Beate Thomas Weiser

Prof. Dr Sigrun Chrubasik

B Beime

Effectiveness and safety of topical capsaicin cream in the treatment of chronic soft tissue pain

Keywords: soft tissue pain, low back pain, capsaicin, placebo, clinical trial, chronic pain chronic pain, soft tissue pain, low back pain, capsaicin, placebo, clinical trial

Effectiveness and safety of topical capsaicin cream in the treatment of chronic soft tissue pain

Improvements in the secondary efficacy measures confirmed the results. Likewise, all outcome measures had significantly more improved in the capsaicin-treated compared to the placebo-treated chronic back pain sufferers. All patients were included in the safety assessments. More adverse events occurred in the capsicum group (n=13) than in the placebo group (n=6). The capsaicin cream was generally well tolerated.

Our results indicate that capsaicin cream is useful in patients with chronic soft tissue pain and is also efficacious in patients with chronic back pain for which effectiveness was already demonstrated in earlier clinical trials.

Introduction

Among painful musculoskeletal conditions, chronic pain of the soft tissues (muscles, fascia, tendons, ligaments, cartilage, synovium, fibrous capsules, organs, and nerves) is as frequent as symptomatic peripheral osteoarthritis. For both, a prevalence of around 9% was calculated from a regional community-based study in Italy [START_REF] Salaffi | Prevalence of musculoskeletal conditions in an Italian population sample: results of a regional community-based study. I. The MAPPING study[END_REF]. Chronic soft tissue pain can be just as debilitating as pain from bony structures. Among the treatment options for chronic musculoskeletal pain, topical NSAIDs have been shown to be as effective as oral NSAIDs (Mason et al., 2004a). Topical capsaicin products have also shown to relieve a variety of pain conditions including osteoarthritis (Cameron et al. 2009a), rheumatoid arthritis (Cameron et al. 2009b) and low back pain [START_REF] Keitel | Capsicum pain plaster in chronic non-specific low back pain[END_REF][START_REF] Frerick | Topical treatment of chronic low back pain with a capsicum plaster[END_REF]. The rationale behind this effect is the capsaicin interaction with neurotransmitter, substance P in sensory C nerve fibres and its interaction with the vanilloid receptor TRPV1, which triggers thermal and inflammatory pain. Repeated capsaicin applications result in desensitization and inactivation of the neurons which trigger pain. Reversible degeneration of epidermal nerve fibres during prolonged capsaicin exposure contributes to the analgesic effect. Nolano and coworkers (1999) have shown that discontinuation of the application of capsaicin after 3 weeks was followed by re-innervation of the epidermis over a 6-week period with a return of all sensations, except cold, to normal levels. This explains why the capsaicin analgesic effect may carry over after the cessation of treatment. Local skin irritation (sensation of warmth and/or pruritus), which is often mild and transient, is common during treatment with capsaicin products and can be easily explained by its mechanism of action, however, systemic adverse events were not observed in any of the clinical trials.

In a meta-analysis of fibromyalgia treatment interventions, a trial with topical capsaicin is listed [START_REF] Rossy | A meta-analysis of fibromyalgia treatment interventions[END_REF]. This pilot study did not find a significant effect. (McCarty et al., 1994). Other information on the use of topical capsaicin in chronic pain caused by soft tissues The test product was "Finalgon® CPD Wärmecreme" , of which 100 g contain 2.2 -2.6 g soft extract of capsici fructus acer (DER 5.5:1 (4-7:1) corresponding to 53 mg capsaicin (0.05%).

The reference product was a cream, identical in viscosity, colour and fragrance to the test product, containing no active substance. The cream was coloured with chinolin yellow to ensure an optical match with the yellowish colour of the test product. The decision which treatment a suitable patient should receive was made according to a computerized randomisation list. Because a burning or itching sensation might have demasked the blinding, the patients were informed that absence of this effect did not provide a definite information of the group to which the patients had been assigned. Patients were instructed to apply the cream as a thin layer and rub it onto the painful area thrice daily over 3 weeks. The study consisted of a treatment phase of 21 days and a facultative follow-up of 14 days.

Selection of patients:

Patients were eligible if all the following criteria were met according to the history of the patients and the physical examination by the physician: age between 18 and 65 years, Caucasian, chronic pain of the soft tissues of the musculoskeletal apparatus, subjective pain at enrolment ≥ 5 (VAS 0-10, 0 no pain, 10 intolerable pain), ability and Concomitant analgesic treatments which could interact with the study medication had to be discontinued before enrolment into the study, Antidepressants, anxiolytics or sedatives, surgical procedures or nerve blocks in the treatment area, systemic opioids, oral corticosteroids, transcutaneous electric nerve stimulation had to be stopped for 4 weeks.

Topical antirheumatics on any part of the body surface for 14 days,non-steroidal antiinflammatory drugs, all topical medicines used in the painful area for 7 days, and topical corticosteroids, baths with aromatic oils for 3 days,. If antihistaminics or muscle relaxants were taken, the dose had to remain unchanged from 7 days before the start of study until its end. Additional analgesics were not allowed except for a short term for acute pain conditions, such as tooth ache.

Outcome Measures: The Pain Sum Score (VAS 0-10) consisting of average pain within the last 24 hours, the worst pain within the past 3 days and average pain within the last 3 days, Secondary outcome measures included the VAS pain subscores and the patients' diaries which contained daily records of the following items: pain intensity before the first cream application and at the moment of maximum pain relief using numeric pain scales (0-10, 0 no pain, 10 intolerable pain), the delay between application of cream and onset and maximum effect, the duration of analgesia and questions on adverse effects (sensation of warmth, itching). Duration of any sick leave since the past visit was also documented after one, two and three weeks as were the efficacy (investigator: excellent, good, adequate, unsatisfactory; patients: free of complaints, symptoms improved, unchanged, worsened) and adverse events (severity, causal relationship, actions taken, outcome), vital parameters (blood pressure, heart rate), skin status, Specific questions concerning the sensations of warmth or pruritus were recorded: Did you experience these sensations during the past week (yes/no)? If so, at how many days (the first day only, up to the second/third day, the whole week, varying)? If so, how long did it last (1-3h, 3-6h, 6-12h, varying)? Intensity of the sensation: Numeric scale (0-10) and a global safety evaluation (investigator: good (no ADR), moderate (tolerable ADR), unsatisfactory (severe ADR), patients: good, moderate, unsatisfactory) -some of the variables also assessed after 5 weeks.

Statistical Analysis

A patient was rated as a responder if the percentage change in the pain sum score relative to the admission value was at least 30 %, i.e. [enrolment value minus final assessment value] / enrolment value x 100 ≥ 30. The 30% decrease was chosen as therapeutically relevant level, following agreement with the study's rheumatological advisers and rheumatologists in independent centres. It is also known that responder rates of 50% or more can be achieved on (Moore et al., 1997). In order to set the difference between the two medications as high as could be reasonably assumed, a minimum responder rate of 65% was expected for the active treatment after 3 weeks, versus a maximum rate of 45% for placebo. The main target of the study was the confirmative proof of a superior efficacy of capsicum cream compared to placebo cream, the between-group difference of the responder rate  was estimated by means of the odds ratio:

OR = ( Verum * (1 - Placebo )) / ( Placebo * (1 - Verum )).
This would result in case of equal responder rates of OR = 1. The following hypotheses for the odds ratio were established:

The null hypothesis postulated: OR ≤ 1 versus an alternative hypothesis of: OR > 1. A onesided 97.5 % confidence interval was calculated for the odds ratio. Only the lower limit was calculated because of the hypothesis of the expected superiority of the capsicum cream. If this limit was above 1, a superior efficacy of the capsicum cream could be shown at a 2.5 % p level.

Before the analysis, the medication groups were tested for homogeneity on enrolment. Twotailed 90 % confidence intervals were calculated for between-group differences quantified by the MANN-WHITNEY statistic or the odds ratio depending on the type of the variable. The following variables entered the test: per protocol collective (yes/no; odds ratio); gender (odds ratio), age, height and weight (MANN-WHITNEY statistics), duration of the disease (MANN-WHITNEY statistics), pre-treatment non-medicinal and analgesic (yes/no, odds ratio), sick leave before enrolment (yes/no, odds ratio), pain subscores and total score (MANN-WHITNEY statistics), duration of treatment until final visit (MANN-WHITNEY statistics), proportion of patients with "very good" to "good" compliance rating (odds ratio).

In case of serious between-group differences in one of these variables with a possible impact The primary efficacy analysis was carried out on all randomised patients who fulfilled the inclusion criteria and who used trial medication at least once (intention-to-treat (ITT) population). Missing values of efficacy variables were replaced according to the method "last observation carried forward". Drop-outs were rated non-responders, unless they stopped treatment because of being free of symptoms, who were rated responders. An according to protocol (PP) analysis was performed on all patients without relevant protocol deviations. The assignment to the PP collective was done for each single case after closing the data base while the study data were still blinded.

We used our previous experience with topical capsaicin [START_REF] Keitel | Capsicum pain plaster in chronic non-specific low back pain[END_REF][START_REF] Frerick | Topical treatment of chronic low back pain with a capsicum plaster[END_REF] to form the present hypothesis. The calculation of the sample size of this study was based on an effect size OR = 2.27, positioned in the range of moderate superiority (benchmark 2.232 according to Tritschler, 2.477 according to Hasselblad-Hedges) using the program nQuery Advisor® Release 2.0 from Statistical Solutions for Fisher's exact test, the counterpart to the exact confidence interval for the odds ratio. The one-sided test had a risk of 0.025 (power 90 %) suggesting group sizes of 138 patients. For quality assurance, the data were monitored.

Results

A total of 282 patients were recruited. One patient from the placebo group was excluded from all analyses because of non-participation in any of the follow-up visits. Table 1 lists the baseline characteristics of the 281 patients who were considered for analysis. The tabulation of all details has been placed on a website (http//www.uniklinikfreiburg.de/rechtsmedizin/live/forschung/phytomedicine/originalartikel.html). Unfortunately, the major part of the patients did not meet the inclusion criteria of chronic soft tissue pain Metric and nominal variables at baseline were comparable between the groups. Ten patients (6 group capsicum, 4 group placebo) had to be excluded from the per protocol analysis because of major protocol violations such as violations of inclusion criteria concerning age, concomitant therapy or concomitant diseases, or poor compliance. The location of the chronic soft tissue pain of the capsicum and placebo groups is presented in Figure 1 and was comparable between the groups as were the baseline pain sum scores and the size of the painful area (Table 1).

Prior non-drug treatments (mainly massages and electrotherapy) had been used by 31.2% of the capsicum and 19.7 of the placebo patients and prior drug treatment by 23.4% and 27.3%, respectively. At inclusion, 87.5% of the capsicum and 93.9% of the placebo patients were fit for work. 58% of the ITT population suffered from concomitant diseases (e.g. hypertension, hyperlipidaemia, thyroid disorders) with no relevant difference between the groups. The patients' reliability was assessed as very good or good in about 90% of the patients in both groups.

After 2 weeks of treatment, the pain sum score had improved significantly more in the capsicum group compared to placebo (Figure 2) . There were also more responders (pain sum score improvement ≥ 30 and 50% (sensitivity analysis) in the capsicum group compared to placebo (Figure 3a andb). The corresponding odds ratios (30%/50%, (CI 97.5% lower limit) after 1, 2 and 3 weeks were 1.0 (0.5)/0.5 (0.1), 2.6 (1.2/1.9 (0.8) and 4.3 (1.9)/2.5 (1.1), F o r P e e r R e v i e w 10 respectively. For the numbers of cases included, the retrospective calculation of the power was 97% after 3 weeks of treatment. The PP analysis supported the results of the ITT analysis reported here (see webpage). The improvements of the pain subscores were greater in the capsicum group (Table 2) as were the numbers of patients with a decrease by more than 3 points of average and worst pain compared to placebo groups. At the end of the optional 2 weeks follow-up after completion of the 3 weeks treatment, 46 patients of the capsicum group and 47 patients of the placebo group were available for additional evaluation of the pain score. No relevant between group difference was observed after stopping the cream application. At the final evaluation, the efficacy of the capsicum treatment was rated as good to excellent in 59.3% of the cases compared to 24.3% for placebo treatment and 21.9 % (capsicum) and 51,5% (placebo) respectively, of the patients complained of symptoms unchanged or worsened. The diary pain score from before the first daily application of cream decreased from week to week, the decrease was more pronounced in the capsicum group.

Capsicum patients documented on more than half of the treatment days an onset of pain relief within the first hour following application of the cream, compared to only 19-23 % under placebo. Only about one quarter of the times these patients reported to have felt no effect compared to 50-70 % under placebo. The absolute numbers of days where the patients reported an analgesic effect were > 70% in the capsicum group than in the placebo group (< 30%). In 80% of these patients the maximum effect was reached within 2 hours after application. In about 50 % of the patients of the capsicum group, the pain relieving effect persisted for 2-4 hours.

The 142 patients with chronic soft tissue and vertebral pain formed a rather homogenous chronic back pain group. Baseline characteristics are presented in Table 1 andFigure 4. Chronic back pain sufferers showed pain sum score improvements (Figure 5) and percentages of responders (Figure 6) in the same range as chronic soft tissue pain sufferers (Figures 2 and3). The incidence of adverse drug reactions (ADRs) was low. Three patients in the capsicum treated group suffered from ADRs with a definite causal relationship to the trial medication (2.1%), which led in all 3 patients to premature termination of treatment. The symptoms reported were an unpleasant local heat sensation in 2 patients and pruritus in the other. No other ADRs with possible or probable relationship to the trial medication occurred. During placebo treatment, no ADRs were recorded, related to the trial medication. The 3 patients who dropped out judged the safety of the capsicum cream as "unsatisfactory" at any visit, while there were no other ratings than "good" in the placebo group. Vital parameters remained unaffected during the course of the study.

The incidence of heat sensation was 83.6 %, 80.0 % and 78.6 % after one, 2 and 3 weeks for the capsicum group and 33.3 % / 32.6 % and 32.6 % for the placebo group, respectively.

Pruritus was seen considerably less frequently, occurring in 18.6 %, 14.3 % and 11.4 % of the capsicum and in 5.0 %, 4.3 % and 3.5 % of the placebo patients after 1, 2 and 3 weeks of treatment. The severity of the heat sensation in the capsicum group was mainly reported as 2 to 6, versus 2 to 3 in the placebo group. The severity of pruritus was predominantly reported as ≤ 3 in both groups. In the capsicum group scores ≥ 3 and were more frequently mentioned at visit 2 than at later assessments.

At admission about 90% of patients in both treatment groups showed a normal skin type.

After one week of cream treatment, 92.1 % (capsicum) and 97.2 % (placebo) of the patients showed no skin irritations. During the 3-week observation period, only two (1.4 %) patients on active treatment and one (0.7 %) on placebo showed skin irritation. No serious reactions were observed. In general, investigators and patients rated the tolerability of the capsicum cream as good. Investigators and patients rated the safety of the capsicum cream as "good" in 93.6 % and 92.1 % of the patients and that of the placebo cream by 97.2 % (investigators and patients).

Discussion

Based on recommendations of meta-analyses and systematic reviews, Kroenke and coworkers (2009) have developed a stepped care approach to the pharmacotherapy of chronic pain. The spectrum of treatments includes: (1) oral analgesics, (2) tricyclic antidepressants or tramadol,

(3) in case of neuropathic pain: gabapentin, duloxetine or pregabalin, (4) in case of fibromyalgia: cyclobenzaprine, pregabalin, duloxetine, or milnacipran, (5) in case of localized neuropathic or arthritic pain: topical analgesics (capsaicin, lidocaine, salicylates) and ( 6) opioids either applied individually or in combination. The domain of topical capsaicin is primarily among those who are unresponsiveness to other treatments. Three to four local applications of capsaicin over at least 3 weeks in case of musculoskeletal pain or 6 weeks in case of neuropathic pain was associated with a significant higher 50% reduction of pain than placebo (Mason et al., 2004b). Our results are in accordance with the findings of this metaanalysis. For chronic low back pain, the numbers needed to treat (NNTs) for a 30 and 50% improvement of the pain sum score after 3 weeks of treatment were calculated to be 6 and 5, respectively [START_REF] Frerick | Topical treatment of chronic low back pain with a capsicum plaster[END_REF]. Such a result was also achieved with topical or oral NSAIDs in the management of chronic musculoskeletal conditions. Mason and coworkers have calculated a NNT of 5 for a 50% reduction of pain due to capsaicin in patients with osteoatrthritis (Mason et al., 2004a).

For the treatment of chronic soft tissue pain, we calculated NNTs of 2.9 for 30% pain sum score reduction and 4.7 50% pain sum score reduction. For comparison with data from the literature we also calculated the NNTs for 50% reduction of average pain within the past 24 hours (NNT 4.1) and worst pain of the past 3 days (NNT 4.4). All these numbers indicate that the proprietary capsaicin cream has a place in the treatment of chronic soft tissue pain as adjunct or as sole treatment. The response to capsaicin in chronic soft tissue pain is as good as that of chronic back pain for which our data gave NNTs of 4.2 (30% pain sum score reduction) and 3.9 (50% pain sum score reduction), respectively. Retrospective power calculation revealed that the number of patients with chronic soft tissue pain was associated with a power of 97% (for a number of 281 patients 99.9%). Even smaller groups than the 64 patients in the capsicum and 66 patients in the placebo group would have been sufficient to demonstrate the superiority of the capsaicin cream versus placebo. The high power is reassuring as concerns the reliability of the result (type two error 2.5%).

In the clinical trials including patients suffering from osteoarthritis so far, 0.025% capsaicin cream was used. The capsaicin plasters for the treatment of chronic low back pain released 11 [START_REF] Keitel | Capsicum pain plaster in chronic non-specific low back pain[END_REF] and 22 [START_REF] Frerick | Topical treatment of chronic low back pain with a capsicum plaster[END_REF] ug capsaicin per square cm over 4 to 8 hours per day. We used a 0.05% capsaicin cream whereas in the trials for neuropathic pain 0.075% capsaicin products were employed [START_REF] Derry | Topical capsaicin for chronic neuropathic pain in adults[END_REF]. It may well be that a higher capsaicin concentration or a longer treatment period might result in better pain relief.

Oral treatment of chronic pain is associated with a relatively high risk of adverse drug reactions which may itself cause morbidity, mortality and extra costs (Piramohamed et al., 2004). The advantage of topical capsaicin treatment over other pharmacotherapy is the low risk of adverse events and the minor nature of those which ocurred. Except for the local warmth and pruritus sensation no adverse events attributable to the capsaicin cream were observed. In general, local capsaicin adverse events occurred in around one third of patients and would not have occurred if these patients were treated with placebo (Mason et al., 2004b). However, in few patients, the skin irritation lead to withdrawal from treatment, in our study this was 2%.

Since chronic pain conditions often require long-term treatment, future studies should assess the effectiveness and safety for topical capsaicin at least over a few months. Treatment over 3 months is described for various other pain conditions, e.g. post herpes zoster pain, or pain due to diabetic neuropathy (Mason et al., 2004b). Future studies on osteoarthritis or back pain should also consider the recommended response criterion or criteria of the Orthopaedic Societies OMERACT / OARSI [START_REF] Pham | Outcome variables for osteoarthritis clinical trials: The OMERACT-OARSI set of responder criteria[END_REF][START_REF] Pham | Osteoarthritis Research Society International set of responder criteria for osteoarthritis clinical trials revisited[END_REF] for better comparability. This Chronic back pain responders (≥ 50% Pain Sum Score improvement)

  . The aim of this study was therefore to evaluate the efficacy and safety of a proprietary capsaicin cream 0.05% in the treatment of chronic soft tissue pain.MethodsDesign of the study: We conducted a randomized, double-blind, placebo-controlled study in 23 outpatient clinics in Germany. The protocol was approved by the Ethics' Committees responsible for the principal investigator and each trial centre. The study was planned and conducted in accordance with the GCP guidance for the European Community and the ICH GCP guidelines which are part of the German Drug Laws. Informed written consent was obtained from all patients.

  the patient to follow the investigator's instructions, i.e. meeting the prerequisites of the study, applying study medication according to the dosage regimen and filling in the questionnaires at the control visits, and granting of written informed consent.Patients were excluded in case of severe co-morbidity, addiction to alcohol or other drugs, pregnancy and lactation, insufficient contraceptive protection, participation in another clinical trial within the last 4 weeks, concomitant psychiatric disorders, a surgical procedure required in the immediate future, inability of the patient to understand the nature, importance and consequences of the study. Special exclusion criteria: muscle rupture, vertebral disk prolapse, spondylolisthesis, spinal canal stenosis, known or clinically proven instability of the spine, spinal fractures, tumours, infections, inflammatory joint conditions, seronegative spondyloarthropathies, osteoporosis as the cause of pain, chronic skin diseases, known hypersensitivity to capsaicin or other ingredients of the cream, anxiety or depressive conditions, ≥ 11 points of HADS D anxiety score or HADS D depression score.

  total score of 30 was assessed at baseline, after one, two, three and five weeks. The primary outcome measure was treatment response, defined as a Pain Sum Score reduction ≥ 30%.

  effect, adequately stratified analyses of the main target parameter were proposed.

  142 patients also suffered from pain due to degenerative spine diseases (chronic back pain) and 9 patients suffered from degenerative diseases of other joints. The ITT population of chronic soft tissue pain consisted therefore of 130 patients (n=64 group capsicum; n=66 group placebo). Seven patients dropped out (capsicum group: n=3 were symptom free shortly after the beginning of treatment, n=2 due to insufficient pain relief, n=1 refused to continue, possibly because of poor tolerance, one patient of the placebo group was symptom free shortly after the beginning of treatment).

  improvements of pain, function and patient's global assessment.Inclusion of a quality of life measure would in addition give information in that important domain.

  Figure 1aBody Diagram

Table 2 Relative pain subscore improvements (%) (Median, 25/75 IQ) *p<0.05,

 2 **p < 0.01, ***p < 0.001

	Week 1	Week 2	Week 3	Week 1	Week 2	Week 3

http://mc.manuscriptcentral.com/ptr

Table 1 Biometrical data, duration of complaints, size of painful area and baseline pain sum score of all patients and those suffering from chronic soft tissue pain and back pain in the capsicum and placebo groups

 1

	(mean ± standard deviation)
			All patients	Chronic Soft Tissue Pain	Chronic Back Pain
			Capsicum	Placebo	Capsicum	Placebo	Capsicum	Placebo
			n=140	n=141	n=64	n=66	n=71	n=71
	Age (ys)		48.6 ± 13	48.9 ± 13	48.9 ± 14	49.9 ± 14	48.1 ± 12.6	48.4 ± 12.1
	Height (cm)		170 ± 9	168 ± 8	170 ± 9.4	167 ± 7.8	170 ± 8.3	168 ± 8
	Weight (kg)		76.3 ± 15.6	72.6 ± 13.7	77.2 ± 17.3	72.9 ± 14.7	76.1 ± 14.1	72.4 ± 13.1
	Sex (males)		59	48	32	22	25	24
	Duration of complaints
	< 2 weeks		8	8	6	2	2	6
	1 month		27	27	13	17	14	9
	2-3 months		23	20	12	13	10	7
	4-12 months		21	25	8	13	11	10
	>1 year		61	61	25	21	34	39
	Size of painful area (cm	2 )	300 ± 309	310 ± 325	269 ± 236	272 ± 304	342 ± 365	358 ± 344
	Pain Sum Score				20.8 ± 3.1	21.2 ± 3.1	20.5± 2.4	20.5 ± 2.6

http://mc.manuscriptcentral.com/ptr Phytotherapy Research

http://mc.manuscriptcentral.com/ptr

Ackknowledgement

The authors sincerely thank Professor Tom Torda/Sydney for his valuable comments prior to submitting the manuscript.


Catherine E Powell

Preconditioning Stochastic Saddle-point Problems List of authors

come

Preconditioning Stochastic Saddle-point Problems

Stochastic Galerkin and stochastic collocation methods are becoming increasingly popular for solving (stochastic) PDEs with random data (coefficients, sources, boundary conditions, geometry). The former usually require the solution of a large, coupled linear system of equations while the latter require the solution of a large number of small systems which have the dimension of the corresponding deterministic problem. Although many studies on positive definite problems with random data can be found in the literature, there is relatively little work on stochastic saddle-point problems.

In this talk, we give an overview of the linear algebra issues involved in applying stochastic Galerkin and collocation schemes to saddle-point problems with random data. We focus on a mixed formulation of a second-order elliptic problem and investigate the efficiency of preconditioning strategies of Schur-complement and augmented type, for use with MINRES. Obtaining preconditioners that are robust with respect to the spatial discretisation parameters, the choice of discretisation on the underlying probability space, as well as the statistical parameters of the random inputs is a tall order.

Stochastic collocation methods, when combined with suitable mixed finite element methods in the physical domain, yield standard deterministic saddle point systems. These are trivial to solve when considered individually; the challenge lies in exploiting their similarities to recycle information and minimize the cost of solving the entire sequence.

Galerkin approximations, which couple mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. However, the matrix blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. For stochastically nonlinear problems, this is compounded by the fact that the matrices are blockdense and the cost of a matrix vector product is non-trivial.

1 University of Manchester 1


Noelia San-José

Ana Gómez-Valdemoro

Félix Saturnino Ibeas

Clemente García

Felipe Serna

Prof. Dr José Miguel García

Colorimetric anion sensing by polyamide models containing ureabinding sites

Keywords: urea binding sites, anion sensing, polyamide model compounds urea-binding sites, anion sensing, polyamide model compounds

Colorimetric anion sensing by polyamide models containing urea binding sites.

Introduction

The development of selective and sensitive materials for the colorimetric detection of anionic, cationic, and neutral molecules is a topic of current scientific interest (1)(2)(3)(4)[START_REF] Sessler | Anion Receptor Chemistry[END_REF][START_REF] Anzenbacher | [END_REF](7). In particular, the sensing of anions has drawn considerable attention because charged chemical species are ubiquitous and play major roles in many chemical and biological processes. Detection is often based on the design and synthesis of receptors (host molecules) capable of selectively binding anionic species (guest molecules) with a concomitant change in a measurable macroscopic property, such as fluorescence or UV/Vis spectra. The field of synthetic anion receptor chemistry is one of the fastest growing disciplines within supramolecular chemistry.

Methodologies based on colorimetric changes of host-guest interactions are especially attractive for their potential use in naked-eye screening applications, especially in situations where conventional techniques are not appropriate or are expensive, or in the development of screening applications that can be used by nonscientist personnel.

The design and syntheses of molecules with selective anionic receptor sites are complex, mainly due to the variety of geometric shapes of anions. The recognition of anions in aqueous environments by biological systems is achieved via hydrogen bonding by highly organised proteins with partially hydrophobic and sterically welldefined complexation sites in the interior of the protein. The protein behaves as a hydrophilic macromolecule, surrounded by water, with partially hydrophobic binding sites where hydrogen bonds, or other feeble interactions, between these sites and the substrates can be formed in the absence of competition with water (8)(9)(10)(11)(12). The complexation properties of such receptor proteins can be mimicked by chemically sophisticated hosts with well-organised binding sites or, alternatively, by slightly hydrophilic synthetic macromolecules with a chemically simple receptor moiety. The latter approach has been scarcely explored, though it has shown promising results in the selective recognition of anionic species in water solutions [START_REF] Anzenbacher | [END_REF](8)(9)(10)(11)(13)(14)(15)(16)(17)(18).

For example, rectangular strips were prepared with polymethacrylate with a pendant pyrylium substructure, showing excellent colorimetric selectivity toward hydrogen carbonate in water. Remarkably, the colour changes in the presence of the anion were observed only in the slightly water-swelled polymeric matrix, and there was no change upon addition of the anion to an aqueous solution of the pyrylium monomer (14,18).

Polymers with host moieties are not only excellent supports but also allow fine alterations of their hydrophilic/hydrophobic characters, allowing for the tuning of their sensing ability.

Our work focused on the development of polymeric optical transducers for analyses. In previous articles, we reported on the syntheses and characterisations of polymer materials with the following host units: crown ether (19)(20)(21), urea (22)(23)(24)(25) or pyrylium (13,18), along with their interaction with cations (26)(27)(28)(29)(30)(31), anions (13,18) or gas molecules (32). Herein, we report four model polyamide compounds (Scheme 1) that exhibit sensing selectivities towards acetate ions; one of these showed colorimetric sensing capabilities toward acetate anions. The models probably mimic polyamides (Scheme 2, P1-P4), polymers derived from diacid monomers (1)(2)(3)(4), which are intermediates in the synthesis of the models. The polymeric materials are supposed to retain the ability to interact selectively with these diamines, such that cheap future sensing devices might be user-friendly naked-eye polyamide film sensors or incorporate a polyamide coating at the end of an optic fiber connected to a portable UV/Vis diode- technological importance, due mainly to their mechanical and electrical properties and outstanding thermal resistances (13,33). Thus, sensing devices prepared using these materials could be utilised in a wide range of thermal and mechanical conditions. These properties make them unique components in the control of industrial chemical concentrations, such as in air, gases, reactors, or waste products. They also have potential in environmental protection and occupational safety and health, through the determination of chemical concentrations.

Experimental

All materials and solvents were commercially available and used as received, unless otherwise indicated. N-Methyl-2-pyrrolidone (NMP) was vacuum-distilled twice over phosphorous pentoxide, and then stored in the presence of 4-Å molecular sieves.

Lithium chloride was dried at 400ºC for 12 hours prior to use. Triphenylphosphite The following salts were used as received in the complexation studies: sodium hydroxide (98%, Panreac), tetrabutylammonium fluoride hydrate (98%, Sigma-Aldrich), sodium bicarbonate (99.5%, Sigma-Aldrich), lithium phosphate (Sigma-Aldrich), lithium acetate (99.99%, Sigma-Aldrich), lithium benzoate (99%, Sigma-Aldrich), potassium oxalate monohydrate (puriss., Riedel-de Haën), sodium ptoluenesulfonate (95%, Sigma-Aldrich), sodium chloroacetate (98%, Sigma-Aldrich), lithium dihydrogenphosphate (99%, Sigma-Aldrich), lithium trifluoroacetate (95%, Sigma-Aldrich), lithium bromide (puriss., Riedel-de Haën), sodium methanesulfonate (98%, Sigma-Aldrich), lithium chloride (anhydrous puriss., Sigma-Aldrich), and tetrabutylammonium iodide (99%, Sigma-Aldrich). Thus, the anion binding studies were carried out using different counterions (lithium, sodium, potassium and tetrabutylammonium). The solvents used in the complexation studies were dry dimethyl sulfoxide (DMSO, 99%, Merck) or dry dimethyl sulfoxide-d 6 (deuteration degree min. 99.8%, Merck).

Polyamide model synthesis

N,N'-Diphenyl-5-(3-phenyl-ureido)-isophthalamide model (M1

). In a 50 mL threenecked flask fitted with mechanical stirring, 20 mmol of aniline, 10 mmol of 5-(3phenylureido)isophthalic acid and 1.4 g of lithium chloride were dissolved in a mixture of 6 mL of pyridine, 22 mmol of TPP, and 20 mL of NMP. The solution was stirred and heated at 110 ºC under a dry nitrogen blanket for 4 hours. Then the system was cooled at room temperature and the solution was precipitated in 300 mL of methanol. The compound obtained was filtered and washed with distilled water and acetone. Then it was extracted with acetone for 24 h in a Soxhlet, and dried in a vacuum oven at 80 ºC overnight (yield: 94%). It was obtained as a semi-crystalline solid (55% crystallinity) that crystallized at 169 ºC. M.p.: 256 ºC. Tg: 122 ºC. (19), 169 (37), 143 (75), 115 (33), 93 (48), 65 (30). FTIR [Wavenumbers (cm (61), 185 (11), 164 (35), 138 (85), 119 (57), 108 (51), 91 (56), 78 (42). FTIR [Wavenumbers (cm (56), 265 (31), 239 (100), 227 (28), 185 (12), 134 (42), 119 (57), 108 (41), 93 (61). FTIR [Wavenumbers (cm -1 )]: ν N-H : 3342; ν C=O : 1669, 1648, 1636; δ N-H : 1599; ν Ar C=C : 1561.

5-(3-Naphthalen

The polyamide model syntheses and acronyms are depicted in Scheme 1.

Polyamide synthesis, characterisation, and membrane preparation.

The reaction of the diacids (1 to 3, Scheme 1) with m-phenylenediamine yielded polyamides, as depicted in Scheme 2. The polymerisation and the polymer characterisation were performed following previously described procedures (22).

The dense polyamide membranes were prepared by casting an N,Ndimethylformamide (DMF) solution containing about 8 wt% of the corresponding polymer on glass plates. The membranes were then dried in an oven at 70ºC for 4 h.

After removing the membranes from the glass plates by distilled water, they were vacuum-dried for 24 hours at room temperature (rt) and reduced pressure. The resulting yellowish transparent dense membranes had a thickness of 20 to 50 µm.

The skinned asymmetric membranes were prepared by the phase inversion precipitation method. The polymeric solutions consisted of 15-20 wt% of cellulose acetate (39.8 wt% acetyl content, average Mn ~30,000, Aldrich), 10 wt% of the corresponding polyamide, and 70 -75 wt% of DMF as the solvent. The polymer water/DMF coagulation bath (up to 30% v/v of water). The membrane was peeled off of the glass plate, washed for 12 h with water to remove all solvents, dried at rt for 24 h and dried further at reduced pressure for 48 h.

Measurements and instrumentation

The 1 H and 13 C NMR spectra were recorded using a Varian Inova 400 spectrometer operating at 399.92 and 100.57 MHz, respectively, with deuterated dimethylsulfoxide (DMSO-d 6 ) as the solvent.

Fast atom bombardment (FAB) and low-resolution electron impact mass spectra (EI-LRMS, 70 eV) were measured using a Micromass AutoSpec Waters mass spectrometer.

Infrared spectra (FT-IR) were recorded using a Nicolet Impact spectrometer.

The UV/Vis spectra were recorded using a Varian Cary3-Bio UV-Vis spectrophotometer.

Differential scanning calorimetry (DSC) data were recorded on a Perkin-Elmer Pyris I analyser from 10 mg of sample under a nitrogen atmosphere at a scan rate of 20ºC/min. The membrane cross section was observed by means of scanning electronic microscopy (SEM). The SEM images were obtained with a JEOL JSM-6460LV, using an accelerating voltage of 20 kV. The polymer samples were cryofractured from film membranes and coated with gold.

Stock solutions of polyamide model compounds and of concentrated solutions of the salts were prepared in DMSO or DMSO-d 6 for the UV/Vis and 1 H NMR and mixing with small variable volumes of the concentrated salt solutions. Although the total volume variation was negligible, it was considered in the analysis of the UV/Vis spectra. The potassium oxalate and lithium trifluoroacetate were not soluble in DMSO, so water solutions were used instead of DMSO solutions. This situation is troublesome because water competes effectively with the urea groups, giving rise to solvated anions, which are unfavourable for creating host:guest complexes. Therefore, the results with these anions may be questionable.

Results and discussion

Four polyamide model compounds were prepared following simple and common synthetic procedures (Scheme 1). The new polyamide model compounds contain one or two urea groups and two aromatic amide linkages, in which the carbonyl groups of the amides act as slightly electron-withdrawing groups and the urea moieties act as feeble electron-donating subunits. The urea group is electron-donating because of its resonance effect and is conversely electron-withdrawing due to its field effect. The resonance effect is predominantly observed in the nitration of N,N'-diphenylurea, during which the urea group directs the nitration mainly to the para-positions and slightly to the ortho-positions but not to the meta (34). As an illustrative example, Figure 1 (35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50)(51)(52)(53)(54), urea groups have been widely exploited in compounds that exhibit sensing behaviour toward anions, both in solution and in the solid state. However, their possibilities in anion-sensing applications have not been completely analysed, partly because of the complex nature of the intermolecular interactions of the urea-containing organic molecules. Aromatic urea derivatives have shown the ability to form oligomeric assemblies, or capsules, in which guests can be encapsulated in solution. These complex capsules persist even in the solid state, as demonstrated by X-ray analysis by alajarín et al (55)(56)(57).

The interactions of the model compounds with different anions (OH -, F -, HCO The other model compounds, M1, M2, and M4 do not show a colorimetric response toward anions, although they exhibit complexing abilities toward anions, as detected by UV/Vis and 1 H NMR. The differences between these models and M3 arise from the strong electron-withdrawing nitro group in the 4-nitrophenylureido moiety in the latter.

Titration of the acetate ion in an M3 solution in DMSO

The UV/Vis spectrum of M3 in a DMSO solution shows an absorption maximum around 346 nm, giving rise to a light beige solution. Upon addition of lithium acetate, the solution turns golden yellow due to the displacement of the λ max to higher wavelengths (Figure 5). The magnitude of the displacement can be correlated with the molar ratio of M3 to acetate ions, allowing for a titration of acetate ions with a detection monitoring the absorbance at 367 nm and plotting these values as a function of the acetate concentration in a DMSO solution of M3 (5x10 -5 M) (Figure 5). From the curvefitting equations, values of the acetate concentration can be obtained as a function of the λ max or the absorbance at 367 nm. Acetate concentrations between 65 ppb and 610 ppb could be estimated by both calculations with mean errors of 8% and 11%, respectively, and with a mean error of 6% when combining both calculations. Figure 6 shows the UV/Vis spectra of a DMSO solution of M3:F -(1:1) with increasing concentrations of acetate. Fitting the curve of the acetate:M3 ratio, which is equal to the acetate:F -ratio, vs. the absorbance at 364 and 479 nm (Figure 7) allows for the calculation of the acetate concentration once the absorbance is known. The calculated acetate concentrations in the range of 0.6 ppm to 2.6 ppm had a mean error of 3%, whereas the mean error for calculated anion concentrations between 5 ppb and 3 ppm was 7%. The latter range of acetate concentrations corresponds to a ratio of M3:F -:AcO -of 1:1:1.

The driving force of the process is largely dependent on the strong acetate:urea interaction (Table 1). The proton mobility associated with the acid/base reaction of the urea group and the naked-like fluoride anion is restricted by the double hydrogen bond of the acetate ion/urea (see inset chemical structure in Figure 9). The acetate counterion (Li + ) could also play a role in the stabilisation of the fluoride anion, giving rise to ion pairs.

Analysing the host:guest interactions by stability constants

The strength of the interaction of the anions (acetate, benzoate, oxalate, ptoluenesulfonate, trifluoroacetate, and chloroacetate) with the polyamide models in DMSO solution were analysed in terms of their stability constants by 1 H NMR [START_REF] Connors | Binding Constants: the Measurement of Molecular Complex Stability[END_REF][START_REF] Al-Soufi | [END_REF](61)(62)(63).

To get insight into the interactions between anions and models in solution, 1 H NMR spectra of the models in DMSO-d 6 (0.05 M) were measured at 20ºC with the addition of increasing quantities of the anions. Upon addition of the anions, chemical shifts of the protons of the urea groups to lower magnetic fields were observed (see With the stoichiometry of the complexes known, the dimensionless stability constants of the formation of an n:m complex (M n A n ) between the anion (A) and the model (M) can be determined by: We studied the interaction of the models with benzoate, acetate, trifluoroacetate, chlorobenzoate, p-toluenesulfonate, and oxalate. As previously outlined, the M1, M2

[ ] [ ] [ ] n m n m nm n m L M A M K A M nL mM = + . (1)
and M3 models have a 1:1 stoichiometry, whereas M4 is characterised by a 1:2 (M4:anion) stoichiometry.

The study of the complex formation between each model:anion pair was performed following the chemical shift variations of the proton signals k, l, q, and r (see Scheme 1) of the models in DMSO-d 6 at 20ºC with increasing anion concentrations.

The 1:1 complex formation between the host (model) and guest (anion) is represented by Eq. ( 2), where M represents the model, A represents the anion, and MA represents the complex with a 1:1 stoichiometry.

MA A M + (2)
Thus, the stability constant is given by Eq. (

] ][ [ ] [ 1 A M MA K = . 3):
(

) 3
Under fast-exchange conditions (59), the observed chemical shift for a model can be expressed by Eq. 4:

MA MA M M obs δ χ δ χ δ + = ∆ , (4)
[ ] [ ] [ MA M M O + = and (6) ] [ ] [ ] [ MA A A O + = . ( 7
)
Considering Eq. ( 3) and the mass balances of M and A (Eqs. ( 6) and ( 7)), the complex concentration can be solved as a function of K 1 and the initial concentrations of

M and A. 0 ] [ ] [ ] [ 1 ] [ ] [ ] [ ) ( 1 2 = + + + - O O O O M A MA K M A MA (8)
From Eqs. ( 8) and (3), Eq. 9 is deduced

      - + + - + + ∆ = ∆ O O O O O O o MA obs M A K M A K M A M ] [ ] [ 4 2 1 ] [ ] [ 1 ] [ ] [ ] [ ) ( ) ( 1 1 δ δ , ( 9
)
which relates the experimental ∆δ obs with the [A] o . Thus, the experimental data can be fitted using a nonlinear least-squared algorithm (64) to obtain the parameters K 1 and ∆δ MA .

The stability constants of the 1:1 complexes were calculated for M1, M2 and M3 with Eq. ( 9) separately using the chemical shifts of the two urea protons (k and l, Scheme 1). The results (Table 1) show that the stability constants calculated using both proton signals are equivalent, as expected, confirming the validity of the calculation model. The host-guest analyses performed by the "naked-eye" and by the UV/Vis technique confirmed that the addition of acetate ions to a solution of M3 in DMSO gave rise to the stronger colour changes. The stability constants for each model obtained by the 1 H NMR analyses revealed that the acetate anion has the strongest interaction for all of the models in DMSO. The strength of the different complexes is related to the electron density surrounding the urea group. Thus, the high electron-withdrawing capability of the nitro group through inductive and mesomeric effects withdraws electron density from the urea group, giving rise to the higher stability constants of the M3:anion complexes. Furthermore, the strength of the interaction of the carboxylate anions with a specific model depends inversely on the pKa of their conjugated bases in DMSO [START_REF]DMSO Reaction Solvent Guide[END_REF], thus the association depends inversely on the localisation of the negative charge over the carboxylic group because of the mesomeric effect and the electron density due to inductive effects.

The order of stability constants of the model:acetate complexes (M3:acetate > M1:acetate > M2:acetate) is in agreement with theoretical calculations (ab initio).

Therefore, gas-phase density functional theory (DFT) calculations at the B3LYP/6-31G* level of theory were performed following a previously described procedure (31,66). The interaction energies of the acetate ion with simplified models of M1, M2 and M3 (i.e., diphenylurea, 1-(naphthalen-2-yl)-3-phenylurea, and 1-(4-nitrophenyl)-3phenylurea) were -224, -219 and -266 kJ/mol, respectively. $: The trifluoroacetate and oxalate anions did not fit well to the mathematical model, because the small water content of the measurement solutions and the salts being insoluble in DMSO, therefore requiring them to be added as concentrated water solutions.

If we consider a 1:2 complex formation between the host (M4 model) and the guest (anion), the complex formation can be represented by Eqs (10) and ( 11

): ] ][ [ ] [ 1 A M MA K MA A M = + and (10) ] ][ [ ] [ 2 2 2 A MA MA K MA A MA = + , (11)
where K 1 y K 2 are stability constants of the complexes. The mass balances are expressed as:

] [ ] [ ] [ ] [ 2 MA MA M M O + + = and (12) F o r P e e r R e v i e w O n l y ] [ 2 ] [ ] [ ] [ 2 MA MA A A O + + = . ( 13
)
Considering the stability constant, Eqs. ( 10) and ( 11) and the mass balances of M and A (Eqs. ( 12) and ( 13)), Eq. ( 14) can be deduced:

0 ] [ ] )[ 1 ) ] [ ] ([ ( ] )[ ) ] [ ] [ 2 ( ( ] [ 1 2 1 2 1 3 2 1 = + + - + + - + O O O O O A A A M K A K A M K K A K K . ( 14
)
Given the concentration of (A), the equilibrium concentrations of M, MA and MA 2 can be calculated as described by Al-Soufi et al. ( 60):

2 2 1 1 ] [ ] [ 1 ] [ ] [ A K K A K M M O + + = , ( 15
) ] ][ [ ] [ 1 M A K MA = , and (16)
] ][ [ ] [ 2 2 A MA K MA = . ( 17
)
The observed chemical shift of a proton, δ obs , is given by the following equation

2 2 MA MA MA MA M M obs δ χ δ χ δ χ δ + + = ∆ , ( 18
)
where M δ , MA δ and δ MA 2 represent the chemical shifts of the free and complexed forms of the model, and χ M , χ MA, and χ MA 2 are their molar fractions. From Eqs. (18) and ( 12), the following expression can be deduced:

] [ ] [ ] [ ] [ 2 2 MA M MA M o MA o MA obs δ δ δ ∆ + ∆ = ∆ . ( 19
)
As has been previously outlined for 1:1 stoichiometry, Eq. ( 19) can be fitted to experimental data considering Eq. (14-17) and using a nonlinear least-squared algorithm (67) to obtain the parameters K 1 , K 2 , ∆δ MA and ∆δ MA 2 . The results of the stability constants are depicted in Table 2. Upon fitting the data of each urea proton, the two model guest molecule with one of the urea groups can affect the other. This is true for benzoate and p-toluenesulfonate, but not for acetate. The distinct behaviour for acetate is reflected in the results obtained using Eq. ( 19) and shown in Table 2. When we tried to obtain the stability constants with the chemical shifts of one of the urea groups, we obtained a correct value for the constant of the interaction with this group but a value near zero for the other urea group. We believe that this fact is due to restricted mobility of the acetate anions because of the stronger host:acetate interaction to those with other anions. Nevertheless, each constant of M4:acetate can be estimated by applying the 1:1 model to each urea group by Eq. ( 9) (Table 3), and the results are similar to those obtained with the 2:1 model. K 2 ) corresponding to the interaction of M4 with acetate calculated from the 1:1 interaction model for each urea group using Eq. ( 9). For comparison, we included the values obtained by the 2:1 model (Eq. ( 19)) in parentheses. See Scheme 1 for proton assignment.

Urea proton k

Urea proton l Urea proton q Urea proton r K 1 528

Outlooks

The amide linkages are the key motif of the polyamide backbones, and are naturally present in the polyamide models. Nevertheless, the amide groups do not play an important role in the complexation. The small downfield displacement of the amide protons upon interaction with acetate ions, as observed by 1 H NMR (Figure 8), arises from modification of the overall electron density of the molecule due to the interaction of the urea motif with the anions, as can be observed in the chemical shift displacements of the amidic protons vs. acetate to M3 molar ratio (Figure 11), where it is observed a change in the slope of the curve corresponding to a M3:acetate molar ratio of 1:1. Nevertheless, and in agreement with pervious results (13,18), a membrane with an adequate hydrophilic/hydrophobic balance can partially inhibit the efficient water/anion interaction, giving rise to the recognition event inside the membrane in a solid/liquid interaction, where the solid is a partially water-swelled matter. The waterswelled organic membrane must have enough water content to allow the water-solvated anions to enter the membrane by diffusion. At the same time, this water content should be lower enough to let lipophilic domains inside the membrane where the recognition process takes place. In polyacrylate and polymethacrylate networks, membranes with a water uptake of 50% w/w have an optimum balance for anion recognition in pure water.

Dense membranes, or films strips, prepared with polyamides P1 to P4 (Scheme 2) were hydrophobic (films prepared with P3 showed a water uptake of 3.6% at 65% rh) and remain silent in water media with increasing quantities of anions. In order to increase the hydrophilicity, the films where prepared with a blend of cellulose acetate and P3 (10% of the latter), achieving a water uptake of 14.5% (at 65% rh).

The blend of the polyamides with cellulose acetate has also been used to obtain asymmetric membranes. The objective was to increase the surface of the materials to test the influence on the recognition processes in water media. The pore control was achieved by means of varying the DMF content of the water/DMF coagulation bath. with much lower guest concentrations in water environments. One way to achieve this is to increase the hydrophilicity of the system by increasing the water affinity of the polymer to be blended with the polyamides.

Conclusions

We have developed new model polyamide compounds that are structurally simple sensors containing the urea receptor group as the selective binding site for anions. One of the models behaves as an effective colorimetric probe for carboxylic acid salts, specifically for acetate. The light beige solutions of M3 in DMSO develop a reddish colour upon addition of basic anions, such as OH -and F -, and a yellowish colour upon adding acetate salts. The titration of acetate can be performed by adding the anion to: a) a model solution in DMSO, giving rise to the development of a yellow colour, or b) a red solution of a mixture of model/fluoride (1/1), turning the reddish solution to yellow due to the displacement of the fluoride by the acetate. The host:guest stoichiometry (polyamide model compound:anion complexes) and the stability constants of complexes were determined by 1 H NMR. The models mimic polyamides, the polymers derived from the diacid monomers (1-4), which are intermediates in the synthesis of the models. The polymers retain the ability to interact selectively with anions in organic media, and cheap future sensing devices might take the form of userfriendly naked-eye polyamide film sensors or might incorporate a polyamide coating at the end of an optic fiber connected to a portable UV/Vis diode-array detector. Further work is needed to achieve a colorimetric response of the polymer materials in water environments.

F

  Moreover, the aromatic polyamides are a class of polymers with great

(

  TPP) was vacuum-distilled twice over calcium hydride, and then stored in the presence of 4-Å molecular sieves. Pyridine was dried under reflux over sodium hydroxide for 24 hours, and distilled over 4-Å molecular sieves. 5-(3-Phenylureido)isophthalic acid (1), 5-(3-naphthylureido)isophthalic acid (2), 5-(3-(4-nitrophenyl)ureido)isophthalic acid (3), and 5-(3-(4´-ureidophenyl)ureidophenyl) isopthalic acid (4) were prepared by previously described procedures (22).

- 1 -

 1 yl-ureido)-N,N'-diphenyl-isophthalamide (M2), 5-[3-(4-Nitro-phenyl)-ureido]-N,N'-diphenyl-isophthalamide (M3), and N,N'-Diphenyl-5-{3-[4-(3-phenyl-ureido)-phenyl]-ureido}-isophthalamide (M4) were prepared and purified in a similar manner to that of M1. M2 was obtained as an amorphous solid that crystallized at 204 ºC. Yield: 92%. M.p.: 256 ºC. Tg: 130ºC.

  onto glass plates and subsequently immersed in a 25±1°C

  . Samples were prepared directly in UV/Vis cuvettes or in NMR tubes by adding the proper polyamide model compound in the DMSO solution

Scheme 1 .Figure 1 .

 11 Scheme 1. Synthesis, chemical structure and acronyms of the polyamide models.

-

  -, benzoate, oxalate, p-toluenesulfonate, CH 2 ClCO 2 -, and I -) in solution were evaluated using NMR and UV/Vis techniques.

Figure 2 Figure 2 . 3 -Figure 3 .Figure 4 .

 22334 Figure2shows the effect of the addition of anions to a solution of M3 in

  100 ppb. A more correct titration curve can be obtained by

Figure 5 . 15 3. 2 .

 5152 Figure 5. A) UV/Vis spectra of solutions of M3:acetate ([M3] = 2x10 -5 M). The arrows indicate the spectra obtained upon increasing the acetate concentration from a M3:acetate ratio of 1:0 to 1:10. B) Titration curves of acetate at 367 nm. vs. acetate concentration; C) λ max vs. acetate concentration.

Figure 6 .Figure 7 .

 67 Figure 6. UV/Vis spectra of M3:F -(1:1) solutions in DMSO upon adding increasing quantities of acetate ions to reach a ratio of M3:F -:AcO -of 1:1:4. Inset: photograph depicting titration. The figure depicts the ratio of fluoride to acetate, with the arrows indicating increasing concentrations of acetate. The dashed line corresponds to the initial spectrum of M3:F -(1:1) without acetate. The concentrations of M3 and fluoride were 5x10 -5 M.

Figure 8 andFigure 8 . 1 HFigure 9 .

 8819 Figure 8 and Figure 9 for the M3:acetate system). The graphical representation of the

Figure 10 .

 10 Figure 10. A) Plot of the observed chemical shifts displacements, ∆δ, of the urea protons vs. benzoate concentration. B) The corresponding Job´s Plot. Solid lines are the results of fitting and applying the corresponding equations.

  where M δ and MA δ represent the chemical shift of the free and complexed model, respectively, and M χ and MA χ are their molar fractions. From the mass balance, the observed chemical shifts can be expressed as ∆δ obs = δ obs -δ M and ∆δ MA = δ MA -δ M .Equation 5 is deduced from Eq. (4).

  ] o is the initial concentration of the model M. The mass balances of M and A are given by ]

  sometimes obtained, indicating that the interaction of a single

Figure 11 .

 11 Figure 11. Chemical shifts of the amidic protons vs. acetate to M3 molar ratio ([M3] = 5x10 -2 M).

Figure 12

 12 Figure 12 depicts an SEM photograph of an asymmetric membrane showing large

Scheme 2 .

 2 Scheme 2. Synthesis, chemical structure and identification of the polyamides.

Figure 12 .

 12 Figure 12. Left: SEM cross-section image of a cellulose acetate/P3 asymmetric membrane obtained by casting a solution of cellulose acetate/P3/DMF (15/10/75 w/w/v) in an H 2 O/DMF (70/30 v/v) coagulation bath. Right: enlargement of a pore.

Figure 1 .Page 37 of 50 URLFigure 2 .Figure 3 .

 15023 Figure 1. Characterisation of the polyamide M3 model by (a) FTIR, (b) 1 H NMR, and (c) 13 C NMR.

Figure 4 .

 4 Figure 4. UV/Vis spectra of M3 solutions in DMSO ([M3] = 2x10 -5 M) with ten equivalents of fluoride, acetate, benzoate and oxalate. The spectra of the M3 solutions with ten equivalents of p-toluenesulfonate, methanesulfonate, trifluoroacetate, and chloroacetate are indistinguishable from the initial solution. The solid line represents the spectrum of M3 in DMSO without salts.

Figure 5 .Figure 6 .

 56 Figure 5. A) UV/Vis spectra of solutions of M3:acetate ([M3] = 2x10 -5 M). The arrows indicate the spectra obtained upon increasing the acetate concentration from a M3:acetate ratio of 1:0 to 1:10. B) Titration curves of acetate at 367 nm. vs. acetate concentration; C) λ max vs. acetate concentration.

Figure 7 .Figure 8 . 1 HFigure 9 .

 7819 Figure 7. Absorbance of DMSO solutions of M3:F -(1:1) at a given wavelength upon addition of increasing quantities of acetate ions, reaching a ratio of M3:F -:AcO -of 1:1:4. The concentration of M3 was 5x10 -5 M.

Figure 10 .Figure 11 . 12 Figure 12 .

 10111212 Figure 10. A) Plot of the observed chemical shifts displacements, Δδ, of the urea protons vs. benzoate concentration. B) The corresponding Job´s Plot. Solid lines are the results of fitting and applying the corresponding equations.

Scheme 1 .

 1 Scheme 1. Synthesis, chemical structure and acronyms of the polyamide models.

Scheme 2 .

 2 Scheme 2. Synthesis, chemical structure and identification of the polyamides.

  FTIR [Wavenumbers (cm -1 )]: ν N-H : 3290; ν C=O : 1663, 1651; δ N-H : 1598; ν Ar C=C : 1541.

	(16).
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l 1 H-NMR (400 MHz, DMSO-d 6 ): δ (ppm), 10.48 (s, 2H); 9.16 (s, 1H); 8.85 (s, 1H); 8.23 (s, 2H); 8.17 (s, 1H); 7.86 (d, y
	J=8.3 Hz, 4H); 7.55 (d, J=7.9 Hz, 2H); 7.42 (m, 4H); 7.35(m, 2H); 7.17 (t, J=7.7 Hz,
	2H); 7.04 (t, J=7.3 Hz, 1H). 13 C-NMR (100,6 MHz, DMSO-d 6 ): δ (ppm), 166.21;
	153.51; 141.11; 140.39; 140.03; 136.94; 129.79; 129.66; 124.77; 123.11; 121.28;
	120.93; 119.40. EI-LRMS m/z: 331 (55), 239 (100), 185 (11), 119 (60), 93 (62), 65
	6

157 ºC. 1 H-NMR (400 MHz, DMSO-d 6 ): δ

	M3 was obtained as a semi-crystalline solid (50% crystallinity) that crystallized
	at 201 ºC. Yield: 87%. M.p.: 271 ºC. Tg: (ppm), 10.49 (s, 2H); 9.63 (s, 1H); 9.42 (s, 1H); 8.28-8.23 (m, 5H); 7.86-7.78 (m, 6H);
	7.42 (m, 4H); 7.17 (t, J=7.5 Hz, 2H). 13 C-NMR (100,6 MHz, DMSO-d 6 ): δ (ppm),
	166.06; 153.05; 147.10; 142.19; 140.45; 139.99; 136.97; 129.66; 126.12; 124.81;
	121.82; 121.58; 121.29; 118.72. EI-LRMS m/z: 331 (37), 239

-1 )]: ν N-H : 3272; ν C=O : 1683, 1649; δ N-H : 1598; ν Ar C=C : 1537.

Table 1 .

 1 Stability constants (K 1 ) corresponding to the interaction of models M1, M2 and M3 with selected anions obtained upon application of Eq. (9). See Scheme 1 for proton assignment. The stability constants are too low to be accurately measured (∆δ AM values are experimental values).

		Urea proton k	Urea proton l
	Anion	K 1	∆δ ∆δ ∆δ ∆δ AM	K 1	∆δ ∆δ ∆δ ∆δ AM
			M1 Model $
	Benzoate	358±115	3.19±0.08	344±114	3.09±0.08
	Acetate	6641±4378	3.18±0.03	6553±4800	3.11±0.03
	Trifluoroacetate	218±96	0.47±0.02	198±80	0.51±0.02
	Oxalate	16±2	1.25±0.04	16±2	1.29±0.04
			M2 Model $
	Benzoate	127±14	2.84±0.04	113±11	2.46±0.03
	Acetate	385±192	2.71±0.09	340±154	2.45±0.08
	Trifluoroacetate	16±3	0.71±0.04	15±3	0.64±0.04
					0.06 &
			M3 Model $
	Benzoate	647±245	3.63±0.03	691±232	3.56±0.05
	Acetate	28889±8317	3.68±0.01	21516±8120	3.59±0.08
	p-Toluenesulfonate	11.8±0.8	0.54±0.01	13±1	0.51±0.02
					0.45 &
	Chloroacetate	33±3	1.71±0.04	28±3	1.45±0.03
	&:

Table 2 .

 2 Stability constants (K 1 and K 2 ) corresponding to the interaction of M4 with selected anions (for proton assignment see Scheme 1). The stability constants corresponding to the M4:oxalate complex are too low to be accurately measured and are not depicted in the table.

	Anions	K 1	K 2	∆δ ∆δ ∆δ ∆δ AM	∆δ ∆δ ∆δ ∆δ MA 2	K 1	K 2	∆δ ∆δ ∆δ ∆δ AM	∆δ ∆δ ∆δ ∆δ MA 2
			Urea proton k			Urea proton l
	Benzoate	379	85	2.3	2.99	436	80	2.15	2.83
	Acetate	778	1.1	2.77	3.05	774	0.4	2.65	3.13
	p-Toluenesulfonate	14	2.9	0.23	0.44	23	2.2	0.21	0.56
			Urea proton q			Urea proton r
	Benzoate	424	121	1.19	2.66	424	121	1.17	2.63
	Acetate	0.005	101	1.71	2.41	0.005	97	1.67	2.35
	p-Toluenesulfonate	27	1.2	0.18	0.72	27	1.1	0.18	0.71

Table 3 .

 3 Stability constants (K 1 and

Table 1 .

 1 Stability constants (K 1 ) corresponding to the interaction of models M1, M2 and M3 with selected anions obtained upon application of Eq. (Error! Reference

	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

source not found.). See Error! Reference source not found

  . for proton assignment. The stability constants are too low to be accurately measured (∆δ AM values are experimental values). $: The trifluoroacetate and oxalate anions did not fit well to the mathematical model, because the small water content of the measurement solutions and the salts being insoluble in DMSO, therefore requiring them to be added as concentrated water solutions.

		Urea proton k	Urea proton l
	Anion	K 1	∆δ ∆δ ∆δ ∆δ AM	K 1	∆δ ∆δ ∆δ ∆δ AM
			M1 Model $
	Benzoate	358±115	3.19±0.08	344±114	3.09±0.08
	Acetate	6641±4378	3.18±0.03	6553±4800	3.11±0.03
	Trifluoroacetate	218±96	0.47±0.02	198±80	0.51±0.02
	Oxalate	16±2	1.25±0.04	16±2	1.29±0.04
			M2 Model $
	Benzoate Acetate Trifluoroacetate Benzoate Acetate p-Toluenesulfonate Oxalate Chloroacetate o r 28889±8317 127±14 385±192 16±3 647±245 11.8±0.8 & 33±3 P e &: F e r	2.84±0.04 2.71±0.09 0.71±0.04 M3 Model $ 113±11 340±154 15±3 3.63±0.03 691±232 3.68±0.01 21516±8120 0.54±0.01 13±1 0.48 & & 1.71±0.04 28±3	2.46±0.03 2.45±0.08 0.64±0.04 0.06 & 3.56±0.05 3.59±0.08 0.51±0.02 0.45 & 1.45±0.03
		R
		e
			v i e
			w
				O n l
				y
					2

Table 2 .

 2 Stability constants (K 1 and K 2 ) corresponding to the interaction of M4 with selected anions (for proton assignment see Error! Reference source not found.). The stability constants corresponding to the M4:oxalate complex are too low to be accurately measured and are not depicted in the table.

	Anions	K 1	K 2	∆δ ∆δ ∆δ ∆δ AM	∆δ ∆δ ∆δ ∆δ MA 2	K 1	K 2	∆δ ∆δ ∆δ ∆δ AM	∆δ ∆δ ∆δ ∆δ MA 2
			Urea proton k			Urea proton l
	Benzoate	379	85	2.3	2.99	436	80	2.15	2.83
	Acetate	778	1.1	2.77	3.05	774	0.4	2.65	3.13
	p-Toluenesulfonate	14	2.9	0.23	0.44	23	2.2	0.21	0.56
			Urea proton q			Urea proton r
	Benzoate	424	121	1.19	2.66	424	121	1.17	2.63
	Acetate	0.005	101	1.71	2.41	0.005	97	1.67	2.35
	p-Toluenesulfonate	27	1.2	0.18	0.72	27	1.1	0.18	0.71

Table 3 .

 3 Stability constants (K 1 and K 2 ) corresponding to the interaction of M4 with acetate calculated from the 1:1 interaction model for each urea group using Eq. (Error! Reference source not found.).

	For

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

& 0.09 & &

& 0.48 & &

Acknowledgements

The financial support provided by the Spanish MICINN (Ministerio de Investigación e Innovación) -FEDER (MAT2008-00946/MAT) and by the Caja de Burgos is gratefully acknowledged.


Crenguta Dordea

Frédéric Brisach

Jaouad Haddaoui

Françoise Arnaud-Neu

Michael Bolte

Alessandro Casnati

Volker Böhmer
email: vboehmer@uni-mainz.de

Tetra-CMPO-Derivatives of Calix[4]arenes fixed in the 1,3-Alternate Conformation

Keywords: Lanthanides, calix[4]arenes, complexation, extraction, crystal structures 25 Lanthanides, calix[4]arenes, complexation, extraction, crystal structures

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Carbamoylmethylphosphine oxides (CMPOs) are efficient extractants for actinides (and lanthanides) and especially the N,N-diisobutyl derivative is used on a technical scale (TRUEX-process). 1 CMPO is a bidentate ligand, and a trivalent cation surely binds more than one molecule of the ligand in its complex. Thus it appeared reasonable to attach several CMPO functions to a common platform. In fact tri-CMPO derivatives of triphenylmethanes 2 and tetra-CMPO derivatives of calix [4]arenes 3 are much more efficient extractants than CMPO itself. This is true for compounds, where the CMPO functions are attached to the wide rim 4 or to the narrow rim 5 Better extraction results are even obtained with linear 6 oligo-CMPOs and with calix [4]arenes bearing only two or three CMPO functions. 7 Calix [4]arenes bearing ether residues equal to or larger than propyl are fixed in one of the four basic conformations (cone, partial cone, 1,2-and 1,3-alternate). Nearly all calix [4]arene derived ligands of the podand-type are based on the cone conformation [START_REF] Arduini | For early examples see[END_REF] where all oxygens (and all p-positions) point in the same direction, although rare examples with other conformations are known [START_REF] Rouquette | Mono-; bis -crownethers derived from the 1,3-alternate conformation are important examples[END_REF][START_REF] Van Der Veen | complexation occurs with the partial cone conformation see[END_REF]11 It should be possible, however, to use also the 1,3-alternate conformation 40 as scaffold 12 , if the ligating groups are attached alternatingly to the wide and to the narrow rim, where an appropriate spacer could bring them to the same level. The 1,3-alternate skeleton is more rigid compared to the cone isomers, which still can change between two "pinched" cone conformations. Considering the fact that a tetra-CMPO derived from a rigid biscrown-3 calix [4]arene is by a factor of 10 more efficient as extractant, 13 this rigidity could be even advantageous.

It should be also worthwhile to attach different ligating functions to the wide and narrow rim of the calix [4]arene in its 1,3-alternate conformation. Picolinamide groups, for instance, should lead to a better selectivity than CMPO functions alone. 14,15,16 Carboxylic acids could potentially act as anion exchanger and reduce the number of nitrate anions to be extracted into the organic phase. Lipophilic carboxylic acids are often used as synergisers for neutral ligands in nuclear waste treatment, 17 while polycarboxylic acids were proven to efficiently complex actinide 18

Results and discussion

Syntheses. Aminogroups attached to the wide rim of a calix [4]arene are usually obtained by reduction of nitrogroups which are often introduced via ipso-nitration. 21 For the attachment at the narrow rim the alkylation by N-(ωbromoalkyl)phthalimides (or by ω-bromoalkylnitriles) followed by hydrazinolysis of the phthalimide groups (or by reduction of the nitrile groups) are convenient methods.

Since selective substitutions are possible only at the wide rim distinguishing phenol and phenolether units [START_REF]similar fashion by substituents at the wide rim[END_REF] the following two pathways (Scheme 1) were checked, starting with the known 1,3-alternate derivative 1. [START_REF] Danila | [END_REF] a): Catalytic hydrogenation leads in excellent yields to the propoxy aniline derivative (amino propyl ether) 2a,b (n=3,4) which was acylated by the active ester AE1 3,24 to give 3b in 84 %. (For n=2 the product could not be sufficiently purified, for n=4 this pathway was not further followed.) Cleavage of the phthalimide by hydrazine (93% 10 of 4b) and acylation of the aliphatic amino groups by AE1 (83% of 5b (n=3)) occurred without problems.

AE1 F C O C O O AE2 N F F F F 1a (n=2) 1b (n=3) 1c (n=4) N O O N O O N O O
Scheme 1 Syntheses of tetra-CMPO derivatives of calix [4]arenes in the 1,3-alternate conformation and of compounds bearing two CMPO and two picolinamide residues. i) H2, Raney-Ni, THF, rt; ii) AE1, CH2Cl2, rt; iii) hydrazine, EtOH, reflux; iv) AE2, CH2Cl2, rt. b): Alternatively (as studied for n=4), cleavage of the phthalimide groups (of 1c) with hydrazine occurred with the simultaneous hydrogenation of the allyl groups to give 81% of 6c. Subsequent acylation by AE1 (75% of 7c) followed by quantitative hydrogenation of the nitro groups to 8c and a second acylation step by AE1 (87% of 5c (n=4)) was also possible without any problems. It should be mentioned that the allyl groups must not be quantitatively hydrogenated during the cleavage of the phthalimide groups, since this will be completed in the next step. It is important, however, that the cleavage of the phthalimide groups is quantitative while nitro groups remain unchanged.

O O O O t-Bu O NH 2 2 t-Bu O NO 2 O O O t-Bu O t-Bu O NH 2 NO 2 O OH t-Bu C C 2 2 2 2 2 2 2 2 O O -Et O C C O O -Et O O O O O O O C C O O -Et O O O O O O O C C O O -Et 2 2 O t-Bu O O O O O O O O C C O OH

10

A direct conversion of 1 to the respective tetra amine (simultaneous or subsequent cleavage of the phthalimide groups and reduction of the nitro groups) was not attempted, although it should be more economic for those cases, where only one acyl-residue has to be attached. Two similar calixarene derivatives combining two picolinamide functions [START_REF]For the extraction of lanthanides; actinides by calixarene-based picolinamides[END_REF] at the narrow rim (attached via three or four methylene groups) with two CMPO functions at the wide rim, were prepared in analogy to pathway b). Acylation of 6b,c with AE2, the hexafluorophenyl ester of α-picolinic acid 27 (64%/83% of 9b,c), reduction of the nitro groups (31%/93% of 10b,c) and acylation with AE1 (65%/47% of 11b,c). The partly striking difference in the yield for the last steps indicates that the reaction conditions are not optimized. These syntheses of extractants 5 and 11 are summarized in Scheme 1.

2 t-Bu O NO 2 O O O 2 t-Bu O NO 2 O O O 2 2 O C C O O -Et 13 2 O NO 2 O O O 2 O NO 2 O O O 2 2 O 17 N O O 2 t-Bu O NO 2 O O O 2 t-Bu O NO 2 O O O 2 2 O 1b' 2 O O O O 2 OH O O O 2 2 O 18 N O O O O t-Bu t-Bu t-Bu t-Bu N O O
In order to (partly) compensate the positive charge of the extracted cation we prepared compound 16, where two CMPO functions are combined with two carboxyl groups. The synthesis starts with the known 28 diester 12 which is fixed in the 1,3-alternate conformation as di-allylether 13 (isolated yield 21%, together with 39% of the isomer in the partial cone conformation). Reduction/hydrogenation and direct acylation of the crude diamine 14 furnished the diester 15 (92%) which was hydrolysed by aqueous LiOH to give the di-CMPO-di-acid 16 with 96% yield. (see Scheme 2). Extraction and Complexation. The extraction of selected cations (lanthanides and thorium) from 1.0 M nitric acid 10 into dichloromethane was studied for two tetra-CMPO compounds (5b, 5c). The respective extraction percentages %E and distribution coefficients D are given in Table 1 and compared with values previously obtained [5][6][7] for calixarenes fixed in the cone conformation and substituted at the wide or narrow rim by four CMPO-functions. These data show that the extraction ability of lanthanides is similar for 5b and 5c (and practically identical for La 3+ , Eu 3+ , Yb 3+ ), but five to twenty times lower than for the analogues in the cone conformation. The stability constants of the complexes, however, are distinctly higher (by factors of 6 to 80) for 5b than for 5c (Table 2).

By all criteria, however, wide as well as narrow rim tetra-CMPOs in the cone conformation are better extractants/ligands than the derivatives in the 1,3-alternate conformation, which in addition are more difficult to synthesize. Both data sets show no real selectivity within the lanthanide series in contrast with the lower and especially the upper rim derivatives.

18 assumes the usual pinched cone conformation where the aryl residues of the phenolic units are bent outwards, including angles of 148.7° and 132.2° with the reference plane defined by the four methylene bridge carbons. Consequently the two ether aryl units are bent towards the cavity, with angles of 114.7° and 108.9°. This conformation (with interplanar angles between opposite calix [4]arene aryl planes of 100.9° and 43.6°) permits a slightly larger distance between the (bulky) ether residues (distances O12-O32 = 4.534 Å and O22-O42 = 3.222 Å).

The orientation of the phthalimide planes is obviously mainly determined by packing requirements, since the (CH 2 ) 4linker (tetramethylene linker) allows for conformational freedom.

Table 3 Comparison of some typical data taken from the crystal structures of three different calix [4]arene derivatives in the 1,3-alternate conformation (1b', 13, 17) and one in the cone conformation (18).

Compounds:

1b' The best plane through the C-atoms of the bridging methylene groups is a suitable reference plane also for the derivatives in the 1,3-alternate conformation, although the deviation from this common plane reaches 0.357 Å for 17, with the four carbon atoms laying alternatingly on both sides (for 13 and 1b' this deviation is 0.243 and 0.076 Å). The shape of the molecule may be characterised then by the angle between the aromatic planes and this reference plane (see Table 3). The deviation of the reference plane from an geometrically exact plane can be characterised also by the angle between two triangles obtained by folding along the diagonal (C1-C3 or C2-C4). These angles are small (4.75°) for 1b', but larger for 13 (15. oxygens (largest for 1b', ∆ = 0.561 Å) and opposite p-aryl carbons (largest for 13, ∆ = 1.155 Å). Most probably also packing effects are responsible for such deviations in the exact shape of the 1,3-alternate skeleton, since especially the p-substitients (t-butyl, nitro, phthalimido) are rather different in size, shape and polarity.

Further distances and angles of the three molecules with 1,3-alternate conformation are compared in Table 3, which contains also data for two compounds (precursors) in the cone conformation. The molecular conformation of compounds 1b', 13 and 17 is compared in Fig. 1, while their packing is compared in Fig. 2.

In the crystal lattice 13•CHCl 3 forms columns with alternating directionality parallel to the b-axis. Within the a-bplane these columns form double layers separated by channels which contain the chloroform molecules (Fig. 2, middle). 17•CHCl 3 forms similar columns along the b-axis, but here the molecules within each column are separated by solvent molecules, which are "included" in the cavity surrounded by the phthalimide and the nitro groups. The larger size of this cavity, compared with 13, may be the reason for this inclusion. Otherwise both structures are rather similar. A description of the packing of 1b' is more complicated. Fig. 2 (left) shows a view along the b-axis in which the included acetone molecules are shown in "ball and stick" representation.

Experimental Part

Syntheses. Compounds 1b, 1c, 2b and 6b were prepared as described earlier.Error! Bookmark not defined. 5,17-Di-t-butyl-11,23-dinitro-26,28-diphthalimidoethyloxy-25,27-di-allyloxy-calix[4]arene 1a. Compound 1a was obtained (together with its partial cone isomer) as described for the analogous compounds 1b and 1c from the dinitro compound. A suspension of 5,17-Di-t-butyl-11,23-dinitro-26,28-diphthalimidoethyloxy-calix [4]arene (2.9 g, 2.98 mmol) in dry DMF (40 ml) and Cs 2 CO 3 (7.8 g, 23 mmol) was stirred under nitrogen at 50 °C for 6 days. Water (50 ml) was added to stop the reaction and after several washing (3x50 ml water) the organic phase was dried over 20 MgSO 4 . The solvent was evaporated under reduced pressure. Colourless crystals (0.05 g, 2%) of the desired 1,3alternate isomer 1a were isolated by column chromatography (chloroform/hexane, 1:2) together with the partial cone isomer (0.27 g, 9 %). Clearly these yields are not optimized. A clear solution of 1a (50 mg, 0.04 mmol) in THF (15 ml) was hydrogenated under normal pressure with Raney-Ni. The catalyst was filtered off, the solvent was evaporated and the dry residue was dissolved in CHCl 3 (3 ml) and reprecipitated with hexane to give 2a as yellow powder (30 mg, 64 %); mp 315°C; 1 H NMR (300MHz, CDCl 3 ) δ 0.57 (t, 6H, 3 J = 7.16 Hz, CH 3 -CH 2 -), 1.23 (s, 18H, t-Bu), 3.17 p-Nitrophenyl carbamoylmethyl diphenyl phosphine oxide (AE1, 0.25 g, 0.65 mmol) was added to a clear solution of diamine 2b (0.28 g, 0.27 mmol) and few drops of NEt 3 in dichloromethane (30 ml). The mixture was stirred at ambient temperature over night. The p-nitrophenol formed during the acylation was extracted with an aqueous solution of NaOH (5 %, 3 x 100 ml). After drying over MgSO 4 the solvent was evaporated under reduced pressure, the residue was dissolved in dichloromethane (5 ml). Compound 3b was obtained as white powder by reprecipitation from hexane; 0. The solvent was evaporated under reduced pressure; the formed residue was dissolved in chloroform (10 ml) and washed three times with water. The organic phase was dried over MgSO 4 , the solvent was evaporated under reduced pressure.

The diamine 8c was obtained as yellow oil which was used in the next step without further purification.

5,17-Di-t-butyl-11,23-dinitro-26,28-dipicolinamidopropoxy-25,27-dipropoxy-calix[4]arene 9b.

The active ester AE2 (0.21 g, 0.73 mmol) was added to a suspension of diamine 6b (0.25 g, 0.3 mmol) in chloroform (30 ml). The mixture was stirred at room temperature until the tlc shows that no starting material was left. The pentafluorophenol formed during the reaction was extracted with a solution of Na 2 CO 3 (5 %, 4 x 100 ml) and the organic phase was dried over MgSO 4. The solvent was evaporated under reduced pressure, the residue was dissolved in dichloromethane (5 ml) from which the desired compound was obtained by reprecipitation with hexane as yellow powder; 0. (20 ml), and the reaction mixture was stirred under hydrogen during 4 days. When the hydrogen uptake was complete the catalyst was filtered off and the solvent evaporated under reduced pressure. The residue was dissolved in chloroform (10 ml) and the desired amine was precipitated with hexane (15 ml) as a white powder, 200 mg, 31 %; m.p. 120 °C; 1 H NMR (300 MHz, CDCl 3 ) δ 0.59 (t, 6H, 3 J = 7.7 Hz, -CH 2 -CH 3 ), 1.01-1.08 (m, 4H, -CH 2 -), 1.22 (s, 18H, t-Bu), 1.55-1.59 (m, 4H, -CH 2 -), 2.89 (bs, 4H, -NH 2 ), 3.17 (1,3alternate) and 13a (partial cone). A stirred suspension of dinitro-calixarene 12 (1.3 g, 1.6 mmol) and Cs 2 CO 3 (4 g, 12.8 mmol) in dry DMF (40 ml) was heated to 40 °C under nitrogen. After 1 h allylbromide (1.1 ml, 12.8 mmol) was added and the reaction was continued for 7 days. The DMF was removed under reduced pressure and the residue was 20 treated with chloroform (25 ml) and water (75 ml). The organic phase was washed with water (2x75 ml), dried (MgSO 4 ) and the solvent was evaporated to give a yellow oil. Analysis by t.l.c showed the presence of two compounds, which were separated and purified by column chromatography (CHCl 3 ) and identified by NMR as the 1,3-alternate and the partial cone isomers.

13 Binding studies. Extraction experiments of lanthanide and thorium nitrates (C M = 10 -4 M) from 1 M nitric acid into dichloromethane were performed at 20 °C. A colorimetric method using Arsenazo-III was applied to determine the concentration of the metal ions before and after extraction in the aqueous phase. The full procedure, the starting materials and the calculations of the distribution coefficients (D) and the percentage extraction (%E) have already been described in detail. 3 Complexation data, i.e. the stoichiometry and the stability constants (log β) of the species formed in methanol, were derived from spectrophotometric titrations at 25 °C. The experimental procedure 30 data processing by program Specfit 31 have been previously published. The ligand concentrations were ranging between 5×10 -5 and 10 -4 M and the ionic strength was settled as constant by using 10 -2 M Et 4 NNO 3 as backgroundelectrolyte.

Stock solutions of lanthanide and thorium nitrates used in extraction and complexation experiments were standardized by complexometric titrations with EDTA in the presence of xylenol orange as coloured indicator.

Crystallographic data. Single crystals were obtained from a solution in acetone/methanol (1b') or chloroform/methanol (13, 17, 18) which was overlaid in a test tube by an excess of methanol. The two molecules of water found per molecule 18 in its crystal structure were obviously uptaken from the atmosphere.

Intensity data were collected on a STOE IPDS II two-circle diffractometer with graphite-monochromated MoK α radiation at 173K. For 13 and 17 an empirical absorption correction was performed using the MULABS 32 option in PLATON. 33 The structures were solved by direct methods using the program SHELXS 34 and refined against F 2 with full-matrix least-squares techniques using the program SHELXL.

Conclusions

Calix [4]arenes fixed in the 1,3-alternate conformation have been used for the first time as basic skeleton for the attachement of four carbamoylmethyl phosphine oxide residues or of two CMPO functions in combination with two picolinamide or (potentially ionizable) carboxylic groups. Preliminary extraction and complexation studies with selected lanthanides showed, however, that this new arrangement does not lead to improved properties in comparison to narrow or wide rim tetra-CMPO derivatives of calix [4]arenes in the cone conformation. Most probably this is not caused by the more rigid scaffold, which might be even beneficial, but to an inappropriate mutual situation of the ligating functions. MDsimulations should be helpful to find their best arrangement.

Fig. 1 Fig. 2

 12 Fig. 1 Molecular conformation of the three 1,3-alternate derivatives 1b', 13, and 17 (from the left to the right)

  00/15.64°) and 17 (21.70/23,12°).

  1a: m.p. 251-253 °C; 1 H NMR (300 MHz, CDCl 3 ) δ 1.78 (s, 18H, t-Bu), 3.72, 3.86 (2d, 4/4H, 2 J = 15.6 Hz, Ar-CH 2 -Ar), 3.79 (m, 8H, O-CH 2 -, -CH 2 -N), 4.10 (d, 4H, 2 J = 5.16 Hz, CH 2 =CH-CH 2 -O-), 5.08 (m, 4H, CH 2 =CH-), 5.75 (m, 2H, CH 2 =CH-), 6.95, 8.28 (2s, 4/4H, ArH), 7.66-7.85 (m, 8H, Phth-H). partial cone isomer of 1a: m.p. 215-217 °C; 1 H NMR (300 MHz, CDCl 3 ) δ 1.00 (s, 18H, t-Bu), 3.24 (d, 2H, 2 J = 13.2 Hz, Ar-CH 2 -Ar), 3.70-4.25 (m, 18H, O-CH 2 -, -CH 2 -N, Ar-CH 2 -Ar, -O-CH 2 -CH=CH 2 ), 4.85, 5.22 (2m, 2/2H, CH 2 =CH-), 5.56, 5.88 (2m, 1/1H, CH 2 =CH-), 6.57, 6.88 (2d, 2/2H, 4 J=2.2 Hz, ArH ), 7.82 (m, 8H, Phth-H ), 8.00, 8.42 (2s, 2/2H, ArH).5,17-Di-t-butyl-11,23-diamino-26,28-diphthalimidoethoxy-25,27-dipropoxy-calix[4]arene 2a. The diamine 2a was prepared as described earlier for 2b. Error! Bookmark not defined.

(

  bt, 4H, -NH 2 ), 3.45-3.60 (2bt, 4/4H, -CH 2 -N, O-CH 2 -), 3.74, 3.84 (2d, 4/4H, 2 J = 12.2 Hz, Ar-CH 2 -Ar), 6.88, 6.95 (2s, 4/4H, ArH), 7.68-7.79 (2m, 8H, Phth-H). 5,17-Di-t-butyl-11,23-diCMPOamido-26,28-diphthalimidopropoxy-25,27-dipropoxy-calix[4]arene 3b.

  -3.22 (m, 8H, -CH 2 -N, -CH 2 -O), 3.51 (t, 4H, 3 J = 7.3 Hz, O-CH 2 -), 3.65, 3.74 (2d, 4/4H, 2 J = 15.8 Hz, Ar-CH 2 -Ar), 6.48, 6.92 (2s, 4/4H, ArH), 7.38 (m, 2H, H b ), 7.80 (m, 2H, H c ), 8.13 (m, 2H, H d ), 8.26 (bt, 2H, -NH), 8.51 (m, 2H, H a ).

°C; 1 H

 1 NMR (300 MHz, CDCl 3 ) δ 0.64 (t, 6H, 3 J = 7.7 Hz, -CH 2 -CH 3 ), 1.09-1.22(m, 26H, 2x-CH 2 -, t-Bu), 1.82-1.86 (m, 4H, -CH 2 -), 3.21-3.73 (m, 20H, -CH 2 -N, -CH 2 -O, Ar-CH 2 -Ar), 6.42 (s, 4H, -NH 2 ), 6.92, 6.97 (2s, 4/4H, ArH), 7.38 (m, 2H, H b ), 7.81 (m, 2H, H c ), 8.18 (m, 2H, H d ), 8.30 (bt, 2H, -NH), 8.54 (m, 2H, H a ). 5,17-Di-t-butyl-11,23-diCMPO-26,28-dipicolinamidopropyloxy-25,27-dipropoxy-calix[4]arene 11b. The final acylation was done as described above e.g. for 6b. 200 mg of diamine 10b (0.20 mmol) dissolved in dichloromethane (30 ml) were reacted with 490 mg (2.4 mmol) of active ester AE1. 170 mg of 11b (65 %) were isolated as white powder; m.p. 108-109 °C; 1 H NMR (300 MHz, DMSO-d 6 ) δ 0.93 (t, 6H, 3 J = 7.2 Hz, -CH 2 -CH 3 ), 1.18 (s, 18H, t-Bu), 1.44 (m, 4H, -CH 2 -), 3.05-3.78 (m, 24H, -CH 2 -N, -CH 2 -O, Ar-CH 2 -Ar, -CH 2 -P), 6.32 (s, 2H, -NH), 6.95, 7.12 (2s, 4/4H, ArH), 7.49-7.99 (m, 22H, m, p-Ph 2 H, o-Ph 2 H, H b ), 8.34 (m, 2H, H c ), 8.50 (m, 2H, H d ), 8.73 (bt, 2H, -NH,), 9.77 (m, 2H, H a ). 5,17-Di-t-butyl-11,23-diCMPO-26,28-dipicolinamidobutyloxy-25,27-dipropoxy-calix[4]arene 11c. 220 mg diamine 10c (0.23 mmol) in dichloromethane (30 ml) were reacted analogously with 325 mg (0.55 mmol) AE1 to yield 160 mg (47 %) of 11c as white powder; m.p. 117-119 °C; 1 H NMR (200 MHz, DMSO-d 6 ) δ 0.60 (t, 6H, 3 J = 7.2 Hz, -CH 2 -CH 3 ), 1.21 (m, 30H, t-Bu, -CH 2 -), 3.06-3.72 (m, 24H, -CH 2 -N, -CH 2 -O, Ar-CH 2 -Ar, -CH 2 -P), 6.90, 7.06 (2s, 4/4H, ArH), 7.40-7.80 (m, 22H, m, p-Ph 2 H, o-Ph 2 H, H b ), 8.11 (m, 2H, H c ), 8.34-8.50 (m, 6H, H d ,-NH, H a ), 9.45 (s, 2H, NH). 5,17-Di-t-butyl-11,23-dinitro-25,27-diallyloxy-26,28-diethoxycarbonylmethoxy-calix[4]arene 13

  :/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

Table 1

 1

	Extraction of selected lanthanides and thorium by tetra-CMPOs 5, c L = 10 -3 M (c L = 10 -4 M); values for a narrow
	and a wide rim tetra-CMPO fixed in the cone conformation are included for comparison.
			La 3+	Eu 3+	Yb 3+	Th 4+
	5b	%E	11	11	9	(17)
		D	0.12	0.12	0.09	(0.2)
	5c	%E	12	10	11	48 (12) a
		D	0.13	0.11	0.12	0.92 (0.13) a
	narrow rim 5	%E	70	68	37	(96)
	(n=3)	D	2.33	2.12	0.58	(24)
	wide rim 6,7	%E	98	64	6.6	(61.8)
		D	49	1.78	0.07	(1.62)
	a Values for c L = 10 -4 M.

Table 2

 2 Stability constants (log β ±σ n-1 ) for complexes of tetra-CMPOs 5 with some lanthanides in methanol at 25 °C (I = 20 10 -2 M, Et 4 NNO 3 ). Data for a wide rim tetra-CMPO in the cone conformation are included for comparison.

			La 3+	Eu 3+	Yb 3+
	5b	1:1	5.5±0.2	4.7±0.2	5.1±0.2
	5c wide rim 26	1:1 1:1	4.2±0.2 6.0	3.9±0.2 5.6	3.2±0.2 3.5
		2:1	10.6	11.0	8.6

,17-Di-t-butyl-11,23-diCMPOamido-26,28-diCMPOamidobutyloxy-25,27-dipropoxy-calix[4]arene 5c. Tetra-

  4 g, 84 %; m.p. 194-196 °C; 1 H NMR (200 MHz, CDCl 3 ) δ 0.65 (t, 6H, 3 J = 7.3 Hz, -CH 2 -CH 3 ), 1.20 (m, 22H, t-Bu, -CH 2 -), 1.51 (m, 4H, -CH 2 -), 3.26-3.69 (m, 24H, Ar-CH 2 -Ar, O-CH 2 -, -CH 2 -P, -CH 2 -N), 6.90, 6.95 powder, 0.2 g, 83 %; m.p. 152 °C; 1 H NMR (400 MHz, CDCl 3 ) δ 0.61 (t, 6H, 3 J = 7.7 Hz, -CH 2 -CH 3 ), 1.20 (m, 22H, t-Bu, -CH 2 -), 2.75 (m, 4H, -CH 2 -), 2.72 (t, 4H, 3 J = 7.0 Hz, -CH 2 -N), 3.20 (t, 4H, 3 J = 7.3 Hz, O-CH 2 -), 3.50, 3.56 (2d, 8H, 2 J = 15.8 Hz, Ar-CH 2 -Ar), 3.61 (m, 8H, O-CH 2 -, -CH 2 -P), 3.76 (d, 4H, 2 J = 10.3 Hz, -CH 2 -P), 6.87, 7.09 (2s, 8H, ArH), 7.38-7.81 (m, 40H, m, p-Ph 2 H, o-Ph 2 H), 8.96, 10.05 (2bs, 2/2H, NH); FD-MS, (M + +H) m/z = 1732.5. 5After the usual work up the diamine 6c was obtained as yellow oil (0.3 g, 81 %). 1 H NMR (400 MHz, CDCl 3 ) δ 0.80 (t, 6H, 3 J = 7.4 Hz, -CH 2 -CH 3 ), 1.23 (m, 22H, t-Bu, -CH 2 -), 1.47 (m, 4H, -CH 2 -), 2.28 (bs, 4H, -NH 2 ), 2.69 (bt, 4H, -CH 2 -), 3.51 (t, 4H, 3 J = 7.4 Hz, N-CH 2 -) 3.59 (bt, 4H, O-CH 2 -) , 3.63, 3.71 (2d, 4/4H, 2 J = 15.2 Hz, Ar-CH 2 -Ar), 6. 98, 7.94 (2s, 4/4H, ArH). 5

	5
		CMPO 5c was prepared as described for 5b. 100 mg (0.078 mmol) diamine 8c in 10 ml dichloromethane were
	10	acylated with and AE1 (72 mg, 0.18 mmol). After purification by column chromatography (CHCl 3 /MeOH 98:2) 0.138 g (87 %) of 5b were obtained as pink powder; m.p. 152 °C; 1 H NMR (400 MHz, CDCl 3 ) δ 0.65 (t, 6H, 3 J = 7.7 Hz, -CH 2 -CH 3 ), 1.01-1.23 (m, 26H, t-Bu, 3x-CH 2 -), 2.93 (t, 4H, 3 J = 7.0 Hz, -CH 2 -N), 3.08, 3.27 (2t, 8H, 3 J = 7.3 Hz, 2xO-CH 2 -), 3.37, 3.67 (2d, 4H, 2 J = 14.3 Hz, CH 2 -P), 3.65, 3.69 (2d, 8H, 2 J = 15.5 Hz, Ar-CH 2 -Ar), 6.92, 7.27 (2s,
		8H, ArH), 7.38-7.74 (m, 40H, m, p-Ph 2 H, o-Ph 2 H), 8.31, 10.20 (2bs, 2/2H, NH).
		5,17-Di-t-butyl-11,23-dinitro-26,28-diaminopropoxy-25,27-dibutyloxy-calix[4]arene 6c. 0.45 g (0.43 mmol) of
		the bis-phthalimide 1c were dissolved in EtOH (30 ml), hydrazine (3 ml) was added and the reaction mixture was
		refluxed for 2 h.
		F o r
		P
		e
		e r
		R
		e
		v i e
		w
		O n l
		y

,17-Di-t-butyl-11,23-dinitro-26,28-diCMPOamidobutyloxy-25,27-dipropoxy-calix[4]arene 7c.

  The acylation of diamine 6c by AE1 was carried out at rt over night as follows: 0.3 g 6c (0.35 mmol) dissolved in dichloromethane (15 ml) were reacted with 0.32 g (0.84 mmol) AE1; 0.35 g (75 %) of diCMPOamide 7c were obtained as white 20 powder; m.p. 142 °C; 1 H NMR (400 MHz, CDCl 3 ) δ 0.80 (t, 6H, 3 J = 7.4 Hz, -CH 2 -CH 3 ), 1.22 (m, 26H, t-Bu, 2x-CH 2 -), 3.13 (m, 4H, -CH 2 -), 3.07 (bt, 4H, -CH 2 -N), 3.48 (2t, 4/4H, 2x O-CH 2 -), 3.66 (d, 4H, 2 J = 13.7 Hz, -CH 2 -P), 3.66, 3.75 (2d, 8H, 2 J = 15.2 Hz, Ar-CH 2 -Ar), 6.96, 7.88 (2s, 4/4H, ArH), 7.38-7.81 (m, 22H, m, p-Ph 2 H, o-Ph 2 H, NH). 5

,17-Di-t-butyl-11,23-diamino-26,28-diCMPOamidobutyloxy-25,27-dipropoxy-calix[4]arene 8c

. A solution of 7c (110 mg, 0.082 mmol) and hydrazine (1 ml) in ethanol (15 ml) was refluxed for 2 h with Raney/Ni as catalyst.

5,17-Di-t-butyl-11,23-dinitro-26,28-dipicolylamidobutyloxy-25,27-dipropoxy-calix[4]arene 9c. Starting from

  CH 2 -CH 2 -CH 2 -CH 2 -, -CH 2 -CH 3 ), 3.46-3.86 (m, 20H, -CH 2 -N, 2x O-CH 2 -, Ar-CH 2 -Ar), 6.96 (s, 4H, ArH), 7.39 (m, 2H, H b ), 7.81 (m, 2H, H c ), 7.92 (s, 4H, ArH), 8.20 (m, 2H, H d ), 8.25 (bt, 2H, -NH), 8.56 (m, 2H, H a ). 5

2 g (64 35 %); m.p. 127-129 °C; 1 H NMR (300 MHz, CDCl 3 ) δ 0.78 (t, 6H, 3 J = 7.35 Hz, -CH 2 -CH 3 ), 1.21 (s, 18H, t-Bu), 1.41 (m, 4H, -CH 2 -CH 2 -CH 2 -), 1.71 (m, 4H, -CH 2 -CH 3 ), 3.45-3.50 (m, 8H, -CH 2 -N, O-CH 2 -), 3.65-3.81 (m, 12H, Ar-CH 2 -Ar, O-CH 2 -), 6.95 (s, 4H, ArH), 7.40 (m, 2H, H b ), 7.80 (m, 2H, H c ), 7.99 (s, 4H, ArH), 8.17 (m, 2H, H d ), 8.29 (bt, 2H, -NH), 8.58 (m, 2H, H a ).

40 diamine 6c (0.25 g), containing traces of compounds which has unhydrogenated allyl groups the desired compound 9c was obtained as the analogous mixture with the respective allyl compounds as described for 9b; 0.25 g (83 %) were isolated as a white powder. The signals belonging to the protons of 9c are listed below.

1

H NMR (300 MHz, CDCl 3 ) δ 0.84 (t, 6H, 3 J = 7.35 Hz, -CH 2 -CH 3 ), 1.21 (s, 18H, t-Bu), 1.20-1.66

(m, 12H, -

,17-Di-t-butyl-11,23-diamino-26,28-di-picolinamidopropyloxy-25,27-dipropoxy-calix[4]arene 10b.

  Raney/Ni was added to a clear solution of 9b (740 mg, 0.70 mmol) in THF

Table 4

 4 ) are disordered over two positions. The C-C bond lengths in the disordered tert-butyl groups were restrained to 1.54(1)Å and the C=C double bonds in the propenoxy groups to 1.30(1)Å. In 18, the water H atoms could not be located and were omitted from refinement. The geometric parameters of the side chain O32 to N324 were restrained to be equal to those of O12 to N124. CIF format have been deposited with the Cambridge Crystallographic Data Centre: reference numbers: 1b' CCDC 734171; 13 CCDC 734172; 17 CCDC 734173; 18 CCDC 734174. Crystal Data and Structure Refinement Details for 1b', 13, 17 and 18; data collection with a Stoe-IPDS-II two-circlediffractometer, solution and refinement with SHELXS and SHELXL.

	Error! Bookmark not defined. Hydrogen atoms were

20

Crystallographic data in -18

≤ k ≤ 18, -23 ≤ l ≤ 22 -13 ≤ h ≤ 13, -18 ≤ k ≤ 17, -25 ≤ l ≤ 25 -13 ≤ h ≤ 13, -21 ≤ k ≤ 19, -24 ≤ l ≤ 24 -15 ≤ h ≤ 15, -29 ≤ k ≤ 29, -23 ≤ l ≤

lanthanide19 cations.20

Crystal structures. Single crystals suitable for X-ray diffraction were obtained for three compounds fixed in the 1,3alternate conformation, 1b' (obtained from 1b by selective hydrogenation of the allyl groups, or by direct alkylation, using propyl-instead of allylbromide), 13, and 17 (a derivative with two different precursor functions for amino groups at the wide rim[START_REF] Danila | [END_REF] ). A crystal structure was also determined for the 1,3-diether 18 in the cone conformation,29 the very first step in the synthesis of 1c.[START_REF]For the extraction of lanthanides; actinides by calixarene-based picolinamides[END_REF]

URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu Supramolecular Chemistry

However, this folding does not entirely explain the strong differences found for the distances of opposite phenolic

Acknowledgements

Financial Support by the European Community (EuroPart) and by the Deutsche Forschungsgemeinschaft (SFB 625) is gratefully acknowledged.

Notes and references

Page 13 of 18 URL: http:/mc.manuscriptcentral.com/tandf/gsch Email: suprachem@mail.cm.utexas.edu


Andy Wathen

Jennifer Pestana

Jennifer Pestana How

How to choose a nonsymmetric preconditioner

How to choose a nonsymmetric preconditioner

List of authors: Andy Wathen and Jennifer Pestana 1

For linear systems of equations with a real symmetric (or complex hermitian) coefficient matrix, there are standard iterative methods of choice: Conjugate Gradients for positive definite problems and MINRES for indefinite problems. For these methods there exist descriptive convergence theories based on eigenvalues, i.e. convergence bounds given in terms of polynomial approximation problems on the eigenvalue spectrum which are often reasonably tight. A key consequence is that it is clear what one is trying to achieve to get fast convergence via preconditioning in these cases, namely well distributed eigenvalues or reduced spectral condition number.

By contrast, existing convergence bounds for Krylov subspace methods such as GMRES for real nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. In this talk we will introduce a desirable mathematical property of a preconditioner which guarantees that convergence of GMRES will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case.

1 Oxford University, UK 1