgroup area per lipid:

(8)

where V_mem is the volume of sample occupied by the membrane and

(9)

Equation 9 has the form of a straight line with intercept equal to the headgroup area of pentanol and slope given by the difference between the lipid and alcohol headgroup areas. Here N_mem is the total number of molecules in the membrane after the water solution is saturated with pentanol.

X-ray diffraction

X-ray scattering studies were performed in-house using an 18 kW Rigaku rotating anode generator (Rigaku, Danvers, MA) (CuK,  = 1.54 Å), a cylindrically bent focusing pyrolitic graphite (002) monochromator and a Bicron point detector (Bicron, Newbury, OH). The in-plane resolution, defined using slits, had a FWHM = 0.01-0.015 Ź and the out-of-plane resolution had a FWHM = 0.14-0.3 Ź; scan stepsize was generally 0.001 Å¹. Additional experiments were carried out at the Stanford Synchrotron Radiation Laboratory on beamlines 6-2 and 10-2 using either a Bicron point detector or a 180-mm MAR image-plate 2-D x-ray detector (Mar Industries, San Diego, CA). In the Bicron experiments, in-plane resolution, again defined by slits, was FWHM = 0.0014-0.0028 Ź and the out-of-plane resolution was FWHM = 0.01-0.02 Ź; scan stepsize was usually 0.0005 Å¹. For the 2-D detector experiments resolution and stepsize were defined by the detector pixel size and the distance from sample to detector. Images were radially averaged to produce powder scans with a stepsize of 0.0007 Å¹ and a radially averaged FWHM of 0.0027 Å¹. Exposure times were typically 1-2 h.

Samples were flame-sealed in either quartz or glass 1.5-mm x-ray capillary tubes (Charles Supper Co., Natick, MA). These capillary tubes were then set on a translation stage for automated data acquisition. We found it necessary to heat and quench some of the samples from the fluid regime in order to obtain a proper "powder" form, i.e., an isotropic distribution of lamellar domains.

Rheology

Constant-stress oscillatory shear-strain experiments were carried out with a Rheometrics dynamic stress rheometer, model 1710C (Rheometrics, Piscataway, NJ), in the cone and plate geometry with a 40-mm diameter plate, a cone angle of 0.04 radians, and a gap size of 0.05 mm. During testing, a small housing was placed around the setup which enclosed pentanol and water-soaked cotton balls in order to minimize evaporation.

Samples were subjected to three different tests. The first test was a dynamic stress sweep test in which the stress is increased from ~0.6-100 dynes/cm² at a frequency of 1 Hz to establish the regime of linear viscoelasticity. Each sample was then tested in a transient single point test within this regime to ensure the sinusoidal strain response followed the sinusoidal stress by a phase angle. Finally, a dynamic frequency sweep test was run at a constant stress over a frequency range of 0.01-10 Hz to determine both the real (storage elasticity) modulus, G', and the imaginary (loss) modulus, G". For each sample, two sets of tests were run. The first set included the dynamic stress sweep test, the transient single point test, and the dynamic frequency sweep test. For the second set of tests, a fresh sample from the same test tube was used and only the dynamic frequency sweep test was run in order to check the reproducibility of the first set of tests. In particular, we wished to ensure that the dynamic moduli were not merely products of alignment produced during the high stresses imposed in a dynamic stress sweep test.

Optical microscopy

Optical glass capillaries (Vitro Dynamics, Rockaway, NJ) of thicknesses ranging from 0.05 to 0.2 mm were filled with sample and flame-sealed. Some capillaries were cleaned first with a 2% solution of PCC-54 concentrate (Pierce, Rockford, IL), then rinsed with spectroscopic grade ethanol, rinsed multiple times with Millipore water, dried and then subjected to 30-90 min of UV light in order to heighten the hydrophilicity of the glass, thus increasing the probability of homeotropic alignment. All samples were observed with an Optiphot 2-Pol microscope (Nikon, Torrance, CA) using polarized light at different magnifications (50-500×). Textures were photographed using a MFX-DX automatic camera and posemeter (Nikon, Torrance, CA). The microscope was also equipped with an FP82 heating stage and an FP80 central processor (Mettler-Toledo Inc., Hightstown, NJ) for temperature annealing experiments.

Freeze-fracture electron microscopy

Freeze-fracture electron microscopy was performed as in Chiruvolu et al. (1994), and the replicas processed as in Fetter and Costello (1986) except that replicas were bathed in ethanol and retrieved on formvar coated microscopy grids. Samples were imaged in a TEM JEOL100CX at an accelerating voltage of 100 keV.

	RESULTS

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Phase diagrams

In this section we show that the addition of any of the PEG-DMPEs to a phase of undulation-stabilized membranes promotes the formation of a lamellar hydrogel phase, labeled L_,g. This hydrogel is distinct from entanglement-based polymeric gels in that it forms only at high water concentrations. Counterintuitively, at low water concentrations where the polymer-lipid concentration is necessarily higher, mixtures possess the rheological properties of a fluid. We also show that PEG-DMPE actually tends to stabilize the lamellar phase, permitting dilution to larger intermembrane separations than are achievable for L phases composed of bare membranes.

To eliminate the effect a changing cosurfactant/surfactant ratio exerts on viscoelastic properties, all samples used in this paper were made with a pentanol-to-lipid molar ratio of 4.0 ± 0.5 (Fig. 2 A). However, due to the non-zero solubility of pentanol in water (Stephen and Stephen, 1963-1979), the actual amount of pentanol retained in the membrane decreases with increasing water concentration, yielding an actual intramembrane pentanol-to-lipid ratio of ~3.0 ± 1.5, samples of higher water concentration having the lowest ratios. However, this higher effective variation in intramembrane pentanol concentration is not the source of the dramatic increase in viscoelasticity between the two lamellar regimes: high water samples made without PEG-DMPE do not gel. When the partial solubility of pentanol is taken into account, a plot of the lamellar period versus membrane volume fraction shows the linear behavior of a 1-D lamellar system (Fig. 2 B). Both the L and L_,g regimes for all the PEG-DMPEs are well described by a bilayer thickness of 27.8 ± 0.1 Å separated by a solvent composed of water, PEG, and trace amounts of dissolved pentanol (Fig. 2 B). Since the membrane thickness is independent of c_PEG, this is the thickness of the bare (no polymer) membrane. This value for the membrane thickness is in excellent agreement with earlier values for DMPC-pentanol membranes of ~28 Å (Safinya et al., 1989).

View larger version (19K): [in this window] [in a new window]   FIGURE 2   (A) Pentanol-to-lipid molar ratio as a function of cPEG for all PEG550, PEG2000, and PEG5000 mixtures used in this paper. This ratio of 4.0 ± 0.5 is the calculated ratio based on the relative amounts of pentanol and lipid weighed into the sample and does not reflect the partial solubility of pentanol in water. When this partial solubility is taken into account, the intramembrane molar ratio is 3.0 ± 1.5, still a small enough variation not to significantly alter the sample viscoelasticity. (B) Intermembrane distance d versus 1/mem (membrane volume fraction of the sample) for all single-phase PEG550, PEG2000, and PEG5000 samples. Regardless of viscoelasticity, polymer-lipid concentration, or polymer molecular weight, the membrane thickness  is stable at 27.8 ± 0.1 Å. The fit is to Eq. 6 with error bars given by Eq. 2; reduced 2 was 1.91 for 174 points. (C) Average area per intramembrane molecule Amem versus the molar fraction of lipid molecules in the membrane. Physically, Amem is a weighted average of the headgroup areas of the lipid and pentanol molecules; the partial solubility of pentanol in water forces this average to vary in a systematic way. We use this variation as described in the Materials and Methods section to obtain an average lipid headgroup area Alipid of 72.8 ± 0.1 Å and an average pentanol headgroup area of 12.7 ± 0.1 Å. Values are from an unweighted fit to Eq. 9; 2 was 1.67 for 314 points.

Noting that the average headgroup area per membrane molecule A_mem is a weighted average between the lipid and pentanol molecules, we use the variation in the intramembrane pentanol concentration to calculate the average headgroup area for each type of molecule. In Fig. 2 C, where A_mem is plotted versus the number fraction of lipid molecules in the membrane, an unweighted, straight-line fit yields a value for A_LIPID of 72.8 ± 0.1 Å², and for A_pentanol of 12.7 ± 0.1 Å², consistent with previous headgroup area measurements from phosphocholine bilayers in the L phase (Reiss-Husson, 1967; Small, 1986) and DMPC/pentanol solutions (Safinya, personal communication, 1995).

Fig. 3 shows the phase diagrams of the PEG-DMPEs as a function of weight fraction water (_water) and mol % PEG-DMPE (c_PEG), while Fig. 4 gives the same information in terms of the intermembrane spacing d and the membrane volume fraction of PEG-DMPE (_memPEG). From Fig. 3 B, mixtures with low c_PEG2000 and _W  0.42 are two-phase. These samples, when viewed in the bulk after moderate centrifugation, show a clear meniscus between an isotropic and a birefringent phase; consistent with this, the small angle x-ray scattering contains both an isotropic liquid peak and lamellar diffraction peaks (data not shown). This boundary corresponds to an intermembrane spacing d of ~53 Å (Fig. 4 B) or a fluid spacing d_w (= d) of 25 Å. The radius of gyration R_g for PEG2000 incorporated in lipid bilayers has been measured to be 25-35 Å (Kuhl et al., 1994), matching the location of this lower two-phase boundary. This behavior is replicated in both the PEG550 and PEG5000 phase diagrams; in Fig. 3, A and C, the lower two-phase boundary occurs at _W ~0.33 and _W ~0.66, respectively, corresponding to fluid spacings of ~16 and 63 Å. By using Flory scaling arguments one can extrapolate the PEG2000 R_g measurement to the other two PEG-DMPEs, obtaining a value for the PEG550 R_g of ~15 Å and for the PEG5000 R_g of ~60 Å. We surmise, therefore, that the PEG-DMPE lamellar regime is stable only when the fluid spacing d_W is large enough to accommodate a swollen polymer coil. Thus the position of the lower two-phase boundary in this type of phase diagram is a simple and accurate method of determining the average polymer extension from a bilayer.

View larger version (30K): [in this window] [in a new window]   FIGURE 3   Phase diagrams of the PEG550, PEG2000, and PEG5000 systems in terms of weight fraction water (W) and mol % PEG-lipid (cPEG). Open circles represent fluids, solid circles represent gels, and diamonds represent two-phase samples. The dotted lines between the fluid and gel regimes are intended to guide the eye. (A) PEG550. Note that the lamellar system is not stable for W  0.33. (B) PEG2000. The lamellar system is not stable for W  0.42. (C) PEG5000. The lamellar system is not stable for W  0.66. For all the PEG-DMPEs there is an inverse relationship between the amount of water required to achieve gelation and the concentration of PEG-lipid.

View larger version (17K): [in this window] [in a new window]   FIGURE 4   Phase diagrams of the PEG550, PEG2000, and PEG5000 systems in terms of intermembrane spacing d and the volume fraction of the membrane occupied by the PEG-DMPE molecules, memPEG. Open circles represent fluids, solid circles represent gels. The dotted lines between the fluid, gel, and two-phase regimes are intended as guides to the eye. (A) PEG550. The lamellar system is not stable for spacings below 42 Å or dw < 14 Å, close to the calculated polymer Rg of 17 Å. (B) PEG2000. The lamellar system is not stable for spacings below 53 Å or dw < 25 Å, consistent with the measured polymer Rg of 25-35 Å. (C) PEG5000. The lamellar system is not stable for spacings below 91 Å or dw  63 Å, approximately the calculated polymer Rg of 60 Å. For all the PEG-DMPEs the intermembrane spacing at which gelation occurs is inversely proportional to the concentration of PEG-DMPE.

Above this lower two-phase boundary, two lamellar regimes with remarkably different viscoelastic and optical properties are observed. These differences are qualitatively demonstrated in Fig. 5. The lower water concentration lamellar regime, which we refer to simply as L, possesses the rheological properties of a low-viscosity fluid (Fig. 5 A, bottom two samples). The L_,g regime, accessed from the L through the addition of either PEG-DMPE or water, retains the lamellar microstructure of the L samples but exhibits a strong gel-like elasticity (e.g., Fig. 5 A, third sample from the bottom). A simple demonstration of this elasticity is the ability to maintain nonspherical shapes for indefinite periods (Fig. 5 B). Viewed between crossed polarizers, an L sample will typically exhibit a uniformly bright, somewhat flat appearance, free of macroscopic features (Fig. 5 C). In contrast, L_,g samples are generally less birefringent than L samples and, when viewed between crossed polarizers in bulk, display macroscopic features reminiscent of the nematic Schlieren texture (Fig. 5 D) (Demus and Richter, 1978; Gray and Goodby, 1984).

View larger version (96K): [in this window] [in a new window]   FIGURE 5   (A) Four samples with cPEG2000  6 mol % and, from bottom to top, increasing water concentration illustrate the fluid-to-gel to two-phase transitions. From bottom to top: fluid sample with water = 0.45 (d = 55 fluid sample with water = 0.68 (d = 110 Å); gel sample with water = 0.78 (d = 165 two-phase sample with water = 0.95. (B) Fluid sample with cPEG2000 = 1.2, water = 0.49 between crossed polarizers. The uniformly bright, somewhat flat appearance of this sample is typical of low-water fluid samples. Fluid samples of higher water content are similarly devoid of macroscopic features, but may be more colorful. (C) Gel sample with cPEG2000 = 6.0, water = 0.85 between crossed polarizers. Macroscopic liquid crystalline defects have replaced the featureless appearance of the fluid sample. (D) Gel sample with cPEG2000 = 6.0, water = 0.79 showing nonspherical bubbles. The ability to maintain arbitrary shapes against surface tension is a qualitative demonstration of elastic properties.

The transition between these two lamellar regimes is fairly broad; average widths in terms of _W are ±0.03 and in c_PEG, ±0.5 mol %. At low PEG-DMPE concentrations, the transition curve in the three phase diagrams of Fig. 3 is roughly

(10)

or, in Fig. 4,

(11)

That is, less PEG-DMPE is required to gel mixtures of higher water concentration. This general behavior immediately differentiates the L_,g from free polymer hydrogels in which the polymer concentration must exceed c*, the polymer mushroom overlap concentration, in order for entanglement and gelation to occur. Remarkably, the L_,g occurs at polymer and water concentrations that preclude the possibility of direct polymer interactions as a gelation mechanism. The L_,g is found in 1 mol % PEG5000 mixtures with measured fluid spacings of ~400 Å (Fig. 4 C and Warriner, 1997); in the PEG550 and PEG2000 systems, d_w of a gel approaches 200 Å (Fig. 4, A and B and Warriner, 1997). These separations are many times greater than the polymer R_g.

Lateral interactions, in particular the mushroom-brush transition (Alexander, 1977; DeGennes, 1976), are also irrelevant in the L-L_,g transition. Using our values for the lipid and pentanol headgroup areas, one can calculate the expected monolayer coverage for these membranes, i.e., the PEG-DMPE concentration c_mono at which the polymer "mushrooms" would first begin to overlap and interact:

(12)

and

(13)

Equating 12 and 13 gives

(14)

For PEG550, c_mono is ~40 mol %; for PEG2000, ~10 mol %; for PEG5000, ~3 mol %. From Fig. 3, the L_,g begins at concentrations up to eight times less than c_mono. Moreover, gel samples have been prepared using a 0.5 M NaCl solution in place of water, ruling out long-range electrostatic interactions in gel formation.

The upper two-phase boundary has the same general shape in all the phase diagrams of Fig. 3, decreasing from _W  0.85-0.90 at low c_PEG to _W  0.70-0.73 at the highest PEG concentrations studied. However, as demonstrated in Fig. 4, the position of the upper two-phase boundary in terms of intermembrane spacing depends strongly on the molecular weight of the polymer lipid used. This effect is explained by recalling that the polymer moiety of the PEG-DMPEs add to the intermembrane spacing in the same way as an equivalent volume of water. Thus PEG5000 samples are actually stable up to intermembrane spacing d > 400 Å, whereas the highest measured spacing for a single-phase sample in the PEG2000 system approaches only 300 Å, and in the PEG550 sample the highest measured spacing is just over 200 Å. Samples just beyond the upper two-phase boundary appear to be composed of a lamellar phase plus excess water; however, no detailed study of this regime has been undertaken.

Rheology

Quantitative rheological tests were performed on 10 PEG2000 samples. Five samples had c_PEG2000 fixed at 6 mol % with the weight fraction of water _W increasing from 0.45 to 0.79; the other five had a nearly constant d-spacing (140 Å ± 15 Å) with c_PEG2000 increasing from 1.73 to 15.36 mol %. For all samples, at least two measurements of the real (elasticity) modulus G' and the imaginary (loss, viscosity) modulus G" were made as described in the Introduction. For the purposes of this section, samples showing gel-like elastic behavior are those for which G' is reliably greater than G" over the full range of frequencies measured. We compare this definition to the one used in the phase diagrams of the previous section.

The evaluation of G' and G" demonstrates the strong elasticity of samples from the gel regime. Fig. 6 shows data taken as a function of frequency (dynamic frequency sweep test) on the line of increasing water samples after a dynamic stress sweep test was performed. For the two lowest _W samples (Fig. 6, A and B), the elastic and viscous components of the dynamic moduli are comparable with G'/G" ~2-3. These samples were classified as fluids in the phase diagram; consistent with that determination, rheological data from such samples tended to be less reproducible than data from gel samples. For example, in Fig. 7 A, two sets of frequency data for a sample classified as fluid taken before and after a dynamic stress sweep test are plotted. Although the data were taken at the same constant stress with the same volume of sample, the moduli differ by almost an order of magnitude between the tests. In other experiments, a crossover frequency (frequency at which the elastic and viscous moduli are equal) was observed in samples after a dynamic stress sweep test was run but was not seen in the same sample if a dynamic stress sweep test was not run. This phenomenon is explained by noting fluids have no shear resistance. That is, during shear, liquid crystalline samples with the viscoelastic properties of a fluid can realign, altering the measured response (Lu and Cates, 1994; Safinya, et al., 1993). Gels, on the other hand, resist shear, yielding a reproducible response; in Fig. 7 B frequency data for a gel sample before and after a dynamic stress sweep test are essentially unchanged.

View larger version (29K): [in this window] [in a new window]   FIGURE 6   Rheological data from four samples with cPEG2000  6 mol % but increasing water concentration. Open symbols are G', the elasticity modulus and closed symbols show G", the viscosity modulus. Note the strong increase in the elastic character of the samples as the water concentration increases and the samples change from fluids to gels. (A) Fluid sample, water = 0.45, G'/G" ~ 2-3. (B) Fluid sample, water = 0.47, G'/G" ~ 2-3. (C) Sample approaching the transition region, water = 0.64, G'/G" ~ 10. Although the G', G" measurement indicates this material is a gel, it flows under its own weight before 5 s elapse, and so was classified a fluid in the phase diagrams. (D) Gel sample, water = 0.79, G'/G" ~ 13.

View larger version (24K): [in this window] [in a new window]   FIGURE 7   Typical data from a fluid and a gel sample before (circles) and after (squares) the dynamic stress sweep test (DSST), which subjects the sample to large stresses (up to 100 dynes/cm2) in order to determine the limits of linear viscoelasticity. Open symbols are G', the elasticity modulus, while closed symbols show G", the viscosity modulus. (A) Fluid sample with cPEG = 6.0, water  0.49. The sample realigned during the DSST, producing much lower values for both viscoelastic moduli in the second test. (B) Gel sample with cPEG = 3.76, water  0.77. Gels, unlike fluids, can resist the high shears imposed in the DSST, yielding a consistent response.

Between the _W = 0.47 and _W = 0.64 samples, elasticity values increase markedly; for _W = 0.79 the elastic portion of the dynamic moduli has increased by more than an order of magnitude from the _W = 0.47 value. Similar results are obtained as a function of increasing PEG-DMPE concentration. In Fig. 8, G' grows by two orders of magnitude as c_PEG increases from 1.73 to 15.36 mol %. Although both the viscosity and elasticity moduli increase between the fluid and gel samples, the elasticity gains are consistently greater: in both lines of samples the ratio of G' over G" grows by an order of magnitude over initial (fluid) values.

View larger version (25K): [in this window] [in a new window]   FIGURE 8   Rheological data of three samples with water  0.77 demonstrates that augmenting sample cPEG leads to gelation as readily as increasing water. Open symbols are G', the elasticity modulus, while closed symbols show G", the viscosity modulus. (A) Sample near the transition region, cPEG2000 = 1.73, G'/G" ~ 4. Although the G', G" measurement indicates this material is a gel, it flows under its own weight before 5 s elapse, and so was classified a fluid in the phase diagrams. (B) Gel sample, cPEG2000 = 3.76, G'/G" ~ 6. (C) Gel sample, cPEG2000 = 15.36, G'/G" ~ 10.

From Fig. 6 we would surmise that the L-L_,g transition at 6 mol % PEG2000 occurs between _W = 0.47 and _W = 0.64, instead of the ~0.70 indicated in the phase diagram. Similarly, from Fig. 8, the transition for _W = 0.78 occurs before 1.73 mol % instead of at 2.0 mol %, as shown in Fig. 3 B. The discrepancy arises from the use of an "operational" 5-s inversion test to classify samples for the phase diagram: i.e., only samples with yield stresses greater than their own weight (as tested by inverting the samples for 5 s and checking for flow) were classified as gels. Under this definition, some gels will be classified as fluids; however, no fluids will be classified as gels. Thus, the 5-s definition delays identification of the L-L_,g transition in terms of c_PEG or _W from the concentrations indicated by quantitative rheologic data, and makes it impossible to attribute particular significance to absolute values of c_PEG and _W at the transition. However, the key, intriguing observation of this study is unaltered by the choice of definition of gelation: for all PEG-DMPE's, the concentration of PEG-DMPE required for gelation is inversely proportional to the water content. Moreover, the arguments employed in the previous section to eliminate polymer entanglement and long-range electrostatic interactions as gelation mechanisms also remain valid regardless of which definition of gelation is used.

X-ray diffraction

Figs. 9 and 10 show small-angle synchrotron x-ray data obtained from unoriented, i.e., "powder," PEG2000 and PEG5000 samples (data for PEG550 are comparable). For the scans in Fig. 9, _W was kept constant to within ±0.05 while c_PEG increased from 0 to just above 15 mol % (PEG2000) or from 0 to 6 mol % (PEG5000). For the scans shown in Fig. 10, c_PEG was fixed at the indicated value while _W increased. The spectra are almost evenly split between samples from the gel and fluid regimes. Regardless of PEG-DMPE molecular weight or concentration, water concentration or viscoelastic properties, the samples display lamellar diffraction patterns and are well-described by a bilayer of 27.8 ± 0.1 Å separated by a solvent composed of PEG, water, and trace amounts of pentanol (Fig. 2 B). Additionally, scans of the interference peak at 1.4 Ź show that the lipid chain interactions remains liquid-like regardless of macroscopic viscoelasticity (Fig. 11). Thus the addition of PEG-DMPE to a flexible L phase dramatically increases bulk viscoelasticity without altering either the local lamellar symmetry or diminishing membrane fluidity. In particular, unlike L_' gels, chain-ordering is not the source of gelation.

View larger version (37K): [in this window] [in a new window]   FIGURE 9   Synchrotron x-ray scattering data as a function of increasing cPEG for PEG5000 (I, A-F and II, A-F) and PEG2000 (III, A-H, IV, A-E) samples. (I) Moderate water content samples from the fluid L regime of PEG5000. (A) cPEG5000 = 0, w = 0.79, d = 153 Å. (B) cPEG5000 = 0.50, w = 0.79, d = 153 Å. (C) cPEG5000 = 0.76, w = 0.79, d = 152 Å. (D) cPEG5000 = 1.60, w = 0.77, d = 145 Å. (E) cPEG5000 = 3.79, w = 0.75, d = 140 Å. (F) cPEG5000 = 5.43, w = 0.72, d = 132 Å. (II) High water content samples drawn equally from the fluid and gel regimes of PEG5000. (A) Fluid, cPEG5000 = 0, w = 0.85, d = 234 Å. (B) Fluid, cPEG5000 = 0.25, w = 0.85, d = 238 Å. (C) Fluid, cPEG5000 = 1.17, w = 0.84, d = 227 Å. (D) Gel, cPEG5000 = 2.09, w = 0.82, d = 204 Å. (E) Gel, cPEG5000 = 3.12, w = 0.81, d = 195 Å. (F) Gel, cPEG5000 = 5.95, w = 0.79, d = 207 Å. (III) Samples of moderate water content drawn equally from the fluid and gel regimes of PEG2000. (A) Fluid, cPEG2000 = 0, w = 0.79, d = 157 Å. (B) Fluid, cPEG2000 = 1.09, w = 0.79, d = 153 Å. (C) Fluid, cPEG2000 = 2.9, w = 0.78, d = 147 Å. (D) Fluid, cPEG2000 = 3.54, w = 0.77, d = 137 Å. (E) Gel, cPEG2000 4.2, w = 0.77, d = 147 Å. (F) Gel, cPEG2000 = 6.01, w = 0.76, d = 137 Å. (G) Gel, cPEG2000 = 7.8, w = 0.75, d = 133 Å. (H) Gel, cPEG2000 = 15.6, w = 0.70, d = 127 Å. (IV) Samples of high water content drawn equally from the fluid and gel regimes of PEG2000. (A) Fluid, cPEG2000 = 0, w = 0.83, d = 202 Å. (B) Fluid, cPEG2000 = 0.91, w = 0.83, d = 195 Å. (C) Gel, cPEG2000 = 2.19, w = 0.82, d = 189 Å. (D) Gel, cPEG2000 = 3.44, w = 0.82, d = 184 Å. (E) Gel, cPEG2000 = 15.32, w = 0.76, d = 151 Å. The number of harmonics, and hence the intermembrane repulsion, is a strong function of the PEG-lipid concentration. However, this repulsion is unconnected to the fluid-gel transition as evidenced by I, A-F, where the number of harmonics steadily increases but the samples retain the viscoelastic response of fluids.

View larger version (41K): [in this window] [in a new window]   FIGURE 10   Synchrotron x-ray scattering data as a function of increasing water for samples without PEG-lipid (I, A-E), and for PEG5000 (II, A-F and III, A-G) and PEG2000 (IV, A-E, V, A-E) samples. (I) All samples are in the fluid L regime with cPEG = 0. (A) w = 0.67, d = 92 Å. (B) w = 0.74, d = 117 Å. (C) w = 0.79, d = 153 Å. (D) w = 0.83, d = 202 Å. (E) w = 0.87, d = 282 Å. (II) All samples are in the fluid L regime. (A) cPEG5000 = 0.31, w = 0.69, d = 98 Å. (B) cPEG5000 = 0.31, w = 0.73, d = 115 Å. (C) cPEG5000 = 0.31, w = 0.75, d = 126 Å. (D) cPEG5000 = 0.31, w = 0.79, d = 152 Å. (E) cPEG5000 = 0.25, w = 0.85, d = 238 Å. (F) cPEG5000 = 0.31, w = 0.87, d = 274 Å. (III) PEG5000 samples drawn equally from the fluid and gel regimes. (A) Fluid, cPEG5000 = 1.61, w = 0.70, d = 109 Å. (B) Fluid, cPEG5000 = 1.62, w = 0.73, d = 122 Å. (C) Fluid, cPEG5000 = 1.60, w = 0.77, d = 145 Å. (D) Fluid, cPEG5000 = 1.65, w = 0.80, d = 165 Å. (E) Gel, cPEG5000 = 1.59, w = 0.82, d = 184 Å. (F) Gel, cPEG5000 = 1.60, w = 0.84, d = 220 Å. (G) Gel, cPEG5000 = 1.67, w = 0.87, d = 279 Å. (IV) All samples are from the fluid regime of PEG2000. (A) cPEG2000 = 0.55, w = 0.64, d = 82 Å. (B) cPEG2000 = 0.55, w = 0.75, d = 125 Å. (C) cPEG2000 = 0.56, w = 0.79, d = 149 Å. (D) cPEG2000 = 0.55, w = 0.85, d = 241 Å. (E) cPEG2000 = 0.55, w = 0.87, d = 281 Å. (V) PEG2000 samples drawn equally from the fluid and gel regimes. (A) Fluid, cPEG2000 = 3.0, w = 0.64, d = 85 Å. (B) Fluid, cPEG2000 = 3.0, w = 0.79, d = 153 Å. (C) Gel, cPEG2000 = 3.0, w = 0.82, d = 193 Å. (D) Gel, cPEG2000 = 3.0, w = 0.84, d = 217 Å. All samples, regardless of viscoelasticity, display a lamellar diffraction pattern. Although gelation occurs readily upon increasing w, the number and strength of harmonics seems relatively unaffected by water content, i.e., by intermembrane distance.

View larger version (21K): [in this window] [in a new window]   FIGURE 11   Typical high-angle rotating anode data for a PEG2000 fluid (open circles) and gel (closed circles) displaying the lipid chain-interference peak at 1.5 Å1 (arrow) and water peaks at 2 and 2.7 Å1. Note that for both samples the lipid peak is broad and liquid-like, indicating that intramembrane molecules are free to diffuse regardless of macroscopic sample viscoelasticity.

Examining Figs. 9 and 10 in more detail, some general trends become apparent. First, the number of harmonics present in a spectrum is an increasing function of the PEG-DMPE concentration, but basically independent of _W. Second, the shape (asymmetry, slope) of the x-ray peaks is also strongly affected by the presence of PEG-DMPE. Third, peaks from gel samples are generally broader than those from fluid samples. Previous theoretical (Caille, 1972; Gunther et al., 1980; Lei et al., 1995) and experimental work on stacked membrane systems (Als-Nielsen et al., 1980; Keller et al., 1991; Safinya, 1989) has demonstrated that changes in the x-ray lineshape reveal changes in material parameters and interactions. In particular, the Caille structure factor, originally developed to describe diffraction from smectic A liquid crystals (Caille, 1972), has been successfully extended to describe scattering from electrostatically (Roux and Safinya, 1988) and undulation-stabilized L systems (Helfrich, 1978; Safinya, 1989; Safinya et al., 1986, 1989) and from the smectic A phase of polymeric liquid crystals (Keller et al., 1991). Here, we apply the Caille theory to a system of polymer-coated, chain-melted, flexible, stacked lamellae.

Although the Caille theory has been extensively tested for undulation-stabilized systems (Roux and Safinya, 1988; Safinya, 1989; Safinya et al., 1986, 1989), this is the first application we are aware of to flexible membranes containing end-anchored polymers. Thus, there is no existing evidence that the Landau-DeGennes Hamiltonian correctly describes the interactions between lamellae of this type of material. There is no prior work to show that the Helfrich undulation repulsion (Helfrich, 1978), which is the dominant interaction between uncharged, flexible membranes, remains important for flexible membranes carrying a polymer coat. We should therefore first explicitly examine the general agreement among the Caille structure factor, the Helfrich theory, and the observed scattering before using these models to probe the relationship between microscopic parameters, interactions, and trends in the bulk viscoelasticity.

Brief review of the Caille x-ray lineshape and Helfrich undulation repulsion

The Caille theory relates the x-ray lineshape to the intermembrane spacing d, bulk compressional modulus B, and bulk bending elasticity K. Analysis of two series of samples spanning the L-L_,g transition, one in the direction of increasing _W, the other in the direction of increasing c_PEG, offers an opportunity to examine material constants (K, d) and interactions (B) in light of the increase in bulk viscoelasticity. The Caille theory begins with the Landau-De Gennes expression for the energy density of a smectic A liquid crystal.

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Here u(r) is the layer displacement in the z direction normal to the layers. Landau and Peierls (Landau, 1965) first showed that for this Hamiltonian, thermally induced mean square layer displacements diverge logarithmically with the domain size L, destroying long-range order. In this case, conventional delta-function Bragg peaks are replaced by power law divergences (Caille, 1972). For a powder sample, profiles of the (00l) reflections have the asymptotic form (Roux and Safinya, 1988; Safinya et al., 1986)

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If 1 < _l < 2, this asymptotic form no longer applies. While there is no theoretical limit on _l, for _l