overview of nmr of bulk polymersoverview of nmr of bulk polymers. introduction • configuration,...

Hans Wolfgang Spiess

Max-Planck-Institut für PolymerforschungMainz, Germany

“NMR Spectroscopy of Polymers”Tutorial

ACS National Meeting

New Orleans, April 6, 2008

Overview of NMR of Bulk Polymers


Introduction •

Configuration, Conformation •

Local Structure & Dynamics •

Phase Behavior •

Supramolecular Organization •

Conclusions •

Basics

Chain Branching

Amorphous & Crystalline Polymers

Core Shell Structures

Functional Polymeric Systems

Scattering and NMR

Chemical Shift

Ranges for

Organic

Compounds

Isotropic Anisotropic (13C)

Analogous for 2H quadrupole coupling

Structure and Dynamics from Solid-State NMR

Dipole-Dipole

Coupling

i

jθij

rij

B0 ( cos )Drij

iji j∝ -3 13

12

2θγ γ

⋅

Orientation of internuclear vectorDistance between nuclei

1H-1H1H-13C 13C-13C1H-15N

20 15 10 5 025

coupling strength [kHz]

Typical

pairs

of nuclei

Structure

Dynamics

2H quadrupolecoupling

static

1 kHz

2 kHz

4 kHz

8 kHz

15 kHz0 4 8-12 -4 12-16

frequency [kHz]16-80 1 2 3 4 5 6 7

time [ms]

Solid State NMR Spectra

2H static spectra

13C MAS spectraSpin

nin

g f

requen

cy

2H quadrupole coupling

Magic-angle spinning (MAS)

rotor is spun around an axis inclined at an angle of θm =54.7° with respect to B0 .

spatial part of interaction tensor averaging by

fast rotationresulting average tensor

How does MAS work ?

in terms of coordinate transformations:

212 (3cos 1)θ − 21 1

2 2sin cos(2 2 ) sin(2 ) cos( )R Rt tβ ω γ β ω γ− − −

rotor modulations with frequencies 2ωR and ωR

θm

ωR

B0

ωR

θ

θmωR

B0 B0 B0

Polyolefin Branching

Short (SCB)< 30 C

Long (LCB) > Me ≈

270 C

pronounced effect on viscosity &

melt processability

MAS-NMR in Melts: Very Low Branch Contents

‘Linear’

PE

∗ α∗B2

∗

α

∗B2 α α

Quantification of 7–8 branches per 10 000 COptimised solution

NMR:

50,000 to 2,000,000 scans (up to 60 days!)Optimised melt-state NMR: 21,500 scans (13 h)

Site SNR Content

per 1000 C

*B2 4.5 0.07

* 3.7 0.05

α 9.4 0.08

Sample:R.H. Grubbs, Caltech

Macromolecules 37, 813 (2004), Macromol. Chem. Phys. 207, 382 (2006).

13 C –

NMR: Conformational Effects

Potential Energy

Polymer Chain

Gamma -

gauche effect:-

5,2 ppm in alkanes

Sensitivity of

13C Chemical Shifts

on Conformation

:

42 40 38 36 34 32 30 28 26 24 22

13C NMR spectrum of PE

Crystalline regions:all-trans

Non-crystallineregions: gauche

Conformational Effects on 13C Chemical Shifts

Self-Assembly and Molecular Dynamics of Peptide- Functionalized Polyphenylene Dendrimers

Short polypeptides (n < 16)High order of columns,

Low order of peptide chains

Long polypeptides (n > 20)Low order of columns,

High order of peptide chains(α-helices)

X-ray Scattering:columnar order

Solid state NMR:Peptide conformation

176.3 ppm 172.4 ppm

13C=O

PLys

G2

F16

N16

G2

F16

N58

α -

helix β -

sheet


Introduction

•

Configuration, Conformations •

Local Structure & Dynamics

•

Phase Behavior •


Conclusions •

Basics

Chain Branching




Scattering and NMR

Motional averaging effects

( ) 2 ( ) ( ) ( ) ( )12 (3cos 1)(3 )ij i j i j

ijD θ= ⋅ − −D Z ZH I I I I

Motional averaging:2 !21

20 0

(3cos 1) sin 0d dπ π

θ ϕ θ θ− =∫ ∫

221

2 (3cos 1)(3 )2 (2 1)Q

e qQI I

θ= ⋅ − − ⋅− ⋅ Z ZH I I I I

h

2 212 (3cos 1 sin cos(2 ))δ θ η θ ϕ= ⋅ − − ⋅CS ZH I

Solid crystal

Plastic crystalLiquid crystal

Anisotropic Chemical Shift

Dipole-Diplole Coupling

Quadrupole Coupling

order parameter Sij :

)1cos3(S 221

ij −θ=

Motional averaging effects

Basics: Two site jumps

(analogous to chemical exchange)

Calculated NMR line shapes resulting from interchange between two NMR frequencies.

Δ

: coupling strengthΩ

: exchange rate

The numerical values apply to 2H NMR of deuterons in C-H bonds

fast

intermediate

slow

ultraslow

Two-site jumps: CSA

1H powder spectrumof H2 O molecules

in crystalline CaSO4 ·2H2 O

δ

1jump

jump

kδ δ τ= = ⋅m

m

m

m

m

m

m

Two-site jump in solid: Different frequencies depending on orientation.

Result in fast motion limit:Averaged interaction tensor

Line shape analysis yields both:Timescale and geometry of motion

Two-site jumps: CSA, DDC and QC

Example: Phenyl180° ring flip:

Reorientation C-H bondsby β

= 120°

Averaged principal axes (1), (2) and (3)

δ = 5/8 δ ; η

= 0.6

Single transition Pake pattern

Anisotropic Chemical Shift

QuadrupoleCoupling

The “solid echo”

experiment

0 0.5 1-0.5-1ω/δ

x

xx

sampling (“dwell”) time

deadtime

Large spectral width (> 250 kHz) requires short dwell and dead times (< 4 µs).

x y

Overcoming the dead-time problem by echo experiments:

spin (“Hahn”) echo solid (“Solomon”) echo

refocuses “linear-spin” interactions refocuses “bilinear-spin” interactions

Motions in the “solid echo”

experiment: Increased dynamic range

Absorption spectra:Line shape changes within one order of magnitude

Solid echo spectra:Line shape changes over several orders of magnitude,

But: loss of signal!

NMR line shapes conveniently calculated by NMR Weblab

http://weblab.mpip-mainz.mpg.de/weblab/weblab.html

NMR Weblab: How to use it


NMR Weblab: Example phenyl flip


Inhomogeneous and homogeneous line broadening

...

Inhomogeneous:CSA, quadrupolar, dipolar two-spin

Due to spin-spin couplings the energy levels of single transitions (resonance lines) are no longer degenerate, but

split into a multitude of levels

Overall resonance consists of indiviual sharp lines and

represents the sum over all different orientations

Homogeneous:dipolar multi-spin

Similar distinction ifwhole line shapes are superimposed due to a distribution of

• different structures, or• different rates, or • different geometry of

motion

Example of heterogeneous rate distribution

narrow broad

distribution

rigid limit

rapid exchange

Superposition of line shapes for different rates

2D-Exchange-Spectroscopy:

Simplicity

Pulse Sequence Spectra

Determine geometry and time scale of motions directly and in real time

Geometry

of Chain Motion in Polymers

POM (crystalline) PEO(disordered) PVAc

(amorphous)

13C 2D Exchange NMR Spectra of Polymers with Different Degrees of Disorder

Helical jumps in polymer crystallites: POM

Timescale and Geometry of Motion

Chain Folding, Chain Diffusion and Drawability

Chain motionMorphology Chain diffusion

Local Collective

Ordered Disordered

SolutionCrystallized

MeltCrystallized

DrawabilitySample:

UHMW-PE (Mw =3.4 M)

Solution

Crystallized:

Drawable

Melt

Crystallized:

Not

Drawable

Timescales of Molecular Dynamics Accessible by NMR

Averaging of dipole-dipole couplings, as detected by spinning sideband experiments: C-H, N-H: REREDOR, REPT-HDORH-H: double-quantum

Destructive interference effects: reduction and loss

of NMR signal

Change of anisotropic interactionsduring experimental “mixing time”:exchange NMR

experiments2H, 13C, 15N: 2D, CODEX etc.

Spin-lattice relaxation:dipole-dipole coupling, quadrupole coupling,anisotropic chemical shift

fast

10010-110-210-310-410-510-610-710-810-9

slowvery fast intermediate

motional correlation time [seconds]

Conformational Dynamics at the Glass Transition

Potential Energy

Gamma - gauche effect:- 5,2 ppm in alkanes

Polymer Chain

Sensitivity of 13C Chemical Shifts

on Conformation :

Chain Dynamics of Atactic Poly(propylene) at the Glass Transition

Correlation Times of Chain Motionfrom different NMR experiments

All data fit on a singleWLF - curve

Conformational dynamics from 2D 13C MAS NMR

Rotational dynamics from 2D 2H NMR

Geometry: ReorientationalAngle Distribution

Conformational transitions,but no defined geometry

Structure Schemes of Syndiotactic

and

Isotactic

Poly-(Methyl-Methacrylate)

syndiotactic isotactic

ω33ω33

localchain-axis

localchain-axis

schematicstructure

crystal structures

schematicstructure

NMR probes local chain-axis

through ω33

n-alkyl-methacrylates contain extended chain segments

Example:PMMA

a-PEMA: Two-step Randomization of Chain Motion

180°

CO

13

ω33<10°

ω22

ω11

180°C

O13

0°

-

180°

180°

CO

13 ω

50°ω

400 K

405 K

411 K

433 K

472 K

394 K

250 200 150 100

298 K

ω

ω

ω22

ω33

ω11

1D 13C NMR: experimentsimulation

melt

melt

melt

gla

ss

Tgrigid + fractionalsidegroup flips

+ anisotropicchain motion

+ randomizationof chain motion

Tg = 354 K

tem

pera

ture

s-PEMA: Conformation and Conformational Dynamics

(mr)

(rr)

CH3 (main chain)

CD3(side chain)

(mm)

T = 303 K

t.tt.g / g.t

g.g

t.g / g.tg.g / g.g

g.g

35 30 25 20 15 10 ppm

[ [

Tg - 50 K

s-PEMA

T = 353 K

Tg + 65 K

Tg = 354 K

T = 418 K

35 30 25 20 15 10 ppm

(mm)(mr)

(rr)

13C MAS NMR

Separation of Dynamic Timescales in PEMA-Melts

Correlation Times from NMR, PCS, Dielectrics

Tg = 338K

α-Relaxation(WLF)

β-relaxation(Arrhenius)

NMR: spatial randomisation

(tI, WLF)

NMR: confromational relaxation

NMRPCSDielectric

2.2 2.4 2.6 2.8 3.0 3.2 3.410-8

10-6

10-4

10-2

100

102

2.0

1000 / T [K-1]

time

[s]

Difference in time scale (factor 50):consistent with length scale 7 repeat units

Tc

Intersegmental

Order in Poly(methacrylates): WAXS(LVDW)

PMMA

PEMA

PBMA

PHMA

Inte

nsity

[a.u

.]

q [nm-1]0 10 20

II/I

WAXS-DataIII

(VDW)

III

?syndiotactic

PEMA

main chain

side chains

dI

dIII

d0

IIId

dII

„layered nano aggregates"

isotacticPEMA

main chains

side chains

monolayer

bilayer

X-Ray patternsreminiscent of

stiff macromoleculeswith

flexible sidechains

X-R

aySca

tter

ing

0

0.4

0.8

1.2

1.6

0 2 4 6 8 10 12# sidechain carbon atoms

Brag

gdi

stan

ces

d [n

m]

I

II

III

(VDW)

(LVDW)

Bragg-Distance

intra chain

inter chain

inter layer

2D DECODER NMR for Ordered Systems

Example: Biaxially stretched PET

2D exchange with sample fliprather than molecular motion

Recoupling CSA: CODEX

CODEX: Centreband-Only Detection of ExchangeApproach: Recoupling the chemical-shift anisotropy (CSA) under MAS

0 1 N/2

~ ~

N/2+n

CP

DD DD

[ ]N/2-1 [ ]N/2-1

tm AQ

nτR

+ + + +- - --

CP DD

exchange during tm ?Complete refocusing of CSA only if there is no exchange during tm !

Advantages:High spectral resolution, short measuring time compared to 2D exchange NMR

θm

ωR

B0

0 1 2 3 4 5 6 7 8 9 10 11 120,0

0,1

0,2

0,3

0,4

0,5

0,6

δ N τR / π

10°/170°

20°/160°

30°/150°

40°/140°

50°/130°

70°-110°

60°/120°

DMST = 288 K

Exc

hang

e in

tens

ity

tmtm

CODEX: reorientation angle

CODEX build-up curves

exchange intensity for a given mixing time depends on the overall duration of recoupling

shape of the curves depends significantly on the reorientation angle


Introduction

•



Phase Behavior •

Supramolecular Organization

•

Conclusions •

Basics

Chain Branching




Scattering and NMR

Phase Separation Probed by Spin diffusion

NMR spectra

Morphology and1H magnetization

MZ

selection spin diffusion

x

A B A

inte

nsi

ty

ωCS

spin diffusion time tm

A B A

B

AA

BA B

NMR spin diffusion experiment

selection diffusion

tm

Chemical shift filters (e.g. DANTE): spectral selection

Dipolar filters (e.g. SR-12): motional selection

Domain Sizes in Phase Separated Polymers

Both Components RigidRigid and Mobile Components

Spin Diffusion in 2D Wideline

Separation Spectra

Interface

spin diffusion

Investigating core-shell particles

structure and particle size can be determined

0 10 20 30 40 50

1.0

0.8

0.6

0.4

0.2

0.0

tm [ms 0.5 ]0.5

I/I0

38 nm76 nm113 nm

(t ms)1/2

d Particle =

contact surface: S source volume: V

withS/V Deff tmSπ

= 2eff

DA DBDDA

=+ DB

magnetization source

magnetization drain

initial slope

I/ I0

1.0

0.8

0.6

0.4

0.2

0.00 5 10 15 20 25

tm [ms0.5 ]0.5


Introduction

•



Phase Behavior

•

Supramolecular Organization

•

Conclusions •

Basics

Chain Branching




Scattering and NMR

Key Elements of Supramolecular Assemblies

shapeπ−π

stac

king

C 1 2 H 2 5

C 1 2 H 2 5

C 1 2 H 2 5

C 1 2 H 2 5

C 1 2 H 2 5

C 1 2 H 2 5

H-bonds

C H2713

N

N

HaO

N

C H27Hd

ON

Hc Hb Hb Hc

O N

NHa

O

N N

Hd

13

n

surfac

es

Challenge: Elucidate Noncrystalline

Structures

Scattering

X-ray- or neutron-scattering

Incident Scattered

Wave

Scattering Diagram / (Reflections)

1/d

r

r

rf- irradiation

DoubleQuantumCoherence

NMR Spectrum

1/r3

Double Quantum NMR

Analogy:In both cases: coherent superposition

of signals from spatially separated centers

1H NMR spectra in solid and liquid state

60 40 20 0 -20 -40 kHz-60

5 0 kHz

20 10 0 kHz

static

30 kHz MAS

solution

θm

ωR

Magic-AngleSpinning (MAS)

rapid isotropictumbling

3 kHz4HRMAS

partialmobility

increasing

spectral

resolution

rigidsolid

dipolarbroadening

high magneticfield: 700 MHz

Dipolar DQ Spectroscopy of a Spin-Pair under MAS

Reconversion DetectionEvolution

t1

Excitation

texc = nexctR

y -yx -x

nexc

y -yx -x

nrec=nexc

x τR

trec = nrectR

θm

ωR

i

j

θij

B0

rij

Dipole-Dipole Coupling:

$ $ $, ,H R T= ⋅2 0 2 0

Space Spin

t2Recoupling Recoupling

Line Narrowing in Solid-State NMR

Hamiltonian of Dipole-Dipole Coupling:

i

j

θij

B0

$ ( cos ) ( $ $ $ $ ), ,Hr

I I I Iij

ij i j Z i Z j i j∝ − − ⋅1

3 1 3312

2 θ γ γ

$ $ $, ,H R T= ⋅2 0 2 0

Space Spin

Magic Angle Spinning:

$,R2 0 0 ωR

θ

θm

ωR

θm

ωR

B0 B0 B0 B0

R R

RF Irradiation:

$,T2 0

00 tC

-x y -y x

tτ ττ τ2τ τ ττ τ2τ τ ττ 2τ

-x y -y x -x y -y

tC

0 t

t

t

(CRAMPS)

(Recoupling)HD,eff.

x y -y-x

τ τ

x y -y-x

τ τ

x y-x

τ

Signal build-up versus rotor-encoding

I

S

REDOR scheme

I

S

t1

Rotor-

encoded

REDOR scheme

trcpl

trcpl trcpl

Two alternative

concepts for measuring recoupled interactions:

• following the signal intensity as a function of the recoupling time (resulting in build-up

or dephasing curves)• recording rotor-encoded

signal (resulting in MAS sideband patterns)

Rotor-encoding of dipolar Hamitonians

IS0

R

D 2 2 sin 2 sinΦ = − β γω

Recoupled

dipolar Hamiltonian:

with dipolar “phases” for first recoupling period:

IS1 R 1t

R

D 2 2 sin 2 sin( t )Φ = − β ω + γω

and for “rotor-encoded”second recoupling period:

t1 2nd recouplingperiod

1st recouplingperiod

ωR

ωR

0Φ0Φ

0Φt1Φ

Leads to Amplitude

Modulation

of Signal

and hence, Sidebands

REDOR-type curves and sideband patterns

ω/ωR

-3 3-1 1 5-5

(i)

(ii)

(iii)

0 1 2 3

DISt rcpl

/ 2π

Inte

nsi

ty (

a.u.)

0.5

0.4

0.3

0.2

0.1

0

(i)(ii)

(iii)

Build-up curves decay

HDOR sideband patternsrobust:

Multispin effects: additional sidebands

Multispin effects and relaxation

Multiple-quantum NMR methods: investigating (supra)molecular

structure

H H''H'

H

H'H''

CH' CH2CH

O

OCH2

H'H

H

H'

rHH

H

H'

rNH

5 3 1 –1 –3 –5

1 H-1 H

hom

onuc

lear

1 H-13

C/15

N h

eter

onuc

lear

internuclear proximities,chemical

shifts

and π-shiftsinternuclear distances

π

7 -79 -9

5 3 1 –1 –3 –57 -79 -9

0.21 nm

0.24 nm

0.27 nm

0.16 nm

0.18 nm

0.20 nm

molecular

dynamics

Multiple Hydrogen Bonds in Natural and Synthetic Systems

DNA

Supramolecular polymers via hydrogen bonds

C H2713

C H2713

N

NN

Hd

Hb Hb Ha

N

N N

N O

Hd

O

HaO

Hc

N

O

Hc

n

R.P. Sijbesma, E.W. Meijer et al., Science, 1997:Thermoreversible linkages through quadruple hydrogen bonding

C H2713

N

N

HaO

N

C H27Hd

ON

Hc Hb Hb Hc

O N

NHa O

N N

Hd

13

n

Keto form Enol form

Watson-Crickbase pairs

DQ NMRspectrum

DQ NMRspectrum

Heat-Induced Tautomeric

Rearrangement: 1H-1H DQ Spectra of Quadruple Hydrogen Bonds

Heating

51015single quantum ω2 [ppm]

10

15

20

25

30

ω1 [p

pm]

a b c d

b- c

b- b

C H2713

N

N

a

O

N

C H27d

ONc b b c

O N

N a O

N N

d

13

n

before heating: keto form

51015single quantum ω2 [ppm]

10

15

20

25

30

ω1

[ppm

]

a b c d

b- b

a- b

C H2713

C H2713

N

NN

d

b b a

N

N N

N O

d

O

aO

c

N

O

c

n

after heating: enol form

Kinetics

of the

Tautomeric

Rearrangement

0

0.2

0.4

0.6

0.8

1

Temperature [K]

keto form

enol form

310 330 350 370 390 410 430

enol

frac

tion

TemperatureDependence

T [K]

10-3/T [K]Plot 0 2.70 2.75 2.80

-13

-12

-11

-10

lnk

375 365 355

EA = (145± 15) kJ/mol

tran

sitio

n ra

te

Arrhenius

TimeDependence

0 5 10 15 20 25

0.4

0.5

0.6

Time [h]

T = 375 Kenol form

keto form

enol

frac

tion

Multiple N-H Distances in the Pyrimidinone

Form

REREDOR 24/24240 pm280 pm~ 65°

107.5REREDOR 6/6107.5 pm201 pm~ 180°

106

201

1H{15N}

recoupling: 1H - detection

-8 -4 0 4 8 ω/ωR

201

107.5

~180°

-8 -4 0 4 8 ω/ωR

280

240

~65°

REREDOR 8/8107 pm

-8 -4 0 4 8 ω/ωR

107

Separator Membranes and NMR

reveal details of proton conductivityon molecular level(site-selective & non-destructive)

provide structural constraints(proton transfer mechanism ?)

PVPA: poly(vinyl

phosphonic

acid)

31P NMR

phosphonic

acid units, local dynamics

1H-31P and 1H-1H NMR

hydrogen bonding

at phosphonic acid units

1H-13C NMR

segment mobilities of alkyl chains, polyvinyl backbone

1H NMR

backbone as well as mobile protons

(local dynamics)

2H NMR

primary process: orientation-dependent rate of movement:time scale and geometry (multi-site jumps)

NMR probes for local structure & dynamics

High proton conductivity under dry conditions at elevated temperatures

PVPA: VT NMR motional narrowing

condensationbackbone unaffected

mobileOH groups

31P

MAS NMR1H

MAS NMR

very narrow

lines in both 1H and 31P spectra

Poly(vinyl phosphonic

acid): PVPA

EA

= 25 kJ/mol2.2 2.4 2.6 2.8 3.0 3.2

-9

-8

-7

-6

-5

Log

(T2* )

1000/T (K-1) -414 12 8 4 0ppm

10 6 2 -2

2015105

-5

ppm

25

0mobile

1H -

1H DQ Spectra

281 K

-416 12 8 4 0ppm

2015105

-5

ppm

2530

0

318 K

motion frozen

**

n

POO

ODD

P-OH proton: mobile

P-OH : mobile proton, hydrogen bondedDynamics of motion involved in proton conduction

20 10 0 -10

413 K

394 K

375 K

356 K

318 K

ppm150 100 50 0 -50 -100 -150

297 K

393 K

353 K

253 K

kHz

230 K

1H MAS spectra

2H solid echo spectra

PVPA: ab initio structure

(model

geometry)

Ab initio calculation based on model geometry (CPMD):

* Elucidation of hydrogen bondings and 1H chemical shift calculation:H-bonding between phosphonic acids on the same chains and between two parallel chainsMD: Proton hopping occurs along chains as well as between chains mediated by hydrogen bonds.calculated δ(P-OH) = 9.7 ppm (exp.: 10.6 ppm)

H-bonding between the chains

H-bonding along the chains

PO

PVPA: Averaging of Deuteron Quadrupole

Coupling

Broad distribution of angles betweeninstantaneous O-H and C-P directions,yetQuadrupole coupling reduced by factor 10 after CPMD run of 15 ps


Introduction

•



Phase Behavior •


Conclusions •

Basics

Chain Branching




Scattering and NMR

Scattering and NMR in Bulk Polymers

incoherent coherent single quantum

double quantum

D

Y

N

A

M

I

C

S

Molecular n-quasielastic n-quasielastic 2D-, 3D-, 4D- exchange

sidebands

Collective n-spin-echo 2D-exchange decay of DQC

S

T

R

U

C

T

U

R

E

Molecular WAXS, WANS chemical shift,sidebands

2D pattern,sidebands

Collective

(packing)

X-ray pole figures, SAXS, SANS

DECODER chemical shift

2D signal pattern

SCATTERING NMR

Advantages of NMR:

- Selectivity, Versatility

- Detailed information on geometryand time scale of dynamics

- Large range of length- and timescales accessible

- Elucidation of supramolecularorganization

- Relation between structure, dynamics and functional behavior

- Limits not reached, e.g. microcoils


K. Schmidt-Rohr, H.W. Spiess, Multidimensional NMR and Polymers, Academic Press, London, 1994

H. W. Spiess, Advanced Solid-State Nuclear Magnetic Resonance for Polymer Science;J. Polym. Sci. A 42, 5031–5044 (2004).

H.W. Spiess, NMR Spectroscopy, in Macromolecular Engineering, edited by K. Matyjaszewski, Y.Gnanou, L. Leibler, WILEY-VCH, Weinheim, Vol. 3, 1937-1965 (2007).

H. W. Spiess, NMR Spectroscopy: Pushing the Limits of SensitivityAngew. Chem. Int. Ed. 47, 639-642 (2008).

References

overview of nmr of bulk polymersoverview of nmr of bulk polymers. introduction • configuration,...

Documents